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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