By Eric Harold Neville
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Vector space homomorphisms. Then f is called a bundle homomorphism. 7. Let (E, π, M ) be a vector bundle of rank n. Let E ⊂ E, and suppose that for any x ∈ M there exists a bundle chart (ϕ, U ) with x ∈ U and ϕ(π −1 (U ) ∩ E ) = U × Rm (⊂ U × Rn , m ≤ n). The resulting vector bundle (E , π|E , M ) is called subbundle of E of rank m. Let us discuss an example: S 1 = {x ∈ R2 : |x|2 = 1} is a submanifold of R2 . If we restrict the tangent bundle T R2 of R2 to S 1 , we obtain a bundle E over S 1 that is isomorphic to S 1 × R2 .
E. for every p ∈ M, expp is defined on all of Tp M. e. by a geodesic of shortest length. Proof. We shall first prove that if expp is defined on all of Tp M, then any q ∈ M can be connected with p by a shortest geodesic. In particular, this will show the implication (iv) ⇒ (v). 2, let p0 ∈ ∂B(p, ρ) be a point where the continuous function d(q, ·) attains its minimum on the compact set ∂B(p, ρ). Then p0 = expp ρV, for some V ∈ Tp M. We consider the geodesic c(t) := expp tV, and we want to show that c(r) = q.
Proof. For v ∈ Tp M, let Λ := {t ∈ R+ : cv is defined on [−t, t]}, where cv is, as usual, the geodesic with cv (0) = p, c˙v (0) = v. It follows from cv (−t) = c−v (t) that cv may also be defined for negative t, at the moment at least for those with sufficiently small absolute value. 2 implies Λ = ∅. The compactness of M implies the closedness of Λ. We shall now show openness of Λ : Let cv be defined on [−t, t]; for example c˙v (t) = w ∈ Tcv (t) M. 2 there exists a geodesic γw (s) with γw (0) = cv (t), γ˙ w (0) = c˙v (t), for s ∈ [0, ε] and ε > 0.