The measure of transitive geodesics on certain three-dimensional manifolds

THE MEASURE OF TRANSITIVE GEODESICS ON CERTAIN THREE-DIMENSIONAL MANIFOLDS

BY /kNNIT TULLER

Introduction. The problem of the existence of transitive geodesics on two-dimensional manifolds of constant negative curvature has been completelysolved by Koebe [1]. These manifolds are obtained by assigning a hyperbolicmetric to the interior of the unit circle in the complex plane and by consideringas identical the points congruent under a Fuchsian group.The question of the measure of the transitive geodesics on such manifolds

has also been treated but has not been completely solved. Using the theoryof continued fractions, Artin [2] proved that almost all geodesics are transitiveif the group is the modular group. Myrberg [3] proved that the same resultholds if the group is of the first kind (i.e., one which ceases to be properly dis-continuous on the unit circle U), has a finite set of generators and has a funda-mental region either lying, with its boundary, entirely inside U or having allits vertices on U. These results are included in the work of E. Hopf [4], whoshows that metrical transitivity holds if the group is of the first kind with afinite set of generators. Metrical transitivity implies that almost all the geo-desics are transitive. The case of an infinite set of generators has not beenconsidered.

Three-dimensional manifolds of constant negative curvature can be obtainedby assigning a hyperbolic metric

482 4 dx + dy + dz)

to the interior of the unit sphere S and considering as identical the pointscongruent to each other under suitable groups of the rigid motions of this space.The groups must be properly discontinuous within S but may or may not beproperly discontinuous on S. An example of such a manifold is the one obtainedby using the Picard group. Recently LSbell [5] gave examples of closed mani-folds of constant negative curvature.As to the properties of geodesics on these manifolds LSbell [5] states that,

by methods analogous to those in his proofs for two-dimensional manifolds [6],it can be shown that the periodic geodesics are everywhere dense among thetotality of geodesics and that there exist transitive geodesics on the manifoldswhich he has set up.

It is the object of this paper to prove that, for certain three-dimensional

Received September 29, 1937.The numbers in brackets refer to the bibliography at the end of the paper.

78

TRANSITIVE GEODESICS ON THREE-DIMENSIONAL MANIFOLDS

manifolds of constant negative curvature, almost all geodesics are transitive.For transitivity, as in the two-dimensional case, it is necessary to assume thatthe group defining the manifold is of the first kind; i.e., that the group is notproperly discontinuous in any domain on S. It will then be shown that underthis hypothesis the periodic geodesics are everywhere dense among the totalityof geodesics. The class of manifolds for which it will be proved that almostall geodesics are transitive is defined by means of a stability property. Hopf [7]has shown that for certain dynamical systems, including those considered inthis paper, almost all motions are either stable or unstableRstable in the sensethat the motion returns arbitrarily close to any one of the positions formerlyoccupied, unstable in the sense that it stays outside any finite domain after afinite length of time. It will be proved that, if almost all geodesics are stable,almost all of them are transitive. This includes the set of manifolds definedby groups of the first kind with a finite set of generators. Whether it includesmanifolds defined by groups of the first kind with an infinite set of generatorsis unknown.The method of proof used in obtaining these results is direct, entirely geo-

metrical and applies equally well to the case of two dimensions.

1. Three-dimensional hyperbolic geometry. This section gives a brief sum-mary of the known properties of three-dimensional hyperbolic space whichform the background for this paper. The proofs are omitted and referencesfor the material used are given.

Consider the Riemannian space of constant negative curvature / -1,given by the metric

(a) dsdx2 dy2 dz2

Z

defining a geometry in the upper half-space z > 0. The angle between twocurves is the ordinary Euclidean angle. The geodesics are circles orthogonalto the plane z 0 [Bianchi, 8, p. 584]. The geodesic surfaces are spheresorthogonal to the plane z 0.The rigid motions of the geometry defined must transform the plane z 0

into itself and also must transform two intersecting spheres orthogonal to z 0into two spheres orthogonal to z 0 and intersecting each other at the sameangle. Hence, the transformation thus determined on the points of z 0must be a conformal transformation. If we introduce into the plane z 0the complex variable x - iy, the desired transformation is the linearfractional transformation in this plane [Bianchi, 8, p. 586]:

r a’ bad-bc O.c + d’

(The class of anti-analytic transformations, a - b ad bc O, x iy

0 ANNITA TULLER

will not be considered in this paper.) From this the analytic form for the rigidmotions in the space itself is very easily developed [Bianchi, 8, pp. 587-588].It assumes the following form"

c,p -t- c3 + d W d(l

Therefore, corresponding to every rigid motion of the space there is a linearfractional transformation of the plane z 0 into itself, and conversely. Astudy of this geometry and of the invariant curves under the different types oflinear fractional transformations is given by Bianchi [8, pp. 588-589].We will, however, transform this hyperbolic space with the metric (a) into

the space with the metric

(b) ds

by transforming the space (x, y, z) into itself so that the upper half-space istransformed into the interior of the unit sphere. This is done by performingin succession the translation

x r sin t cos ,r’ 4 O’ O,r’

, where y rsin0sin,

z r cos 8,

and the translation

x’ x, y’ y, z z- 1.

Thus it will suffice for the purpose of this paper to give a brief description ofthe geometry defined by this new metric and the invariant curves under thelinear fractional transformations in the new hyperbolic space.The metric (b) defines a hyperbolic geometry in the interior of the unit

sphere S. The geodesics, or hyperbolic lines, are now arcs of circles orthogonalto S; the hyperbolic planes are parts of spheres orthogonal to S. The equi-distant surfaces, i.e., the locus of points at a given hyperbolic distance from agiven hyperbolic line, will be called hyperbolic cylinders. The hyperbolicspheres, i.e., the locus of points at a given distance from a given point, are alsoEuclidean spheres not necessarily with the same center. The rigid motions

x x, y y, z z-t- 2,

4(dx + dy + dz2)

the inversion


are given by the four types of linear fractional transformations. These trans-form circles and spheres into circles and spheres respectively and preserveangles.The hyperbolic transformation has two fixed points on the sphere. The

geodesic through these points is the axis of the transformation. The familyof fixed circles on S determines a family of fixed spheres through the two fixedpoints, orthogonal to S, and cutting S in the family of fixed circles. Theinvariant curves of the transformation are arcs of circles through the two fixedpoints. The transforms of a point P under the iterates of a particular hyper-bolic transformation have as limit point one of the fixed points of that trans-formation. The hyperbolic transformation whose axis is AB and which sendspoints toward B will be denoted by TAB. The hyperbolic transformationwhich sends points toward A will be denoted by TsA.The elliptic transformation also has two fixed points on S and an axis which

is the geodesic through these two points. The family of fixed circles on Sdetermines a family of non-intersecting fixed spheres orthogonal to S. Theaxis is fixed point for point. The invariant curves are circles of intersectionof the fixed spheres with the hyperbolic cylinders around the axis. This trans-formation may be thought of as a rotation about the axis. The transforms of apoint P under the iterates of a particular elliptic transformation form a discreteset of points on the invariant circle through P or are everywhere dense on thatcircle according as the angle of rotation is or is not commensurable with 2r.The loxodromic transformation is the product of a hyperbolic transformation

and an elliptic transformation. The invariant curves wind around the hyper-bolic cylinders and cut the circles through the fixed points at a constant anglenot equal to zero or v/2. The transforms of a point P under the iterates of agiven loxodromic transformation have as limit point one of the fixed pointsof that transformation. A notation similar to that used for hyperbolic trans-formations will be used to show which fixed point is being approached.The parabolic transformation has one fixed point on S and a family of fixed

spheres orthogonal to S, tangent to each other at the fixed point and cutting Sin the family of fixed circles. The horospheres, i.e., the Euclidean spheresinternally tangent to S at the fixed point, can also be shown to be fixed. Theinvariant curves are the circles of intersection of the fixed spheres with thehorospheres. The transforms of a point under the iterates of a given para-bolic transformation have as limit point the fixed point of the transformation.

2. Groups of linear fractional transformations. Let G be a group of linearfractional transformations,, a b

ad- bc 1.c d’

Since G defines an isomorphic group of motions of S and its interior into itself,the notation G can be used equally well for this second group. Two points

82 ANNITA TULLER

P and P in or on S are said to be congruent or equivalent if there is a transfor-mation T of G such that T(P) P’. Either point will be called a copy ofthe other.

DEFINITION. The group G will be called discrete if the unit element of Gis not a point of accumulation of other elements of G [van der Waerden, 9,p. 28]. By this we shall mean that there exists a real positive number a suchthat for all transformations of G, other than the identity,

Geometrically this means that given a sufficiently small positive e, there existsno transformation T in G, other than the identity, such that the Euclideandistance between P and T(P) is less than e for all P in or on S.A theorem on discrete groups, which will be used later, is the following:THEOREM 2.1. If there exists a positive number a such that, given any positive, there exists in G two elliptic transformations with fixed points AB and A’B’,

respectively, such that the distances AB and A’B’ are each greater than a, but thedistances AA and BB are each less than , then G is not discrete [F-K, 10, p. 98].

DEFINITION. The group G will be called properly discontinuous in a domainD if each point P of D possesses a neighborhood N(P) which has points incommon with only a finite number of copies of N(P), and if any two non-congruent points P and Q possess neighborhoods N(P) and N(Q) which aresuch that no point of N(P) is congruent to a point of N(Q).

DEFINITION. A fundamental region of a group G in a domain D is an openset having no points in common with any of its copies and which contains inits interior or on its boundary a point congruent to each point P in D [van derWaerden, 9, p. 35].

If a group is properly discontinuous in a domain D, it possesses a fundamentalregion in that domain [van der Waerden, 9, p. 35]. It has also been shown thata necessary and sufficient condition that a group of linear fractional transfor-mations be properly discontinuous in S is that it be discrete IF-K, 10, p. 98].

DEFINITION. A group of linear fractional transformations will be said tobe of the first kind if it is properly discontinuous inside S but not properlydiscontinuous in any domain on S.THEOREM 2.2. The set of transformations in a group of the first kind has fixed

points which are everywhere dense on S.Since the group is properly discontinuous in S, it is discrete and the Euclidean

distance between two congruent points, P and T(P), for generic P (i.e., for Pnot in the neighborhood of fixed points of T), is greater than a suitably chosenpositive number e. Since the group is not properly discontinuous in any domainon S, any neighborhood on S contains at least one pair of congruent points;i.e., we can find a point P and a transformation T such that the Euclideandistance between P and T(P) is less than e. However, it is geometricallyevident that this can happen only in the neighborhood of a fixed point. (See:F-K, 10, pp. 94-98, for detailed description of the linear fractional transforma-tions of a discrete group.)


The groups witl which this paper deals are groups of the first kind. Sucha group has a "normal" fundamental region which consists of a simply con-nected polyhedron bounded by hyperbolic planes, congruent in pairs undertransformations of the group. The transformations relating pairs of congruentfaces of the normal fundamental region generate the group. The normalfundamental region and its copies cover the interior of S. These and furtherproperties of properly discontinuous groups and their fundamental regions aredealt with at length in [F-K 10, pp. 94-150].

If points in S congruent under the transformations of a group G of the firstkind are considered identical, a three-dimensional manifold M is defined. Itis the behavior of the geodesics on this manifold that will be investigated.

3. Periodic geodesics.DEFINITION. A directed geodesic in S is periodic if it passes through two

congruent points with congruent directions.An immediate consequence of this is that the axis of a hyperbolic or loxo-

dromic transformation is periodic.It will be shown that, if G is of the first kind, the periodic geodesics are every-

where dense among the totality of geodesics. The following lemma must firstbe proved.LEM 3.1. Let 11 and I be two arbitrary circular regions on S having no

points in common and bounded by circles 01 and O respectively. If T is a linearfractional transformation such that T(01) 0. and such that the part of the surfaceof S exterior to I1 is transformed by T into the interior of I2, then T is a hyper-bolic or loxodromic transformation with one fixed point in I1 and one in I..

Since I1 and I. have no point in common, the part of the surface of S exteriorto I1 contains I, and hence T transforms a circular region on S into part ofitself. We shall show that any such transformation must have a fixed pointin I2.

If T is an elliptic transformation, then, no matter where the fixed points are,there are fixed circles either cutting one of the bounding circles and not theother or lying entirely in the region between them. Neither case is possibleunder the hypothesis of the lemma. Therefore T is not elliptic.

If T is parabolic, hyperbolic or loxodromic, the transforms of a point P1 on01 under the iterates of T have as limit point a fixed point of T. However,T(P1) P2 on 02. Therefore, further transforms of P1 under powers of Tare inside I2, and thus T has a fixed point inside I. A similar argumentapplied to T-1 gives us a fixed point inside I1. Hence T is either a hyperbolicor a loxodromic transformation of the type desired.THEOREm 3.1. If G is of the first kind and if I1 and I are arbitrary circular

neighborhoods on S with no point in common, there exists a hyperbolic or loxo-dromic transformation of G with one fixed point in I1 and the other in I.

It is convenient to divide the proof into three parts.CASE 1. The group G contains a parabolic transformation T but no elliptic

transformation.

84 ANNIT TULLER

It will first be shown that a hyperbolic or loxodromic transformation of Gcannot have a fixed point in common with a parabolic one. If Q is a fixedpoint of a hyperbolic or loxodromic transformation whose axis is AQ, and if Qis also the fixed point of a parabolic transformation T, then T(AQ) A’Qis the axis of a hyperbolic or loxodromic transformation. We now have twohyperbolic or loxodromic transformations with a common fixed point. This,however, will be shown to lead to a contradiction. By applying powers of T,to a point P on the axis A’Q we get congruent points arbitrarily close, in thehyperbolic sense, to the hyperbolic line AQ. We now construct a sphere S,orthogonal to S and to AQ. Let S T(S). By applying sufficientlyhigh powers of T, and by choosing a proper subset, we get an infinite sequenceof distinct points congruent to P and having as limit point a point on the partof AQ between S and S. However, this is impossible for a properly dis-continuous group.

It will now be shown that the fixed points of parabolic transformations areeverywhere dense on S. Let I be an arbitrary circular region on S. There isa fixed point A in it (Theorem 2.2). If the corresponding transformation isnot parabolic, then it is hyperbolic or loxodromic and its other fixed point Bis not Q. By applying sufficiently high powers of T,, we get Q’, congruentto Q, inside I. Since Q is the fixed point of a parabolic transformation and Iis any region on S, we have the parabolic fixed points everywhere dense on S.We come now to the theorem itself. In I there is a fixed point P of a

parabolic transformation. Let I be a circular region containing all of I butno part of I. Let O be the boundary of I. By applying sufficiently highpowers of Tv we get O transformed into 0’ entirely in I. The part of Sexterior to I will be transformed into a region I, bounded by 0, inside I.This can be seen by considering the behavior under the transformation of anyone point exterior to I. In a similar manner we get 0’ congruent to 0 in I.This time, however, the interior of I is transformed into a region I, boundedby 0, inside I.. We note now that the regions I and I satisfy the con-ditions of the lemma, and the theorem is proved for this case.CtsE 2. The group G contains only hyperbolic and loxodromic ransfor-

mations.Since the fixed points are everywhere dense on S (Theorem 2.2) I contains

a fixed point C of a hyperbolic or loxodromic transformation whose otherfixed point is D and I contains a fixed point C. of a hyperbolic or loxodromictransformation whose other fixed point is D. Let I be a circular neighbor-hood containing I but not I., and bounded by a circle O which passes throughneither D nor D. By applying sufficiently high powers of T, c we get acopy 0’ of O in I bounding a region I in I. Similarly, we get a copy 0’of O in I. bounding a region I in I. However, I’ and I may not satisfythe conditions of the lemma, since the interior of I may correspond to theinterior of I. We will show, however, that there exists a transformation ofthe group which takes the exterior of I into a region I’ inside I. Then I;and I’ will satisfy the conditions of the lemma.

TRANSITIVE GEODESICS ON THREE-DIIIENSIONAL IIANIFOLDS 5

Let A and B be the fixed points of a hyperbolic or loxodromic transformationT.. By taking a region I inside I containing neither A nor B and treatingit as we did I above we get copies A’ and B’ of A and B, respectively, inside I.This can be done since, as has been shown under Case 1, two hyperbolic orloxodromic transformations cannot have a common fixed point. By applyingsufficiently high powers of T,s, we get a copy O of O inside I, bounding aregion I in I such that the exterior of 1 corresponds to the interior of ICASE 3. The group G contains an elliptic transformation.In a manner similar to that of the first part of Case 1, it can be shown that

an elliptic transformation cannot have a fixed point in common with a hyper-bolic or loxodromic transformation unless the whole axis is common.We go on to show that any neighborhood on S contains a pair of fixed points

of an elliptic transformation. Let PQ be the axis of our known elliptic trans-formation. First, let us suppose that the only fixed points in I are elliptic.Then there must be a pair in I belonging to the same transformation. Other-wise, the correspondents of all the fixed points inside any subneighborhood ofI would have a cluster point outside or on the boundary of I, and this con-tradicts Theorem 2.1. Moreover, by applying the same argument to smallerneighborhoods inside I, it is possible to obtain a pair of elliptic fixed pointsarbitrarily close together.

If a fixed point Q’ inside I is parabolic, then by applying sufficiently highpowers of T, we get copies of P and Q inside I, again arbitrarily close. If afixed point inside I is hyperbolic, the other fixed point of the transformationis not P or Q and again we can get copies of P and Q inside I. The same istrue if the fixed point inside I is loxodromic.We come now to the theorem itself. Let 0 be a great circle separating I

and I. Let E be the midpoint of the elliptic axis PQ whose fixed pointsP and Q are in I. Let the plane through E and 0 perpendicular to the axisPQ cut Oa in points A and B, and the boundary of I in points C and D. ThenAB is a diameter of S. Let , t., 0, 0 be the angles in rotation about Ebetween the geodesic rays EC and EA, EA and EB, EB and ED, ED and ECrespectively.

Since we can get P and Q arbitrarily close, we can get E arbitrarily nearthe surface, and thus angles 0, ., and 0 can be made arbitrarily small, whileangle t can be made arbitrarily close to 2r. Therefore, no matter what therotation angle of our elliptic transformation is, we can get E close enough tothe surface of the sphere so that repeated application of the elliptic transforma-tion Te will take the circle O into a circle 0 in I. In a similar mannerwe get a copy 0 of O in I.

If we let I be the region of S bounded by 0a and containing I, the aboveprocess transforms the exterior of Ia into the interior of the region I in Ibounded by 0. Since 0a separates I and I, the interior of Ia will be trans-formed into the interior of the region I in I bounded by 0. I and Isatisfy the conditions of the lemma. This completes the proof.

86 ANNITA TULLER

4. Transitive horospheres.

where

Let E be the set of elements

p" (x, y, z; , 0),

Any such element determines a point P:(x, y, z) inside S and a direction d:(, t)at this point, where is the angle between d and a line z’ through P parallelto the z-axis, and t is the angle between the plane determined by z’d and theplane through P parallel to the (x, z)-plane. The elements will be brieflydesignated by p: (x; d). Any such element determines a directed geodesicsince there is one and only one geodesic through a point P inside S in a givendirection d. We define the distance between two elements pl: (xl; dl) andp.: (x; d.) as

where H(PIP) is the hyperbolic distance between the points P:(x) andP: (x), and a is the least positive angle between d and d..

DEFINITION. Two elements p: (x; dl) and p2: (x.; d) of E are congruentif there is a transformation of G taking P: (xl) into P.: (x.), and the directiond at P into the direction d at P.; i.e., taking P into P and the directedgeodesic determined by p into the directed geodesic determined by p.

DEFINITION. An element of E is periodic if it determines a periodic geodesic.It follows that all the elements on a periodic geodesic are periodic elements.THEOREM 4.1. If G is of the first kind, the periodic elements are everywhere

dense in E.Let N(p) be any neighborhood of any element p: (x; d) of E. The element

p determines a directed geodesic AB. Let I and I, be sufficiently small regionson S about A and B respectively, so that the set of geodesics having initialendpoint in I and terminal endpoint in I. goes through N(p). However, byTheorem 3.1, there is at least one periodic geodesic in that set and hence aperiodic element in N(p).

DEFINITION. A horosphere is a Euclidean sphere internally tangent to S.The point at infinity of a horosphere is its point of contact with S. A horo-

sphere is completely determined by its Euclidean radius and its point at infinity.The horosphere with Euclidean radius r and point at infinity Q will be denotedby h(Q, r). The elements of E on the horosphere will be taken as its pointsinside S with a direction at each point normal to the horosphere and directedoutward. It is clear that an element of E determines a unique horosphere,and that a point A in S and a point B on S determine a unique horospherethrough A with point at infinity at B.We may also note that two horospheres h(Q, rl) and h(Q, r) with Q as point

at infinity cut off equal hyperbolic lengths on the set of geodesics through Q.For, let QAA.R and QBB2S be any two geodesics through Q. The points


A1 and B1 are on h(Q, r), and As and B2 on h(Q, r2). We can find a parabolictransformation TQ such that TQ(A) B. Then T(QR) QS. Since thehorospheres are fixed under T, T(A) B. Hence the hyperbolic dis-tances AA. and BIB are equal.

DEFINITION. A horosphere h(Q, r) is transitive if the totality of elementson it and on all its copies form a set everywhere dense in E.The remainder of this section is devoted to several theorems on the transi-

tivity of horospheres. Where the proofs are completely analogous to thosefor horocycles in the two-dimensionM hyperbolic space, the exact reference isgiven at the end of the statement of the theorem. The proofs are omitted.

DEFINITION. A set of horospheres is transitive if the totality of elementson all copies of all members of the set is everywhere dense in E.]EMMA 4.2. Let P be a point inside S and I a circular domain on S. Let I’

be a circular domain of I, and h(P, I’) the set of horospheres through P with pointsat infinity in I’. If h(P, I’) is transitive for every circular domain I’ of I, thenthere exists an infinite set of transitive horospheres through P with points at infinityin I [Hedlund, 11, Lemma 1.1, p. 532].THEOREM 4.2. If G is of the first ind and if P is an arbitrary point in S

and I an arbitrary circular region on S, there exists a point Q in I such that h(P, Q)is transitive.The theorem will follow from the lemma if we show that the set h(P, I) is

transitive. Let AB be the axis of a hyperbolic or loxodromic transformationof G, and hence a periodic geodesic. It has a copy A’B’ with endpoints in I.Let P0 be a point on AB. It has a copy P on A’B’. By repeated applicationof TA,B, we get copies P, P, of P0 approaching B’. Consider a set ofhorospheres, all passing through P and one through each P’n. As n becomesinfinite, the set of horospheres through P and P’ will have points at infinityQ approaching B’. The angle at which the horospheres cut A’B’ will approachr/2. Therefore, if pn: (x; d) is the element on A’B’ at the point of inter-section with the horosphere through P and P’ and q" (x; d’) is the elementon the horosphere at the point of intersection with A’B’, then, as n becomesinfinite, the angle between d and d: approaches zero, and hence ]] q p 1]approaches zero. Since p0 is any periodic element, and since the periodicelements are everywhere dense in E, the set h(P, I) is transitive.THEOREM 4.3. If G is of the first kind, there exists an infinite set of transitive

horospheres through any point in S.The points at infinity of these form a set everywhere dense on S. This

theorem is an immediate consequence of the preceding one.THEOREM 4.4. If one horosphere with Q as point at infinity is transitive, all

the horospheres with Q as point at infinity are transitive [Hedlund, 11, Theorem 2.1,p. 535].

DEFINITION. A point Q of S will be called h-transitive if all the horosphereswith Q as point at infinity are transitive.THEOREM 4.5. If G is of the first kind, the endpoints of all axes of hyperbolic

88 ANNITA TULLER

or loxodromic transformations of G are h-transitive [Hedlund, 11, Theorem 2.2,

THEOREM 4.6. If G is of the first kind and there are copies of the horosphereh(Q, r) with radii arbitrarily dose to 1, Q is h-transitive [Hedlund, 11, Theorem 2.3,

THEOREM 4.7. If G is of the first kind, Q a point on S, and if there exists onOQ a sequence of points 01,03, such that the limit as n becomes infinite ofthe hyperbolic distance 00, is and such that O has a copy Or, with hyper-bolic distance 00 bounded, then Q is h-transitive [Hedlund, 11, Theorem 2.4,p. 537].THEOREM 4.8. If G is of the first tcind with a fundamental region Ro which

together with its boundary lies entirely inside S, then all points on S are h-transitive.This is an immediate consequence of the preceding theorem.

5. The measure of transitive geodesics.DEFINITION. A directed geodesic is transitive if the totality of elements on

it and on all its copies form a set everywhere dense in E.DEFINITION. An element of E will be called transitive if it determines a

transitive geodesic.With the aid of the preceding theory on transitive horospheres we derive

some theorems concerning the measure of the transitive geodesics on the mani-fold M, defined by considering as identical the points in S congruent underthe transformations of a group G of the first kind. Measure in E is Lebesguemeasure but the space will be considered as a Euclidean space.THEOREM 5.1. The transitive elements of E form a measurable set.Let el, e., be a sequence of positive numbers such that lim e 0. Let

E be covered by a finite set of open cells such that the Euclidean diameter ofeach cell is less than el and such that every cell contains some of E. Let 01be the set of elements of E, each of which determines a geodesic such that everycell contains an element of it or of one of its copies. The set 01 is open; for,if pl is an element of 01 and p is an element in a sufficiently small neighborhoodof pl, the geodesic g determined by p. will stay close enough to that determinedby pl so that every one of our open cells will contain an element of g or of oneof its copies. Let E be now covered by a finite set of open cells each of diam-eter less than and such that every cell contains some of E; and let 0. be theset of elements, each of which determines a geodesic such that every one ofthese new cells contains an element of that geodesic or one of its copies. Con-tinuing in this manner we get sets 01,03,08, An element is transitiveif and only if it belongs to all of these sets. Hence the set of transitive elements

is II o which is measurable since each 0 is open and therefore measurable.nl

THEOREM 5.2. If G is of the first kind with a fundamental region Ro which,together with its boundary, lies entirely within S, any measurable set W in E,where W is of positive measure, determines a transitive set of geodesics.

TRANSITIVE GEODESICS ON THREE-DIMENSIONAL MANIFOLDS 89

We shall consider first a special set O, the set determining a solid cone ofgeodesic rays with vertex at the point P in S of element p" (x; ) and withgenerators determined by the cone of directions through P making an angleless than or equal to 4 with 3. We shall show that the totality of elementson all the geodesic rays determined by is a transitive set.On each ray of the cone consider the element (x8 ds) of E whose point (x)

in S is at a hyperbolic distance s from P. The totality of such points (xs)is on a spherical zone a, of hyperbolic area As. The elements (x ds) havedirections normal to as. As s becomes infinite the hyperbolic radius of thezone becomes infinite. Let s become infinite through a sequence of valuessl < s. < and consider the elements (x 3,) thus determined on thecentral geodesic ray go. Each such element has a copy (x,*; ds) in R0. Sincewe can pick a subsequence of these copies with a limiting element, we shallassume that this is already the subsequence with unique limiting element(x*; 3*) in R0. This element determines a horosphere h(Q, r) which is transi-tive (Theorem 4.8). Let T be the transformation of G which carries (xs 3)

-* and let ao be T(a,). As n becomes infinite a is a sequenceinto (x*; d.),of spherical zones whose hyperbolic radii become infinite. Each as carriesan element e, at its center such that the lim * *e,. is (x*; 3*) and such that a.

approximates more and more of the transitive horosphere h(Q, r). Hence the,limiting position of a is the transitive horosphere h(Q, r). Now let e be anyelement of E, and let N(e) be a neighborhood of this element. Since h(Q, r)is transitive, there exists an element e on h(Q, r) which has a copy in N(e).Let N(e) be a neighborhood of e so small that each element in N(e) has acopy in N(e). If we let On denote the set (x, d,), where the points (xs)are on the spherical zone a,, there exists an N such that for n greater than Nthe set has an element with a copy in N(e) and hence in N(e). Thus thetotality of elements on all the geodesic rays determined by is a transitive set.We now return to the arbitrary set W in E, where W is of positive five-

dimensional measure. We take sections of it by planes determined by fixingparticular (x, y, z). There exists on at least one of these planes r a sectionof W of positive plane measure [Hobson, 12, p. 185], and there is an elementp" (x; 3) of this section at which the two-dimensional metrical density is 1[Hobson, 12, p. 179]. This element determines a geodesic ray gr, and letting sbecome infinite through a sequence of values sl < s. < we again considerthe elements (x d,) on g,. Let C be the hyperbolic sphere with center xand radius s. Given a positive constant A, let a be a spherical zone ofhyperbolic area A, on C, and with center at x,. The set of elements atthe point x and determining rays passing through am forms a circular regionin r with center at p" (x; 3). As n becomes infinite, the radius of the set eapproaches zero, and we have

(1) limm(. W)

1

90 ANNITA TULLER

where the measures are two-dimensional measures. (The cells used in thedefinition of metrical density are squares; but it is easily seen that if the metricaldensity is 1 when squares are employed, the same must be true when circlesare used.)

Let . be the elements of the geodesic rays, with initial point x and passingthrough as, at the points where they intersect as. Let W be the subset of

obtained when only rays determined by the set .W re considered, ndlet w be the set of points on a bearing the elements W. The set w is ameasurable set. We wish to show that

limm(w) 1

where the meure here is in terms of hyperbolic re on a.It cn be ssumed that p:(x; 3) is such that x is at the center O of S, and 3

is such that its is neither 0 nor v. For if the point x is transformed to Oby a rigid motion of the hyperbolic space, ngles re preserved nd hyperbolicareas are preserved. The transformation can evidently be chosen in infinitelymny wys such that the condition on direction is fulfilled. Then for ech n,A is the hyperbolic re of zone of sphere whose equation in spherical co-ordinates is

x rsincos0,

y rsinCsin0,

z rn cos ,where 0 r 1. The hyperbolic metric on this sphere becomes

r (sin dO + d),d#(I r)

and we have

while

m(w) r. (1 r)sin dO d,

A(1 - r)

sin g, dO de,

where both integrals are Lebesgue integrals. Thus

b desin dOm( n)A

sin


f f dO dem(,. W)

Letting M1 and ml be the maximum and minimum, respectively, of sin in,,. W, and M. and m. the maximum and minimum, respeetively, of sin inn, we have

As n becomes infinite, approaches a constant different from 0 or r; namely,the of the element p’(x; 3); and the first and third ratios become equal.Hence,

limm(.W)

limm(w)

-. ra(,) -.. A

From (i),

(2) limm(w,)

1.. A

Again let e be any element of E and let N(e) be a neighborhood of this ele-ment. If we use the notation of the first case considered under this theorem,we see that there exists an element e on h(Q, r) which has a copy in N(e).Let N(eh) be a neighborhood of e so small that each element in N(e) has acopy in N(e). Let A be chosen so that the zone a of h(Q, r) with center at x*and hyperbolic area A contains the point bearing e. The sequence of zonesT,,(a) approaches the zone a uniformly, and for n sufficiently large one of theelements T,,(W,) will lie in N(eh). For if this were not the case, there wouldexist a > 0 and an N such that for n > N the zone T(a) would contain adomain of hyperbolic area greater than and containing no point of the setT(w). The same would hold with respect to the zone a and the set w,and thus

m(w.)lim < 1----,--, m(a,O A"

This contradicts (2). Thus the totality of elements on all the geodesic raysdetermined by W is a transitive set.THEOREM 5.3. If G is of the first kind with fundamental region Ro which,

92 ANNITA TULLER

together with its boundary, lies entirely within S, then almost all elements of E aretransitive.

This theorem will be proved by showing that the complement C(_. On)of the/

set II o of Theorem 5.1 is a zero set. Since C 0 is equal to C(01) + C(02)nl

it will suffice to show that C(O) has zero measure. Let the set ofopen sets in the mesh defined by e be denoted by 0, 0, 0. Let Ebe the set of elements each of which determines a geodesic such that 0 con-tains no element of that geodesic or of any of its copies. Then C(O) EE + E.. The set E is measurable since its complement is open

and therefore meurable. By Theorem 5.2 each E is a zero set. ThereforeC(O) is a zero set. In a similar manner we show that C(O) is a zero set for

i. Hence C(0,)is ofevery measure zero.

The proof of Theorem 5.2, and hence that of Theorem 5.3, depends on thegroup’s being such that all points on are h-transitive. However, we may havegroups of the first kind where all points on are not necessarily h4ransitive.For example, if is a fixed point of two parabolic transformations of the group,not both in the same cyclic subgroup (e.g., the Picard group), the fundamentalregion on a horosphere with as point at infinity, for the subgroup generatedby these two transformations, is of finite hyperbolic area. Hence, the horo-sphere cannot be transitive. We wish, therefore, to extend Theorem 5.3 toinclude such groups of the first kind.

Let u be the space obtained from by considering congruent elementsidentical. This is then the phase space associated with the manifold M. Tothe uniform motion along the geodesics on M there corresponds a steady flowon . The element of five-dimensional volume dm ddd, d being thethree-dimensional hyperbolic element of volume, is invariant under the flow[ef. Hopf, 4, pp. 300-304].DFTO. An element p of u will be called abl [Hopf, 7, "flie-

hende"] if the moion along the geodesic ray determined by i ultimately passesout of and stays out of any finite neighborhood in u once occupied. Thegeodesic ray determined by an unstable element will be called aneodec ray.DEFO. An element p of will be called stable [Hopf, 7, "wiederkehr";

semi-sable in the sense of Poisson] if the motion along the geodesic ray deter-mined by i returns infinitely often to an arbitrarily small neighborhood ofThe geodesic ray deermined by a stable element will be called a stable eo-desic ray.THOR 5.4. I o th rtd ad th nsabl6 elements orm a

o measure ero, almost all eodecs areEach element in determines an infinite set of congruent elements of

f the element of is stable, each of the corresponding elements (x; d) of


will determine a geodesic ray which has on it an infinite sequence of elements(x; d) with lim s and such that each (x; d,) has a copy (x*; d*)with lim(x*; d,*) (x; d). Therefore the geodesic ray ends in a point on S

which is h-transitive (Theorem 4.7). It has been shown that, under the hypoth-esis of this theorem, almost all the elements of E are stable [Hopf, 7, The-orem 1, p. 712]. Then almost all the elements of E are correspondents ofstable elements of E and thus have the property described above.The proof of this theorem now follows closely that of Theorem 5.2 with the

following modifications. In choosing the element p: (x; 3) which is to determinethe central geodesic ray g we choose one which is stable and such that theelement with the same P: (x) but direction opposite to 3 is also stable. Thisexcludes only a set of measure zero; for, since almost all elements of EM arestable, then, of those elements p: (x; d) which are stable, only a zero set canbe such that the elements with the same P: (x) but direction opposite to d arenot stable. This assures us that the limiting element (x*; d*) is inside S (it is pitself) and that the unique horosphere determined by it is transitive. We dothe same in choosing the element p:(x; 3) at which the metrical density is 1.This again excludes only a set of measure zero [Hobson, 12, p. 179]. The restof this proof and that of Theorem 5.3 go through without change and thusTheorem 5.4 is proved.THEOREM 5.5. If G is a group of the first kind with a finite number of gen-

erators, all points on S, with the exception of those which are fixed points of morethan one cyclic parabolic subgroup of G, are h-transitive.A group with a finite number of generators can have only a finite number of

points Q (i 1, ..., m) of the boundary of the fundamental region R0 on S.Each of the points Q is a fixed point of more than one cyclic parabolic sub-group of the given group G IF-K, 10, pp. 125-126] and hence is not h-transitive.If we let Q be the set of points Q (i 1, ..., m) and their copies, this theoremwill prove that all points on S except those of set Q are h-transitive.

If the radii r (i 1, m) of the horospheres h(Q, r) are chosen suffi-ciently near 1, it is geometrically evident that any point of R0 or its boundaryand interior to S will be interior to some one of the set h(Q, r) (i 1, m).If C denotes the set h(Q, r) and all copies of these, any point inside S is in-terior to some member of C.

There exist at least two parabolic transformations of G with fixed point Q.Hence each point of h(Q, r) with the exception of Q has a copy within hyper-bolic distance D of the origin O, where D depends on h(Q, r) and not onthe chosen point on it. If D is a constant denoting the largest of the D, anypoint of the set C which is not on S has a copy within hyperbolic distanceDofO.Now let P be any point on S, not belonging to the set Q. If P is any point

on the ray OP, it lies interior to one of the horospheres of C. But PP cannotlie entirely in any one member of C, for this would imply that P belonged to

94 ANNITA TULLER

the set Q. Hence P’P must intersect one of these horospheres and has on ita point P" with a copy within hyperbolic distance D of O. From Theorem 4.7,P is h-transitive.THEOREM 5.6. If G is a group of the first kind with a finite number of gen-

erators, the unstable elements of EM form a set of measure zero.From Theorem 5.5 all points on S not belonging to set Q are h-transitive

and hence cannot be endpoints of unstable geodesics. Therefore, the unstableelements of EM determine geodesics in S ending in the points of set Q. Theseendpoints form a denumerable set on S. We may have an infinite numberof geodesic rays going out to each point of Q. However, almost all unstablegeodesic rays are also unstable when extended backward through the initialpoint [Hopf, 7, Theorem 2, p. 712]. Hence both ends of almost all geodesicsdetermined by unstable elements must belong to the set Q. Therefore, theunstable elements form a set of measure zero.An immediate consequence of Theorem 5.4 and Theorem 5.6 is the following"THEOREM 5.7. If G is of the first kind with a finite number of generators,

almost all geodesics are transitive.

BIBLIOGRAPHY

1. P. KOEBE, Riemannsche Mannigfaltigkeiten und nichteuklidische Raumformen, fiinfteMitteilung, Sitzungsberichte der Preussischen Akademie der Wissenschaften (1930),pp. 304-364.

2. E. ARTIN, Ein mechanisches System mit quasiergodischen Bahnen, Abhandlungen ausdem Mathematischen Seminar der Hamburgischen Universitt (1924), pp. 170-175.

3. P. J. /YRBERG, Ein Approximationssatzfir die fuchsschen Gruppen, Acta Mathematica,vol. 57 (1931), pp. 389-409.

4. E. HoPF, Fuchsian groups and the ergodic theory, Transactions of the AmericanMathematical Society, vol. 39 (1936), p. 299.

5. F. LSBELL, Beispiele geschlossener dreidimensionaler Clifford-Kleinscher Rdume nega-tiver Krimmung, Berichten der mathematisch-physischen Klasse der SchsischenAkademie der Wissenschaften zu Leipzig (1931), pp. 167-174.

6. F. LSBELL, Die iiberall regul(iren unbegrenzten Fldchen fester Kriimmung, Inaugural-Dissertation, Eberhard-Karls-Universitt zu Tiibingen (1927).

7. E. HOPF, Zwei Sdtze iber den Wahrscheinlichen Verlauf der Bewegungen dynamischerSysteme, M:athematische Annalen (1930), pp. 710-716.

8. L. BIANCHI, Vorlesungen iber Differentialgeometrie, Leipzig, 1899.9. B. L. VAN DER WAERDEN, Gruppen yon Linearen Transformationen, Ergebnisse der

Mathematik und ihrer Grenzgebiete, vol. 4 (1935).10. P. FRICKE AND F. KLEIN, Vorlesungen iber die Theorie der Automorphen Functionen,

vol. 1, Leipzig, 1897.11. G. A. HEDLUND, Fuchsian groups and transitive horocycles, this Journal, vol. 2 (1936),

pp. 530-542.12. E. W. HoBsoN, The Theory of Functions of a Real Variable and the Theory of Fourier’s

Series, vol. 1, Cambridge, 1921.

BRYN MAWR COLLEGE.

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The measure of transitive geodesics on certain three-dimensional manifolds