gladish 2009 - derham cohology essentials

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de Rham cohomology essentials


Carl Gladish

CIMS

October 10, 2008
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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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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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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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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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Outline of Talk

Physical Examples

Contents

Outline of TalkHistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus

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Physical Examples

Solar energy flux

Let M = {(x, y, z) R3|x2 + y2 + z2 R2sun}.
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Physical Examples

Solar energy flux


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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Physical Examples

Solar energy flux


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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Physical Examples

Picture of solar energy flux

d Rh h l i l
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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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Physical Examples


Stokes theorem hints that f is the curl of some vector potential A.

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Physical Examples


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 ...

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Physical Examples

Irrotational 2-d fluid

Let M = {(x, y) R2|x2 + y2 1}.

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Physical Examples

Irrotational 2-d fluid

Let M = {(x, y) R2|x2 + y2 1}.Consider the flow depicted here:

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Physical Examples

To get the circulation of V along a path M we must integratethe 1-form

y

x2 + y2dx +

x

x2 + y2dy

along .

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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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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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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 ...

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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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Physical Examples


v2 = v + d(xy/2)

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Physical Examples


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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Idea of De Rham cohomology

Contents

Outline of Talk

HistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus



O l f lk
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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.


O li f T lk
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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 }


O tli e f T lk
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De Rhams Theorem shows that

Hkde Rham(M)= Linear(Hk(M),R)

where Hk(M) is the k-dimensional singular homology.


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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.


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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 ...


Outline of Talk
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Exterior Calculus

Contents

Outline of Talk

HistoryPhysical ExamplesIdea of De Rham cohomologyExterior Calculus



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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.


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.


Outline of Talk
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Exterior Calculus

Here is a sketch of what d : k(M) k+1(M) is:


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Exterior Calculus


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 ...


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Exterior Calculus


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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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.


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)


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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)


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E terior Calc l s
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Exterior Calculus



If p(M) and q(M) then

d( ) = d + (1)p d


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Exterior Calculus


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Exterior Calculus




d( ) = d + (1)p d

d2 = 0


Outline of Talk

Exterior Calculus
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Exterior Calculus




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


Outline of Talk

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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


Outline of Talk

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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.


Outline of Talk

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M

d =M

.


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 .


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M

d =M

.


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



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M

d =M

.


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 ...


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This pairing immediately gives us a map

Hkde Rham(M) Hom(Hk(M),R).


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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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gladish 2009 - derham cohology essentials

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