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HW02sol - Dierential Geometry 1 Homework 02 1 Let M be a...

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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 γ : I M from some open interval (0 ) I R such that γ (0) = p . (a) Show that a tangent vector ξ γ T p M is defined by the rule f C ( M ) ξ γ ( f ) = ( f γ ) (0) . (b) Show that each tangent vector to M at p has the form ξ γ for a suitable choice of γ . (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 (finite-dimensional 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 ξ γ : C ( M ) R is linear and satisfies the Leibniz rule at p ; this takes far less time to write than to type.
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