Geometry of Manifolds
Geometry of Manifolds MATH 7350
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This 2 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 7350 at University of Houston taught by Andrei Torok in Fall. Since its upload, it has received 50 views. For similar materials see /class/208387/math-7350-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Main topics Math 7350 Most items below come with a de nition examples and maybe proof even if not mentioned explicitly Md denotes a manifold of dimension d an 1 Review of multivariable calculus on open subsets of Euclidean spaces 7 higher order derivatives Taylor polynomials 7 the lnverse Function Implicit Function and Rank theorems normal forms for such maps ie the simplest form up to local diffeomorphisms 2 De nition of manifolds with and without boundary We restrict ourselves to second countable Haus dorff spacesi Ckchartsi Maximal atlas differential structurei Topology through charts if the manifold is not already a topological space New charts amp manifolds from old Theorem Whitney Each Ck manifold admits a compatible C00 structure All these C00 structures are smoothly diffeomorphici Thus could assume from now on that all manifolds are smooth 3 Differentiable maps between manifoldsi Critical singular etci points of a map Submersions immersions embeddingsi The lnverse Function Implicit Function and Rank theorems on manifoldsi 4i Submanifoldsi Preimage of a regular point is a submanifoldi Image of an immersion or embedding 5 The tangent space in local coordinates and associated basis through curves or as derivations for smooth manifolds The differential of a map in each of the above representations 6 General vector bundles local trivializations the transition functions the compatibility condition The tangent bundlei Vector elds in various descriptions 7i ODEls in R and on a manifold The ow localglobal oneparameter group associated to a vector eld 8 The Lie derivative of a tensor eld with respect to a vector eld The Lie derivative is actually associated to the oneparameter group determined by the vector eld The bracket of two vector elds the Lie algebra of smooth vector elds 9 Distributions integrability lf X1X2HiXd are commuting vector elds which are linearly independent at each point then locally there is a chart such that Xk i for 1 S k S d The Frobenius theorem the proof in Spivak is more conceptual 0 Tensors alternating forms for a nite dimensional vector space Bases for tensors and forms The wedge producti H CA3 i The bundle of differential forms of a manifold The exterior differential of a form d2 0i Pullback of a form7 its behavior With respect to the wedge product and exterior differential i Closed and exact forms The Poincare Lemmai i Partitions of unity including locally nite covers7 etc i Orientation of a vector space Orientation of a manifold M is orientable iff there exists a nowhere vanishing n formi Orientation induced on the boundary of an oriented manifold i Integration of kforms on singular kchainsi The boundary of a chain 82 0 The Stokes theorem for chainsi i Integration of compactly supported n forms on an oriented n dimensional manifold The Stokes theorem for manifoldsi i Embedding of compact manifolds in RNi The medium Whitney embedding theorem 1936 M lt gt R2n1i
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