2007_January_DG

2007_January_DG - Diﬀerential Geometry January 2007...

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Unformatted text preview: Diﬀerential Geometry January 2007 Answer SIX questions. Write solutions in a neat and logical fashion, giving complete reasons for all steps and stating carefully any substantial theorems used. 1. Explain how each of the following is a smooth manifold: (a) the unit sphere S n ⊂ Rn+1 ; (b) real projective n-space Pn ; (c) the unitary group U (n). Suggestion: Consider the map f : A → AT A from complex n × n-matrices to Hermitian complex n × n-matrices. 2. Calculate explicitly (and describe geometrically) the time-t ﬂow for the vector ﬁeld ζ on R3 deﬁned by ζ = (z − y ) ∂ ∂ ∂ + (x − z ) + (y − x) . ∂x ∂y ∂z 3. Deﬁne the operators involved and prove the Cartan identity Lξ = (ξ ) ◦ d + d ◦ (ξ ) for the Lie derivative of forms along the vector ﬁeld ξ . 4. A contact form on the 2n + 1-dimensional manifold M is (by deﬁnition) a form θ ∈ Ω1 (M ) such that θ ∧ (dθ)n is nowhere zero. Prove carefully that (the pullback of) θ := x0 dx1 − x1 dx0 + x2 dx3 − x3 dx2 is a contact form on the sphere S 3 ⊂ R4 (using in terms of standard Cartesian coordinates on R4 ). 5. Deﬁne the de Rham cohomology of the smooth manifold M . 1 Prove explicitly (using diﬀerential forms!) that HdR (R3 − {0}) = 0. Suggestion: Express R3 − {0} as the union of two open sets diﬀeomorphic to R3 and with connected intersection. 6. Let ω be a symplectic form on the smooth manifold M . (a) Prove in detail that M is both even-dimensional and orientable. (b) Explain why ω cannot be exact when M is compact. 1 7. Either (a) Deﬁne what is meant by a Lie group G and its Lie algebra g. Discuss the extent to which the structure of G and the structure of g are related. Or (b) Deﬁne the Levi-Civita connexion and the geodesics of a Riemannian manifold. Explicitly determine the geodesics on the standard (‘round’)unit sphere S 2 ⊂ R3 . 2 ...
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.

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2007_January_DG - Diﬀerential Geometry January 2007...

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