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Unformatted text preview: Differential Geometry 1 Homework 02 1. Let M be a smooth manifold. By a smooth curve through p in M we mean a smooth map : I M from some open interval (0 ) I R such that (0) = p . (a) Show that a tangent vector T p M is defined by the rule f C ( M ) ( f ) = ( f ) (0) . (b) Show that each tangent vector to M at p has the form for a suitable choice of . (c) Show that if f C ( M ) has a local minimum at p then f is annihilated by each tangent vector to M at p . 2. Let Y be a (finitedimensional real) vector space of which X is a subspace. Show explicitly that P( X ) is a submanifold of the projective space P( Y ). Now assume that X has codimension one in Y and let g C (P( Y )). Extract as much information as possible from the hypothesis P( X ) = g 1 (0). Solutions (1a) It need only be verified that : C ( M ) R is linear and satisfies the Leibniz rule at p ; this takes far less time to write than to type.; this takes far less time to write than to type....
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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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