Exterior Algebra¶
Let be a real vector space of dimension .
Definition, Alternating algebraic forms:
For each , we define as the space of alternating -linear maps .
Note
- ,
- is the dual space of (the space of covectors)
Definition, Exterior product:
For and , their exterior (wedge) product is given by:
for all . Where the sum is over all permutations of , for which and .
Note
- The exterior product is bilinear, associative,
- anti-commutative: for all and .
Definition, Grassmann Algebra:
Grassmann Algebra is defined by:
This is a anti-commutative graded algebra. Also called Exterior Algebra of
In the case of , we have:
- ,
- ,
- , using Riesz representation theorem,
- , using the map .
Basis¶
Let be a basis of and the associated dual basis for ().
For any increasing permutations , we have:
thus the algebraic -forms , form a basis for and .
Definition, Interior product:
Let be a -form, and . The interior product of and is the -form defined by:
- We have for , and :
Definition, Inner product:
If is has an inner product, then is endowed with an inner product given by:
where the sum is over increasing sequences , and is any orthonormal basis.
Orientation and Volume form¶
Todo
add Orientation and Volume form
Definition, Pullback:
A linear transformation of vector spaces induces a transformation , called the pullback, and given by:
The pullback acts contravariantly: if then,
Let V be a subspace of W. For the inclusion , we can define its pullback : this is a surjection of onto .
If W has an inner product and is the orthogonal projection. We can define its pullback : this an injection of onto .
Let us consider the composition : shortstack{ \ } shortstack{ \ } , and its pullback .
Definition, The tangential and normal parts:
- associates for each its tangential part with respect to :
- associates for each its normal part with respect to .
The tangential part of vanishes if and only if the image of in vanishes.
Let be an oriented inner product space, with volume form . Let . We can define a new linear map as the composition of such as:
and the canonical isomorphism of onto , and using the Riesz representation theorem, there exists an element such that : , i.e.:
Definition, The Hodge star operation:
The linear map which maps onto is called the Hodge star operator.
- If is any positively oriented orthonormal basis, and a permutation, we have
- , thus the Hodge star is an isometry.
- and
- the image of in vanishes if and only if vanishes.
Exterior Calculus on manifolds and Differential forms¶
Let be a smooth manifold, of dimension .
- we denote by the tangent space. This is a vector space of dimension ,
- tangent bundle ,
- Applying the exterior algebra to the tangent spaces, we obtain the exterior forms bundle, whose elements are pairs with and .
- a differential -form is a section of this bundle. This is a map which associates to each an element ,
- if the map is smooth (whenever are smooth), we say that is a smooth differential -form,
- we define the space of all smooth -forms on ,
- ,
- if the map is , we define differential -forms with less smoothness .
Let be a smooth manifold, of dimension .
Exterior product:
if and , we may define as and the Grassmann algebra
Differential forms can be differentiated and integrated, without recourse to any additional structure, such as a metric or a measure.
Exterior differentiation:
For each , can define the -form , such as:
where the hat is used to indicated a suppressed argument.
This defines a graded linear operator of degree , of onto .
We have the following properties:
- ,
- (Pullback) let be a smooth map of onto . Then and ,
- (Interior product) the interior product of a differential -form with a vector field ,
- we obtain a -form by : ,
- (Trace operator) the pullback of is the trace operator
Integration:
- If is an oriented, piecewise smooth -dimensional submanifold of , and is a continuous -form, then th integral is well defined :
- [0-forms] can be evaluated at points,
- [1-forms] can be integrated over directed curves,
- [2-forms] can be integrated over directed surfaces,
- (Inner product) The -inner product of two differential -forms on an oriented Riemannian manifold is defined as :
The completion of in the corresponding norm defines the Hilbert space .
We have the following results:
- (Integration) if is an orientation-preserving diffeomorphism, then
Theorem, Stokes theorem:
If is an oriented -manifold with boundary , then
Theorem, Integration by parts:
If is an oriented -manifold with boundary , then
Sobolev spaces of differential forms¶
As for the classical case, we can define the Sobolev spaces as:
- is the space of differential -forms such that .
- . The associated norm is :
- coincides with ,
- coincides with ,
- for , we have , strictly.
Cohomology and De Rham Complex¶
The De Rham complex is the sequence of spaces and mappings
Since, , we have
If is an oriented Riemannian manifold, we have the following cohomology:
The coderivative operator is defined as:
- we have
- is a graded linear operator of degree .
- is the formal adjoint of whenever or vanishes near the boundary.
- we define the spaces
we have .
- we obtain the dual complex
Cohomology with boundary conditions¶
Let be the subspace of of smooth -forms with compact support. We have .
The De Rham complex with the compact support is
Recall that the closure of in is
The version of the last complex is
Definition, Harmonic forms:
The harmonic -forms are the differential -forms that verify the differential equations
this defines the following space,
We can also define the following space,
As we can see, .
Proposition, Poincaré duality:
There is an isomorphism between the th De Rham cohomology space and the th cohomology space with boundary conditions.
Homological Algebra and Hilbert complexes¶
Homological Algebra¶
- A cochain complex is a sequence of vector spaces and linear maps
- -cocycles ,
- -coboundaries ,
- -cohomology ,
- we say that the sequence is exact, if the cohomology vanishes (i.e. ),
- Given two cochain complexes , a cochain map (such as )
- maps -cochains to -cochains and -coboundaries to -coboundaries, thus induces a map .
Let be two cochain complexes,
The inclusion is a cochain map and thus induces a map of cohomology ,
If there exists a cochain projection of onto , (this leads to ) so .
Thus, is injective and is surjective. Hence, if one of the cohomology spaces vanishes, then so does
Cycles and boundaries of the De Rham complex¶
- -cocycles
- -coboundaries
- each of the spaces of cycles is closed in (), as well in .
- each of the spaces of boundaries is closed in .
- let denotes the orthogonal complement in ,