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##### Hodge Theory

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**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 {e1, . . . , em} is a basis for V, then the elements where I=(i1, . . . , ip) ranges over the set of multi-indices with 1≤i1<. . .<ip≤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 Sp 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 uI 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 {e1, . . . , em} is an orthonormal basis for V, then the corresponding basis for (V) is likewise orthonormal.**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 RV)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 {e1,. . . ,em} be an ordered, orthonormal basis for V such that e1^. . .^em 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:=e1^. . .^em2m(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, v2p(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**A smooth manifold M of dimension m comes equipped with natural smooth vector bundles. Let (U; x1, . . . , xm) be a chart centered at a point q2M.**Tangent and Cotangent Bundles**TM the tangent bundle of M. The fiber TM,q can be identified with the linear span**Tangent and Cotangent Bundles**TM* the cotangent bundle of M. Let {dxi} be the dual basis of the basis {∂xi}. The fiber TM,q* can be identified with the span**Tangent and Cotangent Bundles**p(TM*) the p-th exterior bundle of TM*. The fiber p(TM*)q= p(TM,q*)can be identified with the linear span**Tangent and Cotangent Bundles**(TM*):=mp=0 p(TM*)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(TM*) 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 ii-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 v2Ep-1(M) such that dv=u.**deRham Cohomology Groups**Since d2=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 TM,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 TMTM* and we can naturally define a metric on TM*.**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 • A partition of unity subordinate to {U}2A • 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(TM*)M, where M is embedded in the total space of the line bundle m(TM*) 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(TM*)M has two connected components, non-orientable otherwise. If M is orientable, then the choice of a connected component of m(TM*)M is called an orientation of M which is then said to be oriented.