Hodge Theory Calculus on Smooth Manifolds

by William M. Faucette Adapted from lectures by Mark Andrea A. Cataldo

Structure of Lecture The Exterior Algebra The Star Isomorphism The Tangent and Cotangent Bundles The deRham Cohomology Groups Riemannian Metrics Partitions of Unity Orientation and Integration

The Exterior Algebra Definition and Euclidean Structure

The Setup Let V be an m dimensional real vector space with inner product

The Setup Let be the exterior algebra associated to V.

The Setup If {e 1,..., e m } is a basis for V, then the elements where I=(i 1,..., i p ) ranges over the set of multi-indices with 1≤i 1 <...<i p ≤m forms a basis for  p (V).

The Setup The elements of  p (V) can be seen to be alternating forms on the dual space V* as follows: where S p is the symmetric group and sgn(  ) is the sign of the permutation 

The Setup It is customary to write meaning that the summation is over all ordered multi-indices I as above and u I is a real number.

The Inner Product on  (V) There is a natural inner product on  (V) induced by that on V by defining distinct subspaces  p (V) and  q (V) are mutually orthogonal for p≠q and setting

The Inner Product on  (V) If {e 1,..., e m } is an orthonormal basis for V, then the corresponding basis for  (V) is likewise orthonormal.

The Star Operator on  (V)

The Star Operator The real vector space  m (V) is one- dimensional and  m (V)  has two connected components, that is, two half-lines. However, there is no canonical way to distinguish between the two components. A choice must be made.

The Star Operator The choice gives rise to an isometry between R    V)  m (V). The star operator arises when we want to complete the picture with linear isometries

The Star Operator Definition: The choice of a connected component  m (V) + of  m (V)  is called an orientation of V.

The Star Operator Let V be oriented and {e 1,...,e m } be an ordered, orthonormal basis for V such that e 1 ^...^e m is an element of  m (V) +. This element is uniquely defined since any two such bases are related by an orthogonal matrix with determinant 1.

The Star Operator Definition: The vector dV:=e 1 ^...^e m 2  m (V) + is called the volume element associated with the oriented (V, h, i,   (V) + ).

The Star Operator Definition: The star operator is the unique linear isomorphism Defined by the properties for all u, v2  p (V) and for all p.

The Star Operator The star operator depends on the inner product h, i and the choice of orientation of V.

Tangent and Cotangent Bundles of a Smooth Manifold

Tangent and Cotangent Bundles A smooth manifold M of dimension m comes equipped with natural smooth vector bundles. Let (U; x 1,..., x m ) be a chart centered at a point q2M.

Tangent and Cotangent Bundles T M the tangent bundle of M. The fiber T M,q can be identified with the linear span

Tangent and Cotangent Bundles T M * the cotangent bundle of M. Let {dx i } be the dual basis of the basis {∂ x i }. The fiber T M,q * can be identified with the span

Tangent and Cotangent Bundles  p (T M *) the p-th exterior bundle of T M *. The fiber  p (T M *) q =  p (T M,q *) can be identified with the linear span

Tangent and Cotangent Bundles  (T M *):=  m p=0  p (T M *)  the exterior algebra bundle of M.

deRham Cohomology Groups

Definition: The elements of the real vector space Of smooth real-valued sections of the vector bundle  p (T M *) are called (smooth) p-forms.

deRham Cohomology Groups Let denote the exterior derivation of differential forms.

deRham Cohomology Groups Definition: A complex is a sequence of maps of vector spaces denoted (V ,  so that  i   i  1 =0 for every index i.

deRham Cohomology Groups The vector spaces are called the cohomology groups of the complex.

deRham Cohomology Groups A complex is said to be exact at i if that is, if

deRham Cohomology Groups A complex is said to be exact if it is exact for all i. That is, if

deRham Cohomology Groups A p-form u is said to be closed if du=0. A p-form u is said to be exact if there exists v2E p  1 (M) such that dv=u.

deRham Cohomology Groups Since d 2 =0, every exact p-form is closed, so that the real vector space of exact p-forms is a vector subspace of the real vector space of closed p-forms. So the sequence forms a complex.

deRham Cohomology Groups The deRham cohomology groups are the cohomology groups of the complex That is,

deRham Cohomology Groups The fact that deRham cohomology groups are locally trivial follows from the important Theorem: (Poincaré Lemma) Let p>0. A closed p-form u on M is locally exact.

deRham Cohomology Groups The importance of deRham cohomology is that it is equal to the usual (either simplicial or cellular) cohomology of M: Theorem: (The deRham Theorem) Let M be a smooth manifold. There is a canonical isomorphism of R-algebras

Riemannian Metrics

A Riemannian metric g on a smooth manifold M is a smoothly-varying positive definite inner product g( ,  ) q on the fibers of T M,q of the tangent bundle of M. This means that, using a chart (U; x), the functions are smooth on U.

Riemannian Metrics Equivalently, a Riemannian metric g is a smooth section of the bundle the bundle of symmetric bilinear functions

Riemannian Metrics A Riemannian metric induces an isomorphism of vector bundles T M  T M * and we can naturally define a metric on T M *.

Partitions of Unity

Paritions of Unity Let M be a smooth manifold. A partition of unity on M is a collection {   } of non- negative smooth functions on M such that the sum is locally finite on M and adds up to the value 1.

Paritions of Unity This means that for every q2M there is a neighborhood U of q in M such that    U ¥0 for all but finitely many indices so that the sum is finite and adds to 1.

Partitions of Unity Definition: let {U  }  2A be an open covering of M. A partition of unity subordinate to the covering {U  }  2A is a partition of unity {   } such that the support of each          U  

Partitions of Unity On a non-compact manifold, it is not possible in general to have a partition of unity subordinate to a given covering and such that the functions   have compact support. For example, a covering of R given by the single open set R.

Partitions of Unity Theorem: Let {U  }  2A be an open covering of M. Then there are 1. A partition of unity subordinate to {U  }  2A 2. A partition of unity {  j } j2J, where J≠A in general, such at (i) the support of every  j is compact and (ii) for every index j there is an index  such that supp(  j )  U  

Orientation and Integration

Given any smooth manifold M, the space  m (T M *)  M, where M is embedded in the total space of the line bundle  m (T M *) as the zero section, has at most two connected components.

Orientation and Integration Definition: A smooth manifold M is said to be orientable if  m (T M *)  M has two connected components, non-orientable otherwise. If M is orientable, then the choice of a connected component of  m (T M *)  M is called an orientation of M which is then said to be oriented.

Orientation and Integration Let M be a differentiable manifold of dimension m. Then the following are equivalent 1. M is orientable 2. The Jacobians of the transition functions for T M are all positive 3. There is a nowhere vanishing m-form on M.

Orientation and Integration A collection of charts {(       2  so that the Jacobian of the transition functions x   x   is positive is called an orientation-preserving atlas.

Orientation and Integration Let M be an oriented manifold of dimension m. In particular, M admits an orientation preserving atlas. It is using this atlas and partitions of unity that we can define the operation of integrating m-forms with compact support on M and verify Stokes’ Theorem.

Orientation and Integration Let  be a m-form on M with compact support contained in a chart (U;x). With trivialization  U :U  R m. The m-form (  U -1 )*  =f(x) dx 1  dx m,for a unique compactly supported real-valued function f(x) on R m.

Orientation and Integration Define where dx denoted the Lebesgue measure on R m and the right-hand-side is the Riemann-Lebesgue integral of f(x).

Orientation and Integration Since the atlas is orientation preserving, this definition is independent of the choice of chart U.

Orientation and Integration Let  be a smooth m-form with compact support on M. Let {(      2  be an orientation preserving collection of coordinate systems with orientation preserving trivializations x  :U   R m and {   } be a partition of unity subordinate to the covering {     2 

Orientation and Integration The forms    have compact support contained in U   This quantity is well-defined. Changes the orientation simply changes the signs of the integrals.

Orientation and Integration Complex manifolds are always orientable and are usually oriented using a standard orientation. Using the standard orientation, if  is the (1,1)- form associated with the Fubini-Study metric of the complex projective space P n, then  P n  n =1.

Orientation and Integration  : (Stokes’ Theorem (weak version)) Let M be an oriented manifold of dimension M and u be a (m  1)-form on M with compact support. Then

Orientation and Integration If a Riemannian metric g on the oriented manifold M is given, then we have the notion of Riemannian volume element associated with (M,g) and with the orientation.

Orientation and Integration It is the unique m-form dV on M such that, for every q2M, dV q is the volume element for the dual metric g q * on T M,q *. If the integral  M dV converges, then its value is positive and it is called the volume of the oriented Riemannian manifold.