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de Rham cohomology essentials
de Rham cohomology essentials
Carl Gladish
CIMS
October 10, 2008
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de Rham cohomology essentials
Outline of Talk
History
Contents
Outline of TalkHistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus
Generalized Stokes Theorem and De Rhams Theorem
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de Rham cohomology essentials
Outline of Talk
History
(1899) First general proof of Stokes theorem by HenriPoincare
(1869-1951) Exterior calculus on manifolds is due to ElieCartan
(1903 - 1990) Georges de Rham gave a rigorous proof of DeRhams Theorem in 1931
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de Rham cohomology essentials
Outline of Talk
History
In differential geometry there are two ways to learn things:
Operationally: ie Calculate like this
d(cos(xy)dx) = xsin(xy)dy dx
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de Rham cohomology essentials
Outline of Talk
History
In differential geometry there are two ways to learn things:
Operationally: ie Calculate like this
d(cos(xy)dx) = xsin(xy)dy dx
Rigorous Definitions: ie A cotangent vector is an equivalence
class of germs of C
-functions ...
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Contents
Outline of TalkHistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus
Generalized Stokes Theorem and De Rhams Theorem
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Solar energy flux
Let M = {(x, y, z) R3|x2 + y2 + z2 R2sun}.
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Solar energy flux
Let M = {(x, y, z) R3|x2 + y2 + z2 R2sun}.
The flux of energy from the sun through any surface S M isdetermined by integrating the following 2-form over S:
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Solar energy flux
Let M = {(x, y, z) R3|x2 + y2 + z2 R2sun}.
The flux of energy from the sun through any surface S M isdetermined by integrating the following 2-form over S:
f =E
4
x(x2 + y2 + z2)3/2
dy dz
+y
(x2 + y2 + z2)3/2dz dx
+z
(x2 + y2 + z2)3/2dx dy
d Rh h l l
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Picture of solar energy flux
d Rh h l i l
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de Rham cohomology essentials
Outline of Talk
Physical Examples
This flux is divergence-free (think physically!) everywhere in M soperturbing the surface S by a little (keeping boundary fixed)doesnt change the total flux through it.
de Rh h l esse ti ls
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de Rham cohomology essentials
Outline of Talk
Physical Examples
This flux is divergence-free (think physically!) everywhere in M soperturbing the surface S by a little (keeping boundary fixed)doesnt change the total flux through it.
Stokes theorem hints that f is the curl of some vector potential A.
de Rham cohomology essentials
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de Rham cohomology essentials
Outline of Talk
Physical Examples
This flux is divergence-free (think physically!) everywhere in M soperturbing the surface S by a little (keeping boundary fixed)doesnt change the total flux through it.
Stokes theorem hints that f is the curl of some vector potential A.Indeed, in any small open ball U, this flux is the curl of somevector potential AU:
f|U= AU
But we can not patch together the local potentials AU to get aglobal potential because ...
de Rham cohomology essentials
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Irrotational 2-d fluid
Let M = {(x, y) R2|x2 + y2 1}.
de Rham cohomology essentials
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de Rham cohomology essentials
Outline of Talk
Physical Examples
Irrotational 2-d fluid
Let M = {(x, y) R2|x2 + y2 1}.Consider the flow depicted here:
de Rham cohomology essentials
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de Rham cohomology essentials
Outline of Talk
Physical Examples
To get the circulation of V along a path M we must integratethe 1-form
y
x2 + y2dx +
x
x2 + y2dy
along .
de Rham cohomology essentials
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gy
Outline of Talk
Physical Examples
To get the circulation of V along a path M we must integratethe 1-form
y
x2 + y2dx +
x
x2 + y2dy
along . Greens theorem says that the circulation along a loop L is
C =
L
v dr =
U
x(
x
x2 + y2)
y(
y
x2 + y2)dx dy
=
U
0 dx dy
= 0
if L bounds a disk-like region U.
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Outline of Talk
Physical Examples
This means: If we just consider a disk-like neighbourhood N then
v is given by a gradient N in that neighbourhood.
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Outline of Talk
Physical Examples
This means: If we just consider a disk-like neighbourhood N then
v is given by a gradient N in that neighbourhood.
But this fails globally because ...
de Rham cohomology essentials
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Outline of Talk
Physical Examples
Notice that we can easily make up another velocity field that iscurl-free but doesnt have a potential:
v2 = v + d(xy/2)
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Outline of Talk
Physical Examples
Notice that we can easily make up another velocity field that iscurl-free but doesnt have a potential:
v2 = v + d(xy/2)
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Outline of Talk
Physical Examples
Notice that we can easily make up another velocity field that iscurl-free but doesnt have a potential:
v2 = v + d(xy/2)
Question: what other curl-free flows on M lack a potential?De Rhams answer: essentially v is the only one
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Outline of Talk
Idea of De Rham cohomology
Contents
Outline of Talk
HistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus
Generalized Stokes Theorem and De Rhams Theorem
de Rham cohomology essentials
O l f lk
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Outline of Talk
Idea of De Rham cohomology
The De Rham cohomology of a space M identifies the differentialforms on M where:
is closed (ie d = 0), which is the same as saying that has an anti-derivative in any disk-like neighbourhood
= d for any ( is not exact, it has no globalanti-derivative).
The derivative we are talking about here is the exterior derivative.For now, think of it as a generalization of div, grad and curl.
de Rham cohomology essentials
O li f T lk
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Outline of Talk
Idea of De Rham cohomology
The De Rham cohomology of a space M identifies the differentialforms on M where:
is closed (ie d = 0), which is the same as saying that has an anti-derivative in any disk-like neighbourhood
= d for any ( is not exact, it has no globalanti-derivative).
The derivative we are talking about here is the exterior derivative.For now, think of it as a generalization of div, grad and curl.Specifically,
Hkde Rham(M) ={ k(M)|d = 0}
{ k(M)| = d for some }
de Rham cohomology essentials
O tli e f T lk
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Outline of Talk
Idea of De Rham cohomology
De Rhams Theorem shows that
Hkde Rham(M)= Linear(Hk(M),R)
where Hk(M) is the k-dimensional singular homology.
de Rham cohomology essentials
Outline of Talk
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Outline of Talk
Idea of De Rham cohomology
De Rhams Theorem shows that
Hkde Rham(M)= Linear(Hk(M),R)
where Hk(M) is the k-dimensional singular homology.The upshot is that,modulo the addition of exact forms d, thenumber of linearly independent k-forms on M that are closed butnot exact is exactly the number of independent k-dimensionalholes in the space M.
de Rham cohomology essentials
Outline of Talk
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Outline of Talk
Idea of De Rham cohomology
De Rhams Theorem shows that
Hkde Rham(M)= Linear(Hk(M),R)
where Hk(M) is the k-dimensional singular homology.The upshot is that,modulo the addition of exact forms d, thenumber of linearly independent k-forms on M that are closed butnot exact is exactly the number of independent k-dimensionalholes in the space M.Looking back at our two examples ...
de Rham cohomology essentials
Outline of Talk
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Outline of Talk
Exterior Calculus
Contents
Outline of Talk
HistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus
Generalized Stokes Theorem and De Rhams Theorem
de Rham cohomology essentials
Outline of Talk
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Outline of Talk
Exterior Calculus
Exterior calculus (due to Cartan) on a manifold M deals with thealgebra of differential forms on M
(M) = 0(M) 1(M) . . . n(M).
The exterior derivative is the mysterious map
d : k(M) k+1(M)
that has appeared already as a generalization of div, grad and curl.
de Rham cohomology essentials
Outline of Talk
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Exterior Calculus
Exterior calculus (due to Cartan) on a manifold M deals with thealgebra of differential forms on M
(M) = 0(M) 1(M) . . . n(M).
The exterior derivative is the mysterious map
d : k(M) k+1(M)
that has appeared already as a generalization of div, grad and curl.
de Rham cohomology essentials
Outline of Talk
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Exterior Calculus
Here is a sketch of what d : k(M) k+1(M) is:
de Rham cohomology essentials
Outline of Talk
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Exterior Calculus
Here is a sketch of what d : k(M) k+1(M) is:
By definition, 0(M) is just the smooth functions on M.
If f 0(M) then df is a dual-vector field on M.In particular ...
de Rham cohomology essentials
Outline of Talk
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Exterior Calculus
Here is a sketch of what d : k(M) k+1(M) is:
By definition, 0(M) is just the smooth functions on M.
If f 0(M) then df is a dual-vector field on M.In particular ...If vp is a tangent vector at p represented by the path where(0) = p then
dfp(v) =
d
dtt=0f((t))
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Outline of Talk
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Exterior Calculus
In general, a differential k-form k(M) is something which, foreach p M, provides a multi-linear, alternating function
TpM TpM . . . TpM R.
Intuitively, this provides a way to measure the volume ofinfinitesimal k-dimensional boxes with one corner at p.
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Outline of Talk
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Exterior Calculus
In general, a differential k-form k(M) is something which, foreach p M, provides a multi-linear, alternating function
TpM TpM . . . TpM R.
Intuitively, this provides a way to measure the volume ofinfinitesimal k-dimensional boxes with one corner at p.
A bit of pure algebra shows that for any (finite-dimensional) vectorspace V there is an isomorphism
: V V . . . V {alt. k-linear forms on V}
where
(v1 v2 . . . v
k)(w1, w2, . . . , wk) =
Sk
(1)sign()v(1)(w1)v(2)(w2) . . . v
(k)(wk)
de Rham cohomology essentials
Outline of Talk
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Exterior Calculus
Therefore, given local coordinate functions x1, x2, . . . , xn definednear p, any k-form can be written as
=
1i1
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Exterior Calculus
Therefore, given local coordinate functions x1, x2, . . . , xn definednear p, any k-form can be written as
=
1i1
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Exterior Calculus
Formal properties of d
If f is a smooth function, df is defined as stated above(function on tangent vectors)
de Rham cohomology essentials
Outline of Talk
E terior Calc l s
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Exterior Calculus
Formal properties of d
If f is a smooth function, df is defined as stated above(function on tangent vectors)
If p(M) and q(M) then
d( ) = d + (1)p d
de Rham cohomology essentials
Outline of Talk
Exterior Calculus
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Exterior Calculus
Formal properties of d
If f is a smooth function, df is defined as stated above(function on tangent vectors)
If p(M) and q(M) then
d( ) = d + (1)p d
d2 = 0
de Rham cohomology essentials
Outline of Talk
Exterior Calculus
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Exterior Calculus
Formal properties of d
If f is a smooth function, df is defined as stated above(function on tangent vectors)
If p(M) and q(M) then
d( ) = d + (1)p d
d2 = 0
If U M is open, then
dU
= d(U
)
If : M N is smooth map between manifolds, then
d = d
de Rham cohomology essentials
Outline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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Generalized Stokes Theorem and De Rham s Theorem
Contents
Outline of Talk
HistoryPhysical ExamplesIdea of De Rham cohomologyExterior CalculusGeneralized Stokes Theorem and De Rhams Theorem
de Rham cohomology essentials
Outline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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Ge e a ed Sto es eo e a d e a s eo e
General Stokes Theorem (first proved by Poincare 1899)
M
d =M
.
Proof is quite straightforward but requires careful use of technicaldefinitions.
de Rham cohomology essentials
Outline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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General Stokes Theorem (first proved by Poincare 1899)
M
d =M
.
Proof is quite straightforward but requires careful use of technicaldefinitions.
The important part is that we see that, for the bilinear pairing ofk-forms and k-dimensional submanifolds, d is the adjoint of theboundary operator .
de Rham cohomology essentials
Outline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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General Stokes Theorem (first proved by Poincare 1899)
M
d =M
.
Proof is quite straightforward but requires careful use of technicaldefinitions.
The important part is that we see that, for the bilinear pairing ofk-forms and k-dimensional submanifolds, d is the adjoint of theboundary operator . In fact, Stokes theorem allows us to define abilinear pairing
Hkde Rham(M) Hk(M) R
where
([], [])
.
de Rham cohomology essentialsOutline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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General Stokes Theorem (first proved by Poincare 1899)
M
d =M
.
Proof is quite straightforward but requires careful use of technicaldefinitions.
The important part is that we see that, for the bilinear pairing ofk-forms and k-dimensional submanifolds, d is the adjoint of theboundary operator . In fact, Stokes theorem allows us to define abilinear pairing
Hkde Rham(M) Hk(M) R
where
([], [])
.
Check well-definedness ...
de Rham cohomology essentialsOutline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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This pairing immediately gives us a map
Hkde Rham(M) Hom(Hk(M),R).
de Rham cohomology essentialsOutline of Talk
Generalized Stokes Theorem and De Rhams Theorem
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This pairing immediately gives us a map
Hkde Rham(M) Hom(Hk(M),R).
De Rhams theorem shows that this map is actually anisomorphism. If Hk(M) is finite-dimensional, then
dim(Hkde Rham(M)) = dim(Hk(M))
= number of k dimensional holes.
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