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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 nspace Pn ;
(c) the unitary group U (n).
Suggestion: Consider the map f : A → AT A from complex n × nmatrices to
Hermitian complex n × nmatrices.
2. Calculate explicitly (and describe geometrically) the timet 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 + 1dimensional 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 evendimensional 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 LeviCivita 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.
 Spring '09
 Robinson

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