HW02 - Differential Geometry 1 Homework 02 1 Let M be a...

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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 γ : I → M from some open interval (0 ∈) I ⊂ R such that γ (0) = p. (a) Show that a tangent vector ξγ ∈ Tp M is defined by the rule f ∈ C ∞ (M ) ⇒ ξγ (f ) = (f ◦ γ ) (0). (b) Show that each tangent vector to M at p has the form ξγ 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). 1 ...
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