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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online