Sasakian GeometryCharles P. Boyer and Krzysztof GalickiDepartment of Mathematics and Statistics, University of New Mex-ico, Albuquerque, N.M. 87131PrefaceThis book is a result of a fteen year long collaboration which has led us tounderstand and appreciate the importance of Sasakian manifolds as an integralpart of Riemannian Geometry. In the early nineties neither of us was aware orparticularly interested in Sasakian structures. This rapidly changed in 1992 when,together with Ben Mann, we realized that smooth 3-Sasakian manifolds, which areautomatically Einstein, are orbibundles over positive quaternionic Kahler orbifolds.In the smooth case this bundle was rst described by Konishi. As quaternionicKahler orbifolds are easily manufactured via quaternionic reduction, likewise wewere able to construct large families of smooth compact 3-Sasakian spaces withrelative ease. Searching through the literature afterwards we were surprised that afolklore conjecture attributed to Tanno stated that any 3-Sasakian manifold shouldbe a spherical space form.Soon after we began to understand that the 3-Sasakian geometry, though veryinteresting, was too specialized a case. To a large extent our main motivationhad to do with Einstein metrics. We realized that the 3-Sasakian manifolds weremerely a special case of the more general theory of Sasaki-Einstein manifolds andshifted our focus to that more general case. With time it became clearer and clearerthat Sasakian geometry is one of the richest sources of complete Einstein metrics ofpositive scalar curvature. We began to appreciate that Sasakian geometry, naturallysandwiched between two dierent Kahler geometries, was at least as interestingand important as the latter.It was Nigel Hitchin who rst suggested to us that perhaps it is time to writea modern book on the subject. We started thinking about the project in 2002. Itwas a challenging endeavor: we aimed at writing a monograph which should give afairly complete account of our own contributions to the subject combined with anadvanced graduate level textbook describing the foundational material in a modernlanguage. Unlike the Kahler case, there are very few books of this sort discussingSasakian manifolds.Another diculty which we were to discover later was the rapid development ofthe subject. Some new results, such as the construction of Sasaki-Einstein metricson exotic spheres with Janos Kollar, had to do with the dynamics of our ownresearch program. But many other important results were obtained by others.To complicate things even more, Sasaki-Einstein manifolds appear to play a veryspecial role in the so-called AdS/CFT duality conjecture and they have received anenormous amount of attention among physicists working in this area. Some of theintriguing new results which we decided to include in the book, albeit very briey,indeed came quite unexpectedly from considerations in Superstring Theory.In the end we know our book will not be as complete as we would have wantedor hoped it to be. We nally had to stop writing while knowing all too well that theiiiiv PREFACEbook is neither complete nor perfect. Few books of this sort are. We are pleased tosee so much renewed interest in the subject and hope that our work will be of helpto a number of mathematicians and physicists, researchers and graduate studentsalike.Over the years we have beneted immensely from many discussions with manycolleagues and collaborators about the mathematics contained in this book. Herewe are happy to take this opportunity to thank: B. Acharya, I. Agricola, D. Alek-seevsky, V. Apostolov, F. Battaglia, C. Bar, H. Baum, F. Belgun, L. Berard Bergery,R. Bielawski, C. Bohm, A. Buium, D. Blair, O. Biquard, J.-P. Bourguignon, R.Bryant, D. Calderbank, A. Cap, J. Cheeger, T. Colding, A. Dancer, O. Dearricott,T. Draghici, Y. Eliashberg, J. Figueroa OFarrill, M. Fernandez, T. Friedrich, E.Gasparim, P. Gauduchon, H. Geiges, W. Goldman, G. Grantcharov, K. Grove, M.Harada, T. Hausel, G. Hernandez, L. Hernandez, R. Herrera, O. Hijazi, N. Hitchin,H. Hofer, T. Holm, J. Hurtubise, J. Isenberg, G. Jensen, J. Johnson, A. de Jong,D. Joyce, L. Katzarkov, J. Konderak, M. Kontsevich, J. Kollar, H. B. Lawson,C. LeBrun, B. Mann, P. Matzeu, S. Marchiafava, R. Mazzeo, J. Milgram, A. Mo-roianu, P.-A. Nagy, M. Nakamaye, T. Nitta, P. Nurowski, L. Ornea, H. Pedersen,P. Piccinni, M. Pilca, Y.-S. Poon, E. Rees, P. Rukimbira, S. Salamon, R. Schoen,L. Schwachhofer, U. Semmelmann, S. Simanca, M. Singer, J. Sparks, J. Starr, S.Stolz, A. Swann, Ch. Thomas, G. Tian, M. Verbitsky, M. Wang, J. Wisniewski, R.Wolak, D. Wraith, S.-T. Yau, D. Zagier, and W. Ziller. It is probably inevitablethat we have missed the names of some friends and colleagues to whom we deeplyapologize.We owe our deep gratitude to Janos Kollar. He is not just a collaborator onone of the more important papers we wrote. His continuous help and involvementthroughout writing of this book was invaluable. Much of the material presentedin chapters 10 and 11 was written with his expert help and advice. We also wishto thank Ilka Agricola, Elizabeth Gasparim, Eugene Lerman, Michael Nakamaye,and Santiago Simanca and Thomas Friedrich for carefully reading certain partsof our book and providing invaluable comments and corrections. We thank EvanThomas for helping with computations used to compile the various tables appearingin the appendices. We also thank our graduate students: J. Cuadros, R. Gomez,D. Grandini, J. Kania, and R. Sanchez-Silva for weeding out various mistakes whilesitting in several courses we taught at UNM using early versions of the book.The second author would like to express special thanks to the Max-Planck-Institut f ur Mathematik in Bonn for the generous support and hospitality. Severalchapters of this book were written during K.G.s sabbatical visit at the MPIMduring the calendar year 2004 and also later during two shorter visits in 2005 and2006. Both of us thank the National Science Foundation for continuous support ofour many projects, including this one.Finally, we thank Jessica Churchman and Alison Jones from the Oxford Uni-versity Press for their patience, continuing interest in our work on the project andmuch help in the latter stages of preparing the manuscript for publication.Last but not least we would like to thank our families, especially Margaret andRowan, for support and patience and for putting up with us while we were workingon The Book day and night.Albuquerque, February 2007.ContentsPreface iiiIntroduction 1Chapter 1. Structures on Manifolds 91.1. Sheaves and Sheaf Cohomology 91.2. Principal and Associated Bundles 141.3. Connections in Principal and Associated Vector Bundles 181.4. G-Structures 231.5. Pseudogroup Structures 361.6. Group Actions on Manifolds 38Chapter 2. Foliations 512.1. Examples of Foliations 512.2. Haeiger Structures 522.3. Leaf Holonomy and the Holonomy Groupoid 542.4. Basic Cohomology 592.5. Transverse Geometry 602.6. Riemannian Flows 69Chapter 3. Kahler Manifolds 753.1. Complex Manifolds and Kahler Metrics 763.2. Curvature of Kahler Manifolds 823.3. Hodge Theory on Kahler Manifolds 883.4. Complex Vector Bundles and Chern Classes 923.5. Line Bundles and Divisors 943.6. The Calabi Conjecture and the Calabi-Yau Theorem 102Chapter 4. Fundamentals of Orbifolds 1054.1. Basic Denitions 1054.2. Orbisheaves and orbibundles 1084.3. Groupoids, Orbifold Invariants and Classifying Spaces 1154.4. Complex Orbifolds 1234.5. Weighted Projective Spaces 1334.6. Hypersurfaces in Weighted Projective Spaces 1384.7. Seifert Bundles 144Chapter 5. Kahler-Einstein Metrics 1515.1. Some Elementary Considerations 1525.2. The Monge-Amp`ere Problem and the Continuity Method 1535.3. Obstructions in the Positive Case 160vvi CONTENTS5.4. Kahler-Einstein Metrics on Hypersurfaces in CP(w) 1625.5. Automorphisms and the Moduli Problem 173Chapter 6. Almost Contact and Contact Geometry 1796.1. Contact Structures 1806.2. Almost Contact Structures 1906.3. Almost Contact Metric Structures 1956.4. Contact Metric Structures 1986.5. Structures on Cones 201Chapter 7. K-Contact and Sasakian Structures 2077.1. Quasi-regularity and the Structure Theorems 2077.2. The Transverse Geometry of the Characteristic Foliation 2147.3. Curvature Properties of K-Contact and Sasakian Structures 2197.4. Topology of K-Contact and Sasakian Manifolds 2297.5. Sasakian Geometry and Algebraic Geometry 2367.6. New Sasakian Structures from Old 250Chapter 8. Symmetries and Sasakian Structures 2578.1. Automorphisms of Sasakian Structures and Isometries 2578.2. Deformation Classes of Sasakian Structures 2658.3. Homogeneous Sasakian Manifolds 2728.4. Symmetry Reduction and Moment Maps 2768.5. Contact and Sasakian Reduction 290Chapter 9. Links as Sasakian Manifolds 2999.1. Preliminaries 2999.2. Sasakian Structures and Weighted Homogeneous Polynomials 3009.3. The Milnor Fibration and the Topology of Links 3029.4. The Dierential Topology of Links 3129.5. Positive Sasakian Structures on Links 3199.6. Links of Complete Intersections 326Chapter 10. Sasakian Geometry in Dimensions Three and Five 32910.1. Sasakian Geometry in Dimension Three 32910.2. Sasakian Structures and the Topology of 5-Manifolds 33510.3. Sasakian Links in Dimension Five 35210.4. Regular Sasakian Structures on 5-Manifolds 360Chapter 11. Sasaki-Einstein Geometry 36911.1. Foundations of Sasaki-Einstein Geometry 37011.2. Extremal Sasakian Metrics 37811.3. Further Obstructions to Sasaki-Einstein Structures 38211.4. Sasaki-Einstein Metrics in Dimensions Five 38811.5. Sasaki-Einstein Metrics on Homotopy Spheres 40311.6. The Sasaki-Einstein semi-group 40711.7. Sasaki-Einstein Metrics in Dimensions Seven and Higher 40911.8. Sasakian -Einstein Metrics 417Chapter 12. Quaternionic Kahler and Hyperkahler Manifolds 42112.1. Quaternionic Geometry of Hnand HPn422CONTENTS vii12.2. Quaternionic Kahler Metrics 42812.3. Positive Quaternionic Kahler Manifolds and Symmetries 43412.4. Quaternionic Kahler Reduction 43812.5. Compact Quaternionic Kahler Orbifolds 44312.6. Hypercomplex and Hyperhermitian Structures 45312.7. Hyperkahler Manifolds 45512.8. Hyperkahler Quotients 45812.9. Toric Hyperkahler Metrics 46112.10. ALE Spaces and Other Hyperkahler Quotients 465Chapter 13. 3-Sasakian Manifolds 47313.1. Almost Hypercontact Manifolds and 3-Sasakian Structures 47413.2. Basic Properties 47813.3. The Fundamental Foliations T and TQ 48013.4. Homogeneous 3-Sasakian Manifolds 49113.5. 3-Sasakian Cohomology 49413.6. Symmetry Reduction 49913.7. Toric 3-Sasakian Manifolds 50613.8. Cohomogeneity One 3-Sasakian 7-Manifolds 52213.9. Non-Toric 3-Sasakian Manifolds in Dimension 11 and 15 525Chapter 14. Sasakian Structures, Killing Spinors, and Supersymmetry 52914.1. The Dirac Operator and Killing Spinors 52914.2. Real Killing Spinors, Holonomy and Bars Correspondence 53314.3. Geometries Associated with 3-Sasakian 7-manifolds 53514.4. Geometries Associated with Sasaki-Einstein 5-manifolds 54214.5. Geometric Structures on Manifolds and Supersymmetry 545Appendix A 551A.1. Preliminaries on Groupoids 551A.2. The Classifying Space of a Topological Groupoid 555Appendix B 559B.1. Reids List of K3 Surfaces as hypersurfaces in CP4(w) 559B.2. Dierential topology of 2/(o3o4) and 2/(o5o6) 560B.3. Tables of Kahler-Einstein metrics on hypersurfaces CP(w) 561B.4. Positive Breiskorn-Pham Links in Dimension 5 564B.5. The Yau-Yu Links in Dimensions 5 567Bibliography 569Index 607IntroductionIn 1960 Shigeo Sasaki [Sas60] began the study of almost contact structuresin terms of certain tensor elds, but it wasnt until [SH62] that what are nowcalled Sasakian manifolds rst appeared under the name of normal contact metricstructure. By 1965 the terms Sasakian structure and Sasakian manifold be-gan to be used more frequently replacing the original expressions. For a number ofyears these manifolds were intensively studied by a group of Japanese geometers.The subject did get some attention in the United States mainly due to the papersof Goldberg and Blair. Nevertheless, the main interest in the eld remained in-side Japan, nding it hard to spread out and attract broader attention beyond itsbirthplace either in the United States or in Europe.Over a period of four years between 1965 and 1968 Sasaki wrote a three partset of lecture notes which appeared as an internal publication of the MathematicalInstitute of the Tohoku University under the title Almost contact manifolds, PartI-III [Sas65, Sas67, Sas68]. Put together, the work amounts to almost 500 pages.Even today, after 40 years, the breath, depth and the relative completeness of theSasaki lectures is truly quite remarkable. It is hard to understand why they did notmake it as a monograph in some prestigious Western book series; it is a pity. As itis, the notes are not easily available and, consequently, not well-known1. OutsideJapan the rst and important attempt to give a broader account of the subject wasgiven eight years later by Blair [Bla76a].After 1968 Sasaki himself was less active although he continued to publishuntil 1980. Yet he had already created a new subeld of Riemannian geometrywhich slowly started to attract attention worldwide, not just in Japan. In 1966Brieskorn wrote his famous paper describing a beautiful geometric model for allhomotopy spheres which bound parallelizable manifolds [Bri66]. In 1976 Sasaki[ST76, SH76] realized that Brieskorn manifolds admit almost contact and con-tact structures. (This very important fact was independently observed by severalother mathematicians: Abe-Erbacher [AE75], Lutz-Meckert [LM76], and Thomas[Tho76].) Thirty years later the Brieskorn-Pham links as well as more general linksof weighted homogeneous polynomials, are the key players in several chapters ofour book. Yet again, Sasaki seemed to have had both the necessary intuition anda broad vision in understanding what is and what is not of true importance2.1We became aware of the Sasaki notes mainly because of our work on this book. We obtaineda copy of the lectures in 2003 form David Blair and we would like to thank him for sharing themwith us.2We know little about Sasakis non-mathematical life. He was born in 1912 and what we doknow is from the volume of his selected papers edited in 1985 by Tachibana [Sas85]. There onends a short introduction by S. S. Chern and an essay by Sasaki in which he mostly discusseshis life as a working mathematician. He apparently died almost 20 years ago on August 12th,1987. Sadly, his death passed without any notice, strangely forgotten. We could not nd an12 INTRODUCTIONOver the years Sasakian geometry has taken a back seat to other areas ofRiemannian geometry, most prominently to the study of Riemannian geometrywith reduced holonomy groups. Nevertheless, as we shall see, Sasakian geome-try is closely related to all these other geometries. Although generally a Sasakianmanifold has the generic holonomy oO(n), the Riemannian cone over a Sasakianmanifold does have reduced holonomy making the study of Sasakian geometry quitetractable. A quick perusal of Bergers list of possible irreducible Riemannian holo-nomy groups (see Table 1.4.1 below) shows that there are ve innite series of theseholonomy groups. All ve are related to Sasakian geometry: generally a Sasakianmetric itself has generic holonomy oO(n). and its Riemannian cone has holonomyl(n); the Riemannian cone of a general Sasaki-Einstein structure has holonomyol(n); whereas, the Riemannian cone of a general 3-Sasakian structure has holo-nomy oj(n). The remaining innite series in Table 1.4.1 is the group oj(n)oj(1)of quaternionic Kahler geometry which is closely related to 3-Sasakian geometry aswe discuss shortly. The two remaining irreducible Riemannian holonomy groupsG2 and Spin(7) are related to Sasakian geometry in a less direct way as we discussin Chapter 14. In this same spirit our book attempts to show Sasakian geometrynot as a separate subeld of Riemannian geometry but rather through its inter-relation to other geometries. This is perhaps the most important feature of thesubject. The study of Sasakian manifolds brings together several dierent eldsof mathematics from dierential and algebraic topology through complex algebraicgeometry to Riemannian manifolds with special holonomy.The closest relative of Sasakian geometry is Kahlerian geometry, the importanceof which is dicult to overestimate mainly because of its role in algebraic geometry.But Sasakian geometry also has a very algebro-geometric avor. In fact, there is analgebraic structure on every Sasakian manifold; whereas, Voisin has shown recentlythat there are Kahler manifolds that admit no algebraic structure whatsoever. Tobetter understand the relation between Sasakian and Kahlerian geometries we beginwith the more familiar relation between contact and symplectic geometries. Let(`. . ) be a contact manifold where is a contact form on ` and is its Reebvector eld. It is easy to see that the cone (C(`) = R+ `. = d(t)) issymplectic. Likewise, the Reeb eld denes a foliation of ` and the transversespace Z is also symplectic. When the foliation is regular the transverse space is asmooth symplectic manifold giving a projection called Boothby-Wang bration,and = d relates the contact and the symplectic structures as indicated by(((`). ) (`. . )

(Z. ).English language mathematical obituary honoring his life and work and commemorating his death;apparently, his passing was noted just in a short obituary of a local newspaper. After his deathTokyo Science University where he briey worked after retiring from Tohoku took care of hismathematical heritage with some of his manuscripts placed at the university library. We aregrateful to Professor Yoshinobu Kamishima for this information.INTRODUCTION 3We do not have any Riemannian structure yet. It is quite reasonable to askif there is a Riemannian metric o on ` which best ts into the above diagram.As the preferred metrics adapted to symplectic forms are Kahler metrics one couldask for the Riemannian structure which would make the cone with the warpedproduct metric o = dt2+ t2o together with the symplectic form into a Kahlermanifold? Then o and dene a complex structure . Alternatively, one could askfor a Riemannian metric o on ` which would dene a Kahler metric / on Z via aRiemannian submersion. Surprisingly, in both cases the answer to these questionsleads naturally and uniquely to Sasakian Geometry. Our diagram becomes(((`). . o. ) (`. . . o. )

(Z. . /. J).From this point of view it is quite clear that Kahlerian and Sasakian geome-tries are inseparable, Sasakian Geometry being naturally sandwiched between twodierent types of Kahlerian Geometry. Yet the two fared dierently over the years.Since Erich Kahlers seminal article several dominant gures of the mathematicalscene of the XXth century have, step after step along a 50 year period, transformedthe subject into a major area of Mathematics that has inuenced the evolution ofthe discipline much further than could have conceivably been anticipated by any-one writes Jean-Pierre Bourguignon in his tributary article The unabated vitalityof Kahlerian geometry published in [Kah03]. Sasakian Geometry has not been aslucky. There always has been interesting work in the area, but for unclear reasonsit has never attracted people with the same broad vision, people who would setout to formulate and then work on fundamental problems. Yet, arguably Sasakianmanifolds are at least as interesting as Kahlerian manifolds.Our own research of the last decade has been an attempt to bring Sasakiangeometry back into the main stream. We believe that a modern book on Sasakianmanifolds is long overdue. Most of the early results are scattered, often buriedin old and hard to get journals. They are typically written in an old fashionedlanguage. Worse than that, articles with interesting results are drowned in a vastsea of papers of little importance. There are very few graduate level texts on thesubject. There is the book Structures on manifolds written by Yano and Kon in 1984[YK84], but this book is over 20 years old and treats both Sasakian and Kahleriangeometries as a subeld of Riemannian geometry. Recently, Blair substantiallyupdated his well-known Contact manifolds in Riemannian geometry in the SpringerLecture Notes series [Bla76a] with Riemannian geometry of contact and symplecticmanifolds [Bla02]. Again the major emphasis as well as the techniques used areRiemannian in nature. Our monograph naturally complements Blairs as it employsan entirely dierent philosophy and follows a dierent approach. First we developthe important relations between Sasakian geometry and the algebraic geometry ofKahler (actually projective algebraic) orbifolds. Secondly, our major motivationto begin with was in proving the existence of Einstein metrics. So we have theunderstanding of Sasaki-Einstein metrics as a main goal toward which to work, but4 INTRODUCTIONwe have also come to appreciate the beauty and richness of Sasakian geometry inits own right.Our book breaks more or less naturally into two parts. Chapters 1 through9 provide an introduction to the modern study of Sasakian geometry. Startingwith Chapter 10 the book becomes more of a research monograph describing manyof our own results in the subject. However, the extensive introduction shouldmake it accessible to graduate students as well as non-expert researchers in relatedelds. We have used parts or our monograph as a textbook for advanced graduatecourses. For example, assuming that the students have some basic knowledge ofRiemannian geometry, algebraic geometry, and some algebraic topology, a coursetreating Sasakian geometry starting with Chapter 4 through Chapter 11 is possible,drawing on the rst three chapters as review. Many of the results in Chapters 4-11are given with full proofs bringing the student to the forefront of research in the area.As a guide we use the principle that proofs, or at least an outline of proofs, are givenfor important results that are not found in another book. Our book also containsmany examples, for we believe that the learning process is substantially enhancedby working through examples. We also have exercises scattered throughout thetext for the reader to sharpen her/his skills. Open problems of varying or unknowndiculty are listed, some of which could be the basis of a dissertation. The text isaimed at mathematicians, but we hope it will nd many readers among physicists,particularly those working in Superstring Theory.We begin in Chapter 1 by introducing various geometries that play more orless important roles in the way they relate to Sasakian structures. We espouse thepoint of view that a geometric structure is best described as a G-structure which,in addition, may or may not be (partially) integrable. As Sasakian manifolds are allexamples of Riemannian foliations with one-dimensional leaves, Chapter 2 takes thereader into the world of foliations with a particular focus on the Riemannian case.The literature is full of excellent books on the subject so we just select the topicsmost relevant to us. Chapter 3 reviews some basic facts about Kahler manifolds.Again, we are very selective choosing only what is needed later in describing thetwo Kahler geometries of the Kahler-Sasaki sandwich. Of particular interest isYaus famous proof of the Calabi conjecture.A key tool that allows for connecting Sasakian structures to other geometricstructures is the theory of Riemannian orbifolds and orbifold bundles or orbibun-dles. For that reason Chapter 4 is crucial in setting the stage for an in depth studyof Sasakian manifolds which begins later in Chapter 7. Orbifolds just as manifoldshave become a household name to the well trained geometer. Nevertheless, a lotof important results are scattered throughout the literature, and orbifolds typicallyappear within a specic context. There is a forthcoming book on orbifolds, Orb-ifolds and Stringy Topology [ALR06] by Adem, Leida, and Ruan, but this has aparticularly topological bent. Hence, we take some time and eort to prepare thereader introducing all the basic concepts from the point of view needed in subse-quent chapters. By Chapter 5 we are ready for a second trip into the realm ofKahler geometry. However, now the focus is on Kahler-Einstein metrics, in partic-ular positive scalar curvature Kahler-Einstein metrics on compact Fano orbifolds.We introduce some basic techniques that allow for proving various existence results.We also briey discuss obstructions. Chapter 6 presents the necessary foundationalINTRODUCTION 5material on almost contact and contact geometry. This leads directly to the de-nition of a Sasakian structure introduced at the very end.The study of Sasakian geometry nally begins with Chapter 7. We rst presentthe important structure theorems, and then gather all the results concerning thegeometry, topology, and curvature properties of both K-contact and Sasakian man-ifolds. Most of the curvature results are standard and can be found in Blairs book[Bla02], but our main focus is dierent: we stress the relation between Sasakianand algebraic geometry, as well as the basic cohomology associated with a Sasakianstructure. A main tool used in the text is the transverse Yau Theorem due to ElKacimi-Alaoui. In the companion Chapter 8 we present known results concerningsymmetries of Sasakian structures. We introduce the Sasakian analogue of the bet-ter known symplectic/contact reduction. Then we study toric contact and toricSasakian manifolds and prove several Delzant-type results.Chapter 9 is devoted to the geometry of links of isolated hypersurface singulari-ties as well as a review of the dierential topology of homotopy spheres a la Kervaireand Milnor [KM63]. A main reference for the study of such links is Milnors classictext Singular points of complex hypersurfaces [Mil68], but also Dimcas Singular-ities and topology of hypersurfacs [Dim92] is used. The dierential topology oflinks is a beautiful piece of mathematics, and this chapter oers a hands-on usersguide approach with much emphasis on the famous work of Brieskorn [Bri66]. Ofimportance for us is that when the singularities arise from weighted homogeneouspolynomials the links have a natural Sasakian structure with either denite (posi-tive or negative) or null basic rst Chern class. Emphasis is given to the positivecase which corresponds to having positive Ricci curvature. In Chapter 10 we dis-cuss the Sasakian geometry in low dimensions. In dimension 3 there is a completeclassication. Dimension 5 is large enough to be interesting, yet small enough tohope for some partial classication. We concentrate on the simply connected caseas there we can rely on the Smale-Barden classication. In terms of Sasakian struc-tures our main focus is on the case of positive Sasakian structures. In considerabledetail we describe several remarkable theorems of Kollar which show how positivityseverely restricts the topology of a manifold which is to admit a positive Sasakianstructure.Chapter 11 is central to the whole book and perhaps the main reason and jus-tication for it. Much of this chapter is based on a new method for proving the ex-istence of Einstein metrics on odd dimensional manifolds introduced by the authorsin 2001 [BG01b]. We realized there that links of isolated hypersurface singulari-ties obtained from weighted homogeneous polynomials admit Sasakian structures.Moreover, by using an orbifold version [BG00b] of an old result of Kobayashi, wesuccessfully tied the problem to making use of the continuity method for provingthe existence of Kahler-Einstein metrics on compact Kahler orbifolds. For the au-thors this was the original raison detre for Chapters 4, 5, and 9 of the book.In a series of papers the authors and their collaborators have successfully appliedthis method to prove the existence of Sasaki-Einstein metrics on many 5-manifolds,on odd dimensional homotopy spheres that bound parallelizible manifolds, as wellas on odd dimensional rational homology spheres. Furthermore, our method hasbeen substantially generalized by Kollar who has pushed our understanding muchfurther especially in dimension ve. Although a complete classication is perhaps6 INTRODUCTIONnot within reach we now begin to have a really good grasp of Sasaki-Einstein ge-ometry in dimension 5. In addition, we discuss toric Sasaki-Einstein geometry indimension 5 which began with the work of Gauntlett, Martelli, Sparks, and Wal-dram [GMSW04b], and culminates with the very recent work of Cho, Futaki,Ono, and Wang [FOW06, CFO07] which shows in arbitrary odd dimension thatany toric Sasakian manifold with positive anticanonical Sasakian structure admitsa compatible Sasaki-Einstein metric. We also discuss extremal Sasaki metrics de-ned in analogy with the extremal Kahler metrics, and introduce the Sasaki-Futakiinvariant [BGS06]. In addition to lifting the well-known obstructions of positiveKahler-Einstein metrics we also present some new results due to Gauntlett, Martelli,Sparks, and Yau [GMSY06] involving two well known estimates, one due to Lich-nerowicz, and the other to Bishop. We also present Sasaki-Einstein metrics obtainedvia the join construction described earlier in Chapter 7. We end this long chapterwith a brief discussion of Sasakian--Einstein metrics.Chapter 12 gives an extensive overview of various quaternionic geometries. Themain focus is on the positive quaternionic Kahler (QK) manifolds (orbifolds) andon the hyperkehler manifolds (orbifolds). The reason for such an extensive treat-ment has to do with Chapter 13. The 3-Sasakian manifolds studied there cannotbe introduced without a deeper understanding of these two quaternionic geome-tries, just as Sasakian and Sasaki-Einstein manifolds cannot be studied withoutKahlerian and Kahler-Einstein geometry. The Sasaki-Einstein manifolds of Chap-ter 13 have a completely dierent avor than the ones that appeared in Chapter11. It is not only that these occur only in dimensions 4: + 3, but also that theyhave a somewhat richer geometric structure. In addition the method in whichthe metrics are obtained is completely dierent. In Chapter 11, with some excep-tions, we mostly get our existence results via the continuity method applied to theMonge-Amp`ere equation. Very few metrics are known explicitly, though there areexceptions. Most of the 3-Sasakian metrics we consider are obtained via symmetryreduction similar to the hyperkahler and quaternionic Kahler reduction. Indeed thethree quotients are all related. So the manifolds and the metrics we get are quiteoften explicit and can be studied as quotients. Again, there are some exceptions.Finally, Chapter 14 gives a very brief overview of the rich theory of Killing spinors.There we describe some other geometries and show how they relate to 3-Sasakian7-manifolds and Sasaki-Einstein 5-manifolds. At the end we very briey commenton how Sasakian geometry naturally appears in various supersymmetric physicaltheories. Both Sasaki-Einstein geometry and geometries with exceptional holonomyhave appeared in various models of supersymmetric String Theory fuelling vigorousinterest in them by mathematicians and physicists alike.We also have added two appendices. The rst appendix gives a very brief in-troduction to groupoids and their classifying spaces which are employed in Chapter4, while the second gives many tables listing links of hypersurface singularities thatare used throughout the book.We have compiled a very extensive bibliography. There are various reasons forits size. We should remark that in this day and age of easy internet access, withMathSciNet and Google, it would make no sense to simply compile a bibliographyof every paper with the words Sasaki or Sasakian in the title. Anyone with anaccess to MathSciNet can easily compile such a list of 809 papers3so it would serve3As we checked on February 19, 2007.INTRODUCTION 7no purpose to do it for this book. In a way, in spite of its size, we were very carefuland selective in choosing all bibliography items. Our book brings together so manydierent areas of mathematics, that having good references becomes essential. Insome cases the proofs we give are only sketches and in such instances we wantedto refer the reader to the best place he/she could nd more details. That is quiteoften the original source but not always. We refer to various books, monographs,and lecture notes. We have tried to be both selective and accurate in attributingvarious results with care. This can be at times a hard task. We suspect that wedid not always get the references and proper credits exactly right. This is almostinevitable considering we are not experts in many of the areas of mathematics thatsubstantially enter as part of the book. In any case, we apologize for any omissionsand errors; these are certainly not intentional4.Finishing this book was a challenging task. We had to deal with an increasingnumber of new and interesting results appearing every few months. It is always a bitdangerous to include material based on articles that have not yet been published.To make the book as up-to-date as possible we took the risk to include some ofthese new results without giving proofs. On the other hand we are happy to seethat the eld is active and we very much hope that our book will become somewhatoutdated in a few years.4There are many examples in the literature where incorrect attributions are made. A recentcase that we just uncovered is that of nearly Kahler manifolds which are usually attributed toGray [Gra69b, Gra70], yet they were discovered 10 years earlier by Tachibana [Tac59]. Thefact that any nearly Kahler 6-manifold is Einstein is also attributed to Gray in [Gra76], yet itwas proven earlier by Matsumoto [Mat72]. See Section 14.3.2.CHAPTER 1Structures on ManifoldsA unifying viewpoint for doing dierential geometry involving dierent struc-tures is that of a G-structure, where G can stand for geometric or more appro-priately a Lie group. A (rst order) G-structure is just a reduction of the bundleof frames of a manifold from the general linear group to a subgroup. For example,from the point of view of G-structures a Riemannian metric on a manifold corre-sponds to a reduction of the frame bundle with group G1(n. R) to the orthonormalframe bundle with subgroup, the orthogonal group O(n. R). Many other exampleswill be given below. It will be important to have at our disposal the general theoryof connections in principal and associated bre bundles of which the G-structuresmentioned above are special cases. However, before embarking on the study of suchstructures we give a short review of sheaves and their cohomology groups. Sheavesare more general than bundles, but they are a bit too general for describing geomet-ric structures on manifolds. They are mainly used to pass from local informationto global information.1.1. Sheaves and Sheaf CohomologySheaves, which were invented by Jean Leray as a prisoner of war during WorldWar II, have become an important tool in geometry. Here we give a very brief tourof sheaf theory referring to the literature [Bre97, GR65, Hir66, Wel80] for moredetail and proofs. Our presentation follows [Wel80] fairly closely.Denition 1.1.1: Let A be a topological space. A presheaf on A is an assignmentto each nonempty open set l A a set T(l) together with maps, called restrictionmaps,UV : T(l)T(\ )for each pair of open sets l and \ with \ l that satisfy the conditions UU = idUand UW = UV VW whenever, \ \ l.Very often the sets T(l) have some additional algebraic structure, such as agroup, a ring, or a module structure. In this case we assume that the restrictionmaps preserve the algebraic structure. So a presheaf is a contravariant functorfrom the category of open sets of a xed topological space with inclusion maps asmorphisms to the category of groups, rings, or modules whose morphisms are thehomomorphisms of that category.910 1. STRUCTURES ON MANIFOLDSA morphism of presheaves 1 : T( on A is a family of homomorphisms1U : T(l)((l) such that the diagram(1.1.1)T(l) hU ((l)

UV

UVT(\ ) hV ((\ )commutes, where \ l A. If the homomorphisms 1U : T(l)((l) aremonomorphisms then T is a subpresheaf of (.Given a presheaf T over A we can consider the direct limitTx = limxUT(l)with respect to the restriction maps UV. Clearly, Tx inherits whatever algebraicstructure the sets T(l) have. Tx is called the stalk of T at r. and for r l thereis a natural projection Ux : T(l)Tx by sending an element : T(l) to itsequivalence class :x in the direct limit. :x is called the germ of : at r.Denition 1.1.2: A presheaf T is called a sheaf if for every collection li of opensubsets of A with l = li. the presheaf T satises the following two conditions(i) If :. t T(l) and UUi(:) = UUi(t) for all i. then : = t.(ii) If :i T(li) and if for any , with lilj = the equality UiUiUj(:i) =UjUiUj(:j) holds for all i. then there is an : T(l) such that UUi(:) = :ifor all i.A morphism of sheaves is a morphism of the underlying presheaves. In par-ticular, an isomorphism of sheaves is a morphism of sheaves such that the all themaps 1U : T(l)((l) are isomorphisms. Not every presheaf is a sheaf. Forexample consider C and the presheaf that assigns to every open set l C thealgebra of bounded holomorphic functions B(l) on l. Since there are no boundedholomorphic functions on all of C this presheaf violates condition (ii) of denition1.1.2 above. Nevertheless, as we shall see shortly we can associate a sheaf to anypresheaf in a fairly natural way. First we give an important example of a presheafthat is a sheaf.Example 1.1.3: Let A and Y be topological spaces and consider the presheaf (X,Yon A which associates to every open subset l of A the set (X,Y (l) of all continuousmaps from l to Y. The restriction map UV is just the natural restriction, i.e. if1 (X,Y (l) and \ l. then UV (1) = 1[V. It is easy to see that (X,Y satisesthe two conditions of Denition 1.1.2 and thus denes a sheaf on A. An importantspecial case is obtained by taking Y to be either of the continuous elds F = R orC. In this case we note that (X,F has the structure of an F-algebra. Two particularcases of interest are when A is either a smooth real manifold or a complex manifold.In these cases we are more interested in the subsheaf cX (X,R of smooth functionsand the subsheaf OX (X,C of holomorphic functions, respectively. Such sheavesare called the structure sheaf of A.Denition 1.1.4: An etale space over a topological space A is a topological spaceY together with a continuous map : Y A that is a local homeomorphism.1.1. SHEAVES AND SHEAF COHOMOLOGY 11For each open set l A we can consider the set (l. Y ) of continuous sectionsof . that is, the subset of 1 (X,Y (l) that satisfy 1 = idU. Then the assignmentthat assigns to each open subset l of A the set of continuous sections (l. Y ) formsa subsheaf of (X,Y which we denote by oX,Y. We shall now associate to any presheafT an etale space T. Then by taking sections of T we shall get a sheaf. Dene T by(1.1.2) T = .xXTx.Let : TA denote the natural projection that sends any :x Tx to r. Wedene a topology on T as follows: for each : T(l) dene a map : : lT by :(r) = :x. This satises : = idU. Then we let the sets : [ l is open in A. : T(l)be a basis for the topology on T. Then and all functions : are continuous, and itis easy to check that is a local homeomorphism, so that T is an etale space overA. But we have already seen that the set of sections of an etale space form a sheaf.We denote the sheaf associated to the etale space T by oX,Y. Thus, beginningwith a presheaf T we have associated a sheaf oX,Y. This sheaf is called the sheafassociated to or generated by the presheaf T.Exercise 1.1: Show that if one starts with a sheaf T then the sheaf associated toT is isomorphic to T.Denition 1.1.5: A ringed space is a pair (A. /) consisting of a topologicalspace A together with a sheaf of rings / on A. called the structure sheaf. (A. /)is a locally ringed space if for each point r A, the stalk /x is a local ring.All ringed spaces considered in this book will be locally ringed spaces, so weoften omit the word locally and just refer to a ringed space. We now consider thestructure sheaves cM and OM on a real or complex manifold `. respectively. Wedenote by / the sheaf c on a real manifold or the sheaf O on a complex manifold.Denition 1.1.6: Let (A. /) be a ringed space with structure sheaf / given by c(or O), respectively. A sheaf T of /-modules is said to be locally free of rankr if A can be covered by open sets l such that there is an isomorphism of sheavesT[U /[Ur times /[U. A locally free rank 1 sheaf is called an invertible sheaf.Let : 1` be an F-vector bundle over a smooth manifold `. As in thecase of an etale space the subset of 1 (M,E satisfying 1 = idM denes asubsheaf of (M,E, called the sheaf of germs of continuous sections of 1. Actually,we are more interested in the sheaf of germs of smooth sections of 1 which wedenote by c(1). On a complex manifold ` a C-vector bundle can have a specialtype of structure.Denition 1.1.7: Let ` be a complex manifold. A complex vector bundle :1` on ` is said to be holomorphic if 1 is a complex manifold with holomorphic, and the transition functions for 1 can be taken to be holomorphicfunctions.Of course, the transition functions being holomorphic is equivalent to the localtrivializations being holomorphic. Then we haveProposition 1.1.8: Let ` be a real or complex manifold. There is a one-to-onecorrespondence between smooth vector bundles on ` and locally free cM-sheaves on12 1. STRUCTURES ON MANIFOLDS`. Similarly, when ` is a complex manifold, there is a one-to-one correspondencebetween holomorphic vector bundles on ` and locally free OM-sheaves on `.The correspondence is given by associating to a holomorphic vector bundle1 the sheaf of germs of holomorphic sections of 1. Because of Proposition 1.1.8we shall often use interchangeably the concepts of vector bundles and locally freesheaves. Of particular interest is the rank one case in the holomorphic categorywhich gives a one-to-one correspondence between holomorphic line bundles andinvertible OM-sheaves.As mentioned previously the power of sheaf theory is as a tool in passing fromthe local to the global. This is accomplished by the way of sheaf cohomology theorywhich we now briey describe.1.1.1. Sheaf Cohomology. There are several approaches to sheaf cohomol-ogy theory. There is the axiomatic approach used in [GR65], the derived functorapproach using the resolution by discontinuous sections due to Godement [God58]and espoused in [Bre97, Wel80], and nally the Cech theoretic approach describedin [GH78b] and [Kod86]. We prefer to begin with the latter and then present theso-called Abstract de Rham Theorem using resolutions as in [Wel80].For simplicity we assume that A is paracompact and Hausdor. Let T be asheaf A, and let U = lii be a locally nite cover by open sets. Dene the setCp(U. T) of p-chains of T by(1.1.3) Cp(U. T) =i0,...,ipT(li0 lip).where we assume that the indices ij are distinct. We denote elements of Cp(U. T)by i0,...,ip. The coboundary operator p : Cp(U. T)Cp+1(U. T) is dened by(1.1.4) (p)i0,...,ip+1 =p+1j=0(1)ji0,...,

ij,...,ip+1[Ui0Uip.where as usual i means remove that index. This gives rise to a cochain complexC0(U. T) 0C1(U. T) 1 Cp(U. T) pCp+1(U. T) and we dene the cohomology groups Hp(U. T) byHp(U. T) = kerpImp1.If Vis a cover of A rening U. there are homomorphisms : Hp(U. T)Hp(V. T).Thus, we can take the direct limit of the cohomology groups Hp(U. T) as the coverbecomes ner and ner, so we dene the cohomology group with coecients in thesheaf T by(1.1.5) Hp(A. T) = limUHp(U. T).Note that generally the groups Hp(U. T) depend on the cover, but there are certainspecial covers known as acyclic covers for which we have the following theorem ofLerayTheorem 1.1.9: If the cover U is acyclic in the sense that Hq(li0 lip. T)vanishes for all c 0 and all i0. . . . . ip, then H(`. T) H(U. T).1.1. SHEAVES AND SHEAF COHOMOLOGY 13In practice, one always computes H(U. T) for an acyclic cover and then usesthe above theorem.Example 1.1.10: Consider the sphere o2=(r1. r2. r3) R3[ 3i=1(ri)2= 1 andthe constant sheaf T = R. together with two covers. The rst is the stereographiccover dened by two open sets l = o2 (0. 0. 1). This cover is not acyclicsince l+ l has the homotopy type of a circle. The second cover is a coverconsisting of a cover of the lower hemisphere by 3 open sets together with theupper hemisphere. This gives an acyclic cover from which one can compute thecohomology groups Hi(o2. R) of o2. See Example 9.3 of [BT82].The sheaf cohomology groups have some nice properties. For any sheaf T thecohomology group H0(A. T) is the space of global sections T(A) of T. and a sheafmorphism 1 : /B induces a group homomorphism 1q : Hq(A. /)Hq(A. B).Moreover, a short exact sequence of sheaves,0/B(0induces a long exact sequence in cohomology,(1.1.6)0H0(`. /)H0(`. B)H0(`. () H1(`. /) Hq(`. /) .where is the well-known connecting homomorphism.To proceed further we need some denitions.Denition 1.1.11: A sheaf T is soft if for any closed subset o A, the restrictionmap XU : T(A)T(o) is surjective. T is ne if for any locally nite open coverli of A there is a partition of unity of T. i.e. there is a family of sheaf morphismsi : TT such that i = 1. and i(Tx) = 0 for all r in some neighborhood ofA li.The importance of soft sheaves is the followingProposition 1.1.12: If T is a soft sheaf, then Hq(A. T) = 0 for all c 0.Fine sheaves are special cases of soft sheaves, viz.Proposition 1.1.13: Fine sheaves are soft.Example 1.1.14: The structure sheaf cM of a real smooth manifold is ne, thussoft. The structure sheaf OM of a complex manifold is neither ne nor soft, nor arethe constant sheaves.Denition 1.1.15: Let T be a sheaf on A. A resolution of T is an exact sequenceof sheaves Tiof the form0TT0T1 Tk .The resolution is acyclic if Hq(A. Ti) = 0 for all c 0 and all i 0.A resolution of a sheaf T is conveniently written in the shorthand notation0TT. By Proposition 1.1.12 if the sheaves Tiof a resolution are softsheaves, then the resolution is acyclic. Thus, a resolution by soft or ne sheaves isacyclic. A resolution of sheaves gives rise to a cochain complex C = C(T) onglobal sections(1.1.7) 0T(A)T0(A) d0T1(A) d1 dk1Tk(A) dk .14 1. STRUCTURES ON MANIFOLDSIn general this sequence is exact only at T(A). but we do have dk+1 dk = 0. Thecohomology groupHk(C) = ker dkim dk1of this complex is called the /thderived group of C. We are now ready for theAbstract de Rham Theorem:Theorem 1.1.16: Let T be a sheaf over a paracompact Hausdor space A, andlet 0TT be an acyclic resolution of T. Then there is a natural isomorphismHq(A. T) Hq(C) = ker dkim dk1.This is a very powerful theorem of which the usual de Rham theorem is a specialcase. We give this as an example below. The Dolbeault Theorem is another specialcase which will be given in Chapter 3.Example 1.1.17: de Rhams Theorem. Let ` be a smooth manifold and letcpdenote the sheaf of germs of sections of the exterior bundles p` with c0= c.Let T = R the constant sheaf on `. Then by the well-known Poincare Lemma weget a resolution of the constant sheaf R. viz0Rc0c1 cn1cn0.Since the sheaves cpare ne this resolution is acyclic, so by the abstract de RhamTheorem 1.1.16 we have an isomorphismHk(`. R) Hk

ker dkim dk1

= HkdeRh(`. R).Moreover, Hk(`. R) can be identied with the /thsingular cohomology group bytaking an acyclic resolution of the constant sheaf R by sheaves of singular cochains.See [Wel80].1.2. Principal and Associated BundlesWe begin by considering principal bre bundles. The main reference here isthe classic text of Kobayashi and Nomizu[KN63, KN69].Denition 1.2.1: A principal bundle 1 over ` with Lie group G consists of:(1) A pair of smooth manifolds 1 and ` together with a smooth surjection : 1 `.(2) A smooth free action of the Lie group G on 1, 1 : 1 G 1 given by theright action (n. o) 1an = no such that the quotient is `, i.e., ` = 1G.(3) 1 is locally trivial. More precisely, every point j ` has a neighborhood land a dieomorphism : 1(l) l G dened by (n) = ((n). (n)). where : 1(l) G satises the compatibility condition (no) = (n)o.As usual the manifold 1 is called the total space, the manifold ` is called the basespace, and for j `, 1(j) is called the bre at j. If we x a point n0 1(j)the map sending o G to n0o 1(j) identies 1(j) with G. Moreover, oneeasily sees that this map is a dieomorphism from G to 1(j) whose inverse is .When we want to emphasize the base space and the group we write 1(`. G) for 1,or we refer to 1 as a principal G-bundle over `. Notice that for any open subsetl `. 1(l) is a principal G-bundle over l. Another example is the trivialG-bundle when 1 is just the product ` G.1.2. PRINCIPAL AND ASSOCIATED BUNDLES 15Now let l be an open cover of ` such that for each the map :1(l)l G is a dieomorphism, and satises the compatibility conditionof Denition 1.2.1. Then if n 1(l l) we see that (no)((no))1=(n)((n))1for all o G. Hence, we can dene a smooth map : l lG by(1.2.1) ((n)) = (n)((n))1.The smooth maps are called transition functions for the principal bundle1(`. G). and one easily sees that for each j l l l they satisfy thecocycle condition(1.2.2) (j) = (j)(j).The appellation cocycle condition comes from the fact that equivalence classes oftransition functions are elements of the sheaf cohomology set H1(`. (). where (denotes the sheaf of germs of maps from opens sets of ` to G. Here two transitionfunctions and t are equivalent if there are maps : lG such thatt(j) = (j)(j)((j))1. and the inverse element in G is given by (j) =((j))1. The reader is referred to Section 1.1 below for more details on sheaftheory and its cohomology. We then have the well-knownTheorem 1.2.2: Let l be an open cover on the smooth manifold ` and letG be a Lie group. Suppose further that on each nonempty intersection l lthere exist smooth maps : l lG satisfying (1.2.2). Then there is aprincipal G-bundle 1(`. G) with transition functions . Moreover, there is abijective correspondence between isomorphism classes of principal G-bundles andelements of H1(`. ().The rst statement in the theorem says that the transition functions determinethe principal bundle, and the last statement says that the set H1(`. () classiesprincipal G-bundles over `. The denition of an isomorphism of principal bundleswill be given below. It is only in the case that G is Abelian, for example, G = o1,the circle, that H1(`. () has itself the structure of an Abelian group, and morestandard techniques such as the exponential sequence can be used to relate this tothe integral cohomology of `. We will discuss this later in more detail.Perhaps the most important example of a principal bundle is:Example 1.2.3: The linear frame bundle 1(`). Recall that a frame n =(A1. . . . . An) at a point j ` is a basis of the tangent space Tp`. We let 1(`)denote the set of all frames at all points of `. Then : 1(`)` is a principalbundle on ` with group G1(n. R). The action of G1(n. R) on 1(`) is just givenby matrix multiplication from the right, that is (n. ) n for n 1(`) and G1(n. R). The local triviality can be seen by choosing a local coordinate chart(l; r1. . . . . rn) on ` and writing the vectors Ai of the frame n in local coordinatesasAi =jAjirj.where the Aji are the components of a non-singular matrix (Aji ) of smooth functionson l. Then the dieomorphism : 1(l)l G1(n. R) is given by () =((). (Aji ())). We remark that a point n 1(`) can be viewed as a linear map16 1. STRUCTURES ON MANIFOLDSn : RnTp` with j = (n) by n(ci) = Ai. where c1. . . . . cn denotes thestandard basis of Rn.Exercise 1.2: Let G be a Lie group and H a closed Lie subgroup. Show that theLie group G can be viewed as a principal H-bundle G(GH. H) over the homoge-neous manifold GH.Now suppose we are given a principal G-bundle 1(`. G) over `, and a prin-cipal H-bundle Q(. H) on . A homomorphism of principal bundles consists of asmooth map 1 : Q1 together with a Lie group homomorphism / : HG suchthat for all n Q and o H. 1 satises 1(no) = 1(n)/(o). This condition impliesthat 1 maps bres to bres; hence, there is a smooth map 1 : ` such thatthe diagram(1.2.3)Qf 1

Q

Pf `commutes. If in addition 1 is an embedding of smooth manifolds and / is a groupmonomorphism then we say that 1 : Q1 is an embedding of principal bundles.This implies that the map 1 : ` is also an embedding. In the case that = ` and 1 is the identity map on `. then Q(`. H) is called a subbundleor a reduction of 1(`. G) to the group H. We also say that Q(`. H) is a reducedsubbundle of 1(`. G). It may not be possible to reduce a principal bundle 1(`. G)to a given subgroup H. In general, there are topological obstructions to doing so. Weshall see many examples of this below. If also / is an isomorphism then Q(`. H)and 1(`. G) are isomorphic as principal bundles. We also say that 1(`. G) istrivial if it is isomorphic to the product ` G. Observe that the local trivialityconditions says that every point j ` has a neighborhood l such that 1(l) istrivial. Concerning the transition functions we haveProposition 1.2.4: A principal G-bundle 1(`. G) can be reduced to a Lie sub-group H if and only if there is an open cover l of ` with transition functions taking their values in H.Proof. Suppose that 1(`. G) can be reduced to a Lie subgroup H. Then thereis a principal H-bundle Q(`. H) together with a smooth bundle map 1 : Q1such that the diagram (1.2.3) commutes and 1(no) = 1(n)o for all o H G. Onthe open set 1Q (l) Q(`. H) the maps Q and P are related by Q = P 1.It follows that for any 1 there exists an o G and n Q such that = 1(n)oand P() = Q(n)o. Hence, the transition functions P satisfy(1.2.4) P() = P ()(P())1= Q (n)o o1(Q(n))1= Q (n)(Q(n))1.and thus, have their values in H.Conversely, given transition functions P : l lG which take theirvalues in the Lie subgroup H. a standard result says that P is smooth as amap into H. Thus, by Proposition 1.2.2 there is a principal H-bundle Q(`. H)with transition functions P. To construct the bundle map 1 we dene maps1 : 1Q (l)1P (l) by putting 1 = 1P Q. One easily sees that 1 = 11.2. PRINCIPAL AND ASSOCIATED BUNDLES 17on l l. and so denes a global bundle map 1 : Q1 with the requisiteproperties. Let 1(`. G) be a principal bundle and 1 a G-manifold, that is, a manifoldtogether with a smooth action of the Lie group G. We denote this action by leftmultiplication as r o1r. where r 1 and o G. Then we have the productaction on 1 1 dened by (n. r) (no. o1r). The quotient space (1 1)Gby this action is a smooth bre bundle 1(`. 1. G. 1) called the bundle associatedto 1(`. G) with bre 1. Note that the projection map E is dened as follows: let[n. r] denote equivalence class of the pair (n. r) 1 1. where (n. r) is equivalentto (nt. rt) if there is an o G such that (nt. rt) = (no. o1r). Then dene theprojection map E : 1` by E([n. r]) = P(n). One easily checks that thisis well dened. In the case that 1 is a vector space, say Rk, and G = G1(/. R)the associated bundle 1 = 1(`. Rk. G1(/. R). 1) is called a vector bundle (of rank/) associated to the principal bundle 1(`. G1(/. R)). or just a real vector bundle.Similarly, if the bre 1 is Ckwith group G = G1(/. C). then 1 is called a complexvector bundle. We shall often combine these an refer to a real or complex vectorbundle as an F-bundle, where F = R or C. A vector bundle of rank one is called aline bundle.Exercise 1.3: Use E to dene a dierential structure on 1(`. 1. G. 1) thatmakes it a smooth bre bundle with projection map E and bre 1.Exercise 1.4: Show that the tangent bundle T` of ` is a vector bundle associ-ated to the principal bundle 1(`). More generally show that the tensor bundlesTrs`. exterior bundles p`. and symmetric bundles op` are vector bundles as-sociated to the principal bundle 1(`).One often describes structures on manifolds by tensor elds. Examples arecomplex structures, symplectic structures, Riemannian structures. These arise assmooth sections of certain bundles over ` which are associated to the frame bundle1(`) or its subbundles. Recall that a section of a bre bundle E : 1`is a (smooth) map : : `1 such that E : = 1lM. Unless otherwise statedsections will be smooth. Vector bundles always have sections (e.g., the zero section);whereas, a principal bundle has a section if and only if it is trivial. This is easy toverify directly, but also follows from Theorem 1.2.5 below. By a local section of 1we shall mean a section of the bundle 1E (l) for some open set l in `.Theorem 1.2.5: A principal G-bundle 1(`. G) is reducible to a closed subgroup Hif and only if the associated bundle 1(`. GH. G. 1) admits a section : : `1.Moreover, there is a bijective correspondence between such sections : and subbundlesQ(`. H) of 1(`. G).Proof. Suppose 1(`. G) is reducible to a closed subgroup H and let 1 :Q1 be the subbundle. The associated bundle 1 can be identied with thequotient space 1H by mapping the equivalence class [n. oH] 1 to the equivalenceclass [no] 1H. Let : 11 = 1H denote the natural projection. Then it iseasy to see that 1 is constant on the bres of Q. Thus, we can dene a section: : `1 by setting :(r) = (1(n)). where P(1(n)) = r.Conversely, let : : `1 = 1H be a section, and consider 1 as a principalH-bundle over 1 = 1H with transition functions / : \\H. where \ isan open cover of 1. Then we can dene transition functions for 1(`. G) with values18 1. STRUCTURES ON MANIFOLDSin H by setting = / :. Thus, by Proposition 1.2.2 1(`. G) is reducible tothe subgroup H. The correspondence between sections and H-subbundles can beseen to be 1-1. Let 1(`. G) be a principal bundle on a smooth manifold ` with G = G1(/. R),and let 1 be an associated real vector bundle of rank /. Reductions of 1(`. G) tosubgroups G G1(/. R) correspond to adding certain structures to its associatedvector bundle 1. In general, there are obstructions to be able to do this. For exam-ple, a reduction to the group G1(/. R)+corresponds to choosing an orientation on1. which can be done if and only if the bundle 1 is orientable. Such obstructions areoften given in terms of so-called characteristic classes [MS74]. Obstruction theoryfor general bre bundles is expounded in the classic text of Steenrod [Ste51] andalso in [Hus66]. 1 is orientable if and only if its rst Stiefel-Whitney class n1(1)vanishes. See Appendix A. Many more examples of such obstructions are discussedfor G-structures in Section 1.4. Another example of such a reduction is the reduc-tion of 1(`. G1(/. R)) to its maximal compact subgroup O(/. R). Since G1(/. R)is homotopy equivalent to the orthogonal group O(/. R) there are no obstructionsto performing this reduction. This corresponds to choosing a Riemannian metricon 1. If 1 is also orientable then we can reduce further to the special orthogonalgroup oO(/. R). The corresponding principal bundle is now 1(`. oO(/. R)).Another type of structure comes from lifting instead of reduction. Thus, since1(oO(/. R)) = Z2 the group oO(/. R) has a two-sheeted universal covering groupcalled Spin(/). So given a 1(`. oO(/. R)) or one of its associated vector bundles 1one can ask whether the bundle 1(`. oO(/. R)) can be lifted to a covering bundle1(`. Spin(/)) with group Spin(/)? When such a covering exists the bundle 1 issaid to have a spin structure. We refer to [LM89] for a full treatment of spinstructures. The exact sequence of Lie groups 0Z2Spin(/)oO(/)0induces the coboundary map n2 : H1(`. oO(/))H2(`. Z2) whose image isjust the second Stiefel-Whitney class n2(1) described briey in Appendix A. Soan oriented vector bundle 1 admits a spin structure if and only if n2(1) = 0.Moreover, the distinct spin structures on 1 are in one-to-one correspondence withthe elements of H1(`. Z2). We say that a smooth oriented manifold ` is a spinmanifold if T` admits a spin structure. So ` is spin if and only if n1(`) =n2(`) = 0. Spin manifolds admit certain vector bundles that do not exist on non-spin manifolds, namely, those whose bres are representations of Spin(n) that arenot representations of oO(n) or G1(n. R). Such vector bundles are called spinorbundles and its sections are called spinor elds. Evaluation at a point gives aspinor. Notice that spinor bundles depend on a choice of Riemannian metric. Astudy of these representations involves Cliord algebras which will not be treatedin any detail in this text, and will not appear until Chapter 14. See [LM89] for athorough treatment.1.3. Connections in Principal and Associated Vector BundlesIn this section we briey review the fundamentals of the theory of connections.As the reader undoubtedly knows there are various settings and denitions for aconnection. We begin with the most general as well as most abstract, namely thatof connections in a principal bre bundle. We then discuss connections in associatedvector bundles, and the relationship to the former. Both formulations have theiradvantages and will be used in the sequel. This brief discussion will also allow us to1.3. CONNECTIONS IN PRINCIPAL AND ASSOCIATED VECTOR BUNDLES 19set our notation and terminology for the remainder of the text. Again the standardreference for much of this material is the classic text of Kobayashi and Nomizu[KN63, KN69].Let 1 be a principal bundle over ` with group G. Let n 1 and consider thetangent space Tu1 at n. Let Gu be the vertical subspace of Tu1 consisting of allvectors that are tangent to the bre 1((n)). The dierential of the restrictionof the map to the bre through n identies Gu with the tangent space TeG atthe identity c G, and thus with the Lie algebra g of G.Denition 1.3.1: A connection in 1 is an assignment to each n 1 a com-plimentary subspace Hu to Gu Tu1, called the horizontal subspace, that for alln 1 satises:(i) Tu1 = GuHu.(ii) Hua = (1a)Hu for all o G.(iii) Hu depends smoothly on n.In words a connection is a G-equivariant choice of compliment to the verticalthat varies smoothly with n. If A Tu1 then A and /A denotes the verticaland horizontal components of A, respectively. We can describe a connection infancierterminology as follows. We dene the vertical subbundle of T1 by 11 =uP Gu. and the horizontal subbundle by H1 =uP Hu. Then we have an exactsequence of G-modules0 11 T1 O 0.Then a connection in 1 is a splitting of this exact sequence as G-modules, and thusgives a decomposition of the tangent bundle T1 as G-modules, viz.T1 = 11 H1.For a Lie group G acting smoothly on a manifold ` there is well-known homo-morphism from the Lie algebra g of G to the Lie algebra A(`) of smooth vectorelds on `. and if G acts eectively (which we assume) this is a monomorphism.Given g we let denote its image in A(`). If we specialize to the case of theright action of G on a principal G-bundle 1, the vector eld is tangent to thebres 1((n)) at each point n 1, and is called the fundamental vertical vectoreld on 1 associated to g. Evaluation of at a point n 1 gives a vectorspace isomorphism of the Lie algebra g with the vertical tangent space Gu at n.Denition 1.3.2: Given a connection on 1, we dene the connection formassociated to to be the 1-form on 1 with values in the Lie algebra g by setting(A) equal to the unique g such that ()u is the vertical component of A.Clearly, satises (A) = 0 if A is horizontal. In the sequel we are particularlyinterested in the case of a circle bundle. In this case the Lie algebra g R. andit is common to take 1 as a generator of g. Actually a connection 1-form denes aconnection as is seen by the followingProposition 1.3.3: The connection form satises(i) () = for all g.(ii) 1a = oda1 for all g.Conversely, given a g valued 1-form on 1 which satises conditions (i) and (ii),there is a unique connection in 1 whose connection form is .20 1. STRUCTURES ON MANIFOLDSWe shall often refer to a connection 1-form as simply a connection. Bylocalizing and using partitions of unity one hasTheorem 1.3.4: Every principal bundle 1(`. G) admits a connection.Let 1(`. G) be a principal bundle and \ be a nite dimensional vector spaceover F = R. C or the quaternions1Q. Let r(1. \ ) denote the set of smooth :-formson 1 with values in \. that is the set of smooth sections of the bundle r(1)\. Thisset has some natural algebraic structures. It is an innite dimensional vector spaceover F as well as a C(1) RF module. Consider a representation : G nt \on \ . A pseudotensorial form of degree : and of type (. \ ) is :-form on 1 withvalues in \ such that1a = (o1) . is called tensorial if is pseudotensorial and satises (A1. . . . . Ar) = 0 if atleast one of the vector elds A1. . . . . Ar is vertical. In particular, a pseudotenso-rial 0-form, which is automatically tensorial, is just a smooth G-equivariant mapfrom 1 to \. We denote by Tr(1. \ ) the closed subspace of tensorial r-forms. Nowsuppose that is a connection in 1(`. G). Then the connection form of ispseudotensorial of type (od. g), but not tensorial. Notice, however, Proposition1.3.3 implies that the dierence t of any two connections is tensorial. Givenany pseudotensorial form we can dene a tensorial form by taking its horizontalprojection, i.e., /(A1. . . . . Ar) = (/A1. . . . . /Ar). Generally d is only pseu-dotensorial, even if is tensorial. However, the exterior covariant derivative 1dened as the horizontal projection(1.3.1) 1(A1. . . . . Ar+1) = d(/A1. . . . . /Ar+1)is always tensorial. In particular, if is a connection 1-form, 1 is a tensorial 2-form of type (od. g) called the curvature 2-form of and usually denoted by . It isa smooth section of 2(1) g. and satises the famous Cartan structure equations(1.3.2) = 1 = d + 12[. ].as well as the well-known Bianchi identities(1.3.3) 1 = 0.The meaning of the bracket in the Equation (1.3.2) is [. ](A. Y ) = [(A). (Y )].Generally, the exterior covariant derivative of any tensorial form is1 = d + [. ].A connection or is said to be at if its curvature vanishes.Tensorial forms can be described alternatively in terms of associated bundles.Let 1 = 1 G\ be the F-vector bundle on ` with standard bre \ and associatedto the F-representation of G. Then there is a 1-1 correspondence between tensorial:-forms Tr(1. \ ) of type (. \ ) and smooth sections of the bundle r(`) 1 asfollows: By xing a point n 1 the natural projection 1 \ 1 gives an F-linear isomorphism from the vector space \ to the bre 1(u). Then if A1. . . . Arare tangent vectors at (n) ` we dene the section of r(`) 1 by(A1. . . . . Ar) = n(A1. . . . . Ar).1We shall need all three types of vector spaces in the sequel. Some care must be takenwhen working with quaternionic vector spaces, due to the noncommutativity of Q. For example,quaternionic vector spaces are considered by multiplication from the left.1.3. CONNECTIONS IN PRINCIPAL AND ASSOCIATED VECTOR BUNDLES 21where A denotes any vector on 1 that projects to A on `. One checks that is independent of the choices made. In particular, smooth G-equivariant functions1\ correspond to smooth sections of 1.Now a connection in 1 induces a connection in the vector bundle 1 and,more generally, in the vector bundles r(`) 1. We denote the C(`)-moduleof smooth sections of r(`) 1 by r(`. 1).Denition 1.3.5: A (Koszul) connection on an F-vector bundle 1 is an F-linear map on sections: : (1) 1(`. 1) satisfying the Leibnitz rule (1:) =1: +d1 :. where 1 C(`) and : is a smooth section of 1.Given a connection on 1(`. G) and an associated vector bundle 1 = 1 G\.we can dene a connection on 1 by setting = n1n1.Conversely, suppose we have an F-vector bundle 1 with associated principal bundle1(`. G) and F-representation of G on the standard bre \ so that 1 = 1 G\.Then given a connection on 1. we obtain a connection on 1 as follows: Lets = (:1. . . . . :k) be a G-frame of local sections of 1. If A Tx`. then the subspaceHu of 1u with (n) = r dened byHu = sA [ Xs = 0is a G-equivariant complement to the vertical subspace Gu. This denes the con-nection in 1. The Koszul connection in the associated bundles is often referredto as the covariant derivative, and a section : of 1 is said to be covariantly constantif : = 0.As in Proposition 1.3.3 it is convenient to express the connection in terms ofa 1-form. However, this can only be done locally in terms of a local trivializationof the vector bundle 1. Given a local trivialization 1[U l \. we can write(1.3.4) [U = d +U.where the exterior derivative d represents the at connection on l and U is a g-valued 1-form on l. Given another such trivialization on the open set \ `. andtransition functions o : l \G relating the two trivializations in the overlap,one obtains the relation(1.3.5) W = o1Uo +o1do.Conversely, given a cover of ` by open sets l with g-valued 1-forms U on eachopen set in the cover satisfying Equation (1.3.5) in the overlaps, one can reconstructthe connection on 1. This formulation is essentially Cartans denition of con-nection, whereas, the formulation given in Proposition 1.3.3 is due to his studentEhresmann.There is a natural extension of Equation (1.3.4) to the bundles r(`) 1. Weshall use the notation in [DK90] and write the exterior covariant derivative asdA = d + : r(`. 1)r+1(`. 1).Here the symbol denotes a family of g-valued 1-forms on open sets of ` satisfyingEquation (1.3.5) which act linearly via the representation : g1nd \ on thelocal sections of 1 obtained from a local trivialization of 1. The curvature form22 1. STRUCTURES ON MANIFOLDS = 1 on 1 corresponds by the isomorphism n to the smooth section 1A= dAof 2(`) g. The Cartan structure Equations (1.3.2) then take the form(1.3.6) 1A= dA = d+ .where here again we follow the convention in [DK90] using the wedge product inlieu of the brackets to emphasize as an endomorphism of \ via local trivializa-tions.Exercise 1.5: Show that if we write the g-valued connection 1-form as =idriand its curvature 2-form 1A=1ijdridrjin a local coordinate chart(l; r). Equation (1.3.6) can be written as1ij = jri irj+ [i. j].A connection in 1(`. G) allows us to dene the notion of parallel translationof a bre of 1 along any piecewise smooth curve in `. This is done as follows: Letr0. r1 ` and let : [0. 1]` be a curve in ` with (0) = r0 and (1) = r1.Now at each point n 1 there is a vector space isomorphism : T(u)`Hu.So xing n0 1(r0) we can lift horizontally to a unique piecewise smoothcurve in 1 such that (0) = n0 and (t) = ( (t)) for all t [0. 1]. This givesan isomorphism of bres 1((t)) 1(r0). called parallel translation along .Now suppose that is a loop at r0, then parallel translation gives automorphismsof the bre 1(r0). By composing loops and running the loop backwards, we seethat the set of such automorphisms form a group. Moreover, if we x a pointn 1(r0) this group can be identied with a connected Lie subgroup of G,called the holonomy group through n and denoted by Holu. If we restrict ourselvesto loops at (n) that are null homotopic, then we obtain a normal subgroup Hol0uof Holu known as the restricted holonomy group through n. The groups Hol0u andHolu enjoy the following properties:(i) If n. 1 can be joined by a horizontal curve then Holu = Holv andHol0u = Hol0v.(ii) If = no for o G. then Holu = AdaHolv and H0(n) = AdaHol0u.A fundamental result in the theory of holonomy groups is the so-called Reduc-tion Theorem which we now state. Of course, we refer to [KN63] for its proof:Theorem 1.3.6: Let ` be a smooth connected manifold and 1(`. G) be a prin-cipal G-bundle with a connection . Let n 1 be an arbitrary point and let 1(n)denote the subset of points in 1 that can be joined to n by a horizontal curve. Then1(n) is a reduced subbundle of 1 with structure group Holu, and the connection restricts to a connection on 1(n).The subbundle 1(n) is called the holonomy bundle through n. and we call sucha connection reducible. It has its values in the Lie algebra holu of Holu. A well-known theorem of Ambrose and Singer [AS53] characterizes the Lie algebra holuas precisely the Lie algebra spanned by the curvature v(A. Y ). where 1(n)and A. Y are arbitrary horizontal vectors at . Conversely, if Q(`. H) 1(`G)is principal subbundle corresponding to a Lie subgroup H G, a connection ina principal bundle Q(`. H) can be extended to a connection in 1(`. G).1.4. G-STRUCTURES 231.4. G-StructuresIn this section we describe structures on manifolds from the unifying viewpointof G-structures. There are several texts where this point of view is expounded[Kob72, Mol77, Sal89, Ste83]. Here we consider only rst order G-structures,that is, reductions of the bundle of linear frames on `. Here is the precise denition.Denition 1.4.1: Let G G1(n. R) be a subgroup, then a G-structure on ` isa reduction of the frame bundle 1(`) to the subgroup G. The G-structure is saidto be integrable if every point of ` has a local coordinate chart (l; r) such thatthe local section

r1. . . . . rn

of 1(`) is a local section of the reduced bundle 1(`. G). Such a coordinate chartis called admissible.Let us return to the bundle of linear frames 1(`) of a smooth manifold `.On 1(`) there is a canonical Rn-valued 1-form dened as follows: as seen inExample 1.2.3 we can view any point n 1(`) as a vector space isomorphismn : RT(u)`. So can be dened by(1.4.1) '. A` = n1A.where A Tu1(`) and '. ` denotes the natural pairing between the tangent bun-dle to the bundle of linear frames T1(`) and its dual cotangent bundle T1(`).If 1(`. G) is a G-structure on `. i.e., a subbundle of 1(`). then we can restrict to 1(`. G) giving a canonical 1-form on the G-structure. The canonical 1-form ona G-structure 1(`. G) behaves functorially under the action of the general lineargroup G1(n. R) on 1(`). Indeed we haveLemma 1.4.2: For any o G G1(n. R) the transformation rule holds:1a = o1.Proof. For any vector eld A on 1(`. G), we have'1a. A` = '. 1aA` = (no)11aA = o1n1A = 'o1. A`. Generally, there are topological obstructions to the existence of G-structures.To see that such a reduction does not always exist let G = c the identity group.Then an c-structure on ` is nothing but a global frame or parallelism of `. Butit is well-known that global frames do not always exist, that is that ` may notbe parallelizable. For example, the 2-sphere o2does not even have one nowherevanishing vector eld let alone a global frame. Even if there is a G-structureon ` there may not be an integrable G-structure. For example, as seen belowevery manifold ` admits many O(n. R)-structures, but a compact ` admits anintegrable O(n. R)-structure only if ` is covered by a torus (see Example 1.4.7below). On the other hand every G1(n. R) structure on ` is integrable. Thefollowing proposition is evident.Proposition 1.4.3: A G-structure 1(`. G) is integrable if and only if there is anatlas of coordinate charts (l; r())I on ` whose Jacobian matrices

x()ix()j

i,jlie in G at all points of l l.24 1. STRUCTURES ON MANIFOLDSRecall from Section 1.3 that associated to any connection in a principal bundlethere is the fundamental curvature 2-form. So for any connection in 1(`), usuallycalled a linear connection, we have its curvature = 1. However, since the linearframe bundle 1(`) has a canonical 1-from associated to it, we have another2-form associated to any linear connection . namely = 1. called the torsion2-form. In the case of linear connections we can add to Cartans structure Equation(1.3.2) the so-called First Structure Equation(1.4.2) d + = .We can also add to the Bianchi identities (1.3.3), the First Bianchi identities(1.4.3) 1 = .Exercise 1.6: Show that for linear connections the usual expressions for the torsionand curvature tensorsT(A. Y ) = XY YA[A. Y ]. 1(A. Y )7 = XY7YX7[X,Y ]7are related to the corresponding Cartan expressions byT(A. Y ) = n(2(A. Y)). 1(A. Y )7 = n(2(A. Y))(n17).where A. Y are the horizontal lifts of the vectors A. Y T(u)`, respectively,and n 1(`) is any point.This entire discussion holds for any G-structure 1(`. G) 1(`) with a con-nection . We shall often refer to a linear connection in a G-structure 1(`. G) asa G-connection. A linear connection with = 0 is said to be torsion-free. Clearlythe notions of parallel translation and holonomy apply to the case of G-structures.We now want to put Theorem 1.2.5 to work by seeing how certain naturaltensor elds dene G-structures on `. Suppose that T0 is an element of the tensoralgebra T (Rn) over Rnand that G is the largest closed Lie subgroup of G1(n. R)that leaves T0 invariant. Viewing a point n 1(`) as a vector space isomorphismn : RnT(u)`. we obtain an induced isomorphism n : T (Rn)T (T(u)`) ofthe tensor algebras. Then the image T = nT0 is a section of the tensor algebrabundle T (`). If T0 is a tensor of type (:. :) then T is a section of the tensor bundleTrs`. In any case because of the invariance of T0 under G, the tensor eld T denesa section of the associated bundle 1(`)G. In this case we say that 1(`. G) is aG-structure dened by the tensor T0.Proposition 1.4.4: Let 1(`. G) be a G-structure dened by the tensor T0. Then1(`. G) is integrable if and only if there exists an atlas of charts (l; r())Ion ` such that the corresponding tensor eld T = nT0 has constant componentson l.Proof. () Let 1(`. G) be integrable and let (l; r) be a coordinate chart,then the frame n = ( x1. . . . . xn) belongs to 1(`. G). Let ci denote the standardbasis for Rnand cj its dual basis. Then xi= n(ci). and drj = n(cj). So ifT0 = ti1irj1jsci1 cir cj1 cjsis a tensor of type (:. :), we haveT = nT0 = ti1irj1jsri1 rirdrj1 drjs.Hence, T has constant components on l.1.4. G-STRUCTURES 25() Conversely, suppose that (l; r) is a coordinate chart such that T = nT0has constant components. So T is a constant section of the associated bundle1(l)G. Then there is a linear transformation of coordinates on l to new coor-dinates (n1. . . . . nn) such that the frame n = ( y1. . . . . yn) belongs to 1(`. G).Thus, the G-structure 1(`. G) is integrable. One easily sees that if a G-structure 1(`. G) is dened by a tensor eld T and is a G-connection, then T is covariantly constant with respect to . or equivalently,T = 0. Generally, the integrability condition of Denition 1.4.1 can be veryrestrictive, and we will discuss various levels of integrability. First, it is easy toshow thatProposition 1.4.5: An integrable G structure admits a torsion-free connection.It is now convenient to consider a much less restrictive denition of integrability.Denition 1.4.6: A G-structure 1(`. G) is said to be 1-integrable if it admitsa torsion-free G-connection.So an integrable G-structure is automatically 1-integrable. The failure of theexistence of torsion-free G-connection can be seen as the rst order obstruction tointegrability, hence, the name 1-integrable. The question of uniqueness of a torsion-free connection, assuming one exists, is related to prolongations of G-structureswhich we shall briey treat below (See Denition 1.6.8). We now wish to considermany examples of G-structures. Our rst example is a good example where non-integrable G-structures are of more interest than the integrable ones.Example 1.4.7: Riemannian metrics. We consider a reduction of the framebundle 1(`) to the orthogonal groupO(n. R) = G1(n. R) [ t = 1ln .The reduced bundle O(`) 1(`) is called the orthonormal frame bundle of `.Theorem 1.2.5 says that such a reduction is equivalent to a choice of section of theassociated bundle 1(`. G1(n. R)O(n. R). 1(`)) = 1(`)O(n. R). We show thatsuch a section is just a Riemannian metric on `. As mentioned previously eachpoint n 1(`) gives an isomorphism of the standard vector space Rnwith thetangent space T(u)`. Let '. ` denote the Euclidean inner product on Rn. and letA. Y T(u)`. Thenou(A. Y ) = 'n1A. n1Y `denes an inner product on T(u)`. Furthermore, if o O(n. R) we haveoua(A. Y ) = '(no)1A. (no)1Y ` = 'o1n1A. o1n1Y `= 'n1A. n1Y ` = ou(A. Y ).where the second to the last equality holds by the invariance of '. ` under O(n. R).This shows that ou is constant along the bres of O(`). and thus, is a section of1(`)O(n. R). Thus, the associated bundle 1(`)O(n. R) can be identied witha subbundle of the vector bundle Sym2T` of symmetric covariant 2-tensors on`. The choice of n modulo O(n. R) corresponds to a choice of Riemannian metricon `. Since manifolds are paracompact the standard partition of unity argumentshows that such sections and hence, such reductions always exist.When are O(n)-structures integrable? According to Proposition 1.4.4 this oc-curs when the metric tensor o has constant components in some coordinate chart26 1. STRUCTURES ON MANIFOLDS(l; r). and then by a change of coordinate, say (\ ; n) the metric can be broughtto the formo =i(dni)2in \. By the Second Fundamental Theorem of Riemannian geometry this happensprecisely when the Riemann curvature tensor vanishes. For example, if ` is a com-pact 2-dimensional manifold, then it is either a torus or a Klein bottle dependingon whether it is orientable or not. This is quite restrictive. On the other handthe First Fundamental Theorem of Riemannian geometry says that there existsa unique torsion-free Riemannian connection, denoted g , called the Levi-Civita connection. So all Riemannian G-structures are 1-integrable. Similarly, areduction to the group oO(n. R) = O(n. R) G1+(n. R) corresponds to orientedRiemannian geometry. In particular, one can consider the parallel translation de-ned by the Levi-Civita connection and its associated holonomy group which is asubgroup of the structure group O(n. R) (oO(n. R) in the oriented case). Since thisconnection gis uniquely associated to the metric o, we denote it by Hol(o), andrefer to it as the Riemannian holonomy group or just the holonomy group whenthe context is clear. Indeed, it is precisely this Riemannian holonomy that playsan important role in this book. Now on a Riemannian manifold (`. o) there is acanonical epimorphism 1(`)Hol(o)Hol0(o). in particular, if 1(`) = 0 thenHol(o) = Hol0(o). In 1955 Berger proved the following theorem [Ber55] concerningRiemannian holonomy:Theorem 1.4.8: Let (`. o) be an oriented Riemannian manifold which is neitherlocally a Riemannian product nor locally symmetric. Then the restricted holonomygroup Hol0(o) is one of the following groups listed in Table 1.4.1.Table 1.4.1: Bergers Riemannian Holonomy GroupsHol0(o) dim(`) Geometry of ` CommentsoO(n) n orientable Riemannian generic Riemannianl(n) 2n Kahler generic Kahlerol(n) 2n Calabi-Yau Ricci-at Kahleroj(n) oj(1) 4n quaternionic Kahler Einsteinoj(n) 4n hyperkahler Ricci-atG2 7 G2-manifold Ricci-atojin(7) 8 ojin(7)-manifold Ricci-atWe will encounter all the geometries listed in this table throughout this book.Most of them will already be introduced in this chapter as G-structures. Orig-inally, Bergers list included ojin(9) but Alekseevsky proved that any manifoldwith such holonomy group must be symmetric [Ale68]. In the same paper Bergeralso claimed a classication of all holonomy groups of torsion-free ane (linear)connections that act irreducibly. He produced a list of possible holonomy rep-resentations up to what he claimed was a nite number of exceptions. But hisclassication had some gaps discovered 35 years later by Bryant [Bry91]. An in-nite series of exotic holonomies was found in [CMS96] and nally the classicationin the non-Riemannian ane case was completed by Merkulov and Schwachhofer[MS99]. We refer the reader to [MS99] for the proof, references and the history ofthe general ane case. In the Riemannian case a new geometric proof of Bergers1.4. G-STRUCTURES 27Theorem is now available [Olm05]. An excellent review of the subject just priorto the Merkulov and Schwachhofers classication can be