By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2
The proof is elementary. 1. Let M be a compact Riemannian manifold, p, q ∈ M. Then there exists a geodesic in every homotopy class of curves from p to q, and this geodesic may be chosen as a shortest curve in its homotopy class. Likewise, every homotopy class of closed curves in M contains a curve which is shortest and geodesic. Since the proof is the same in both cases, we shall only consider the case of closed curves. 2. 4. Let γ0 , γ1 : S 1 → M be curves with d(γ0 (t), γ1 (t)) ≤ ρ0 for all t ∈ S 1 .
For this, let S ∈ O(n + 1) be the reﬂection across that E. Together with cv , Scv is also a geodesic through p with tangent vector v. 2 implies cv = Scv , and thus the image of cv is the great circle, as claimed. 1. We introduce a metric on T 2 by letting the projection π be a local isometry. For each chart of the form (U, (π|U )−1 ), we use the Euclidean metric on π −1 (U ). 4 Riemannian Metrics 27 are Euclidean isometries, the Euclidean metrics on the diﬀerent components of π −1 (U ) (which are obtained from each other by such translations) yield the same metric on U.
A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map expp is deﬁned on all of Tp M, or, in other words, if any geodesic c(t) with c(0) = p is deﬁned for all t ∈ R. We can now state the Theorem of Hopf–Rinow. 1. Let M be a Riemannian manifold. t. 1). (ii) The closed and bounded subsets of M are compact. (iii) There exists p ∈ M for which expp is deﬁned on all of Tp M. 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.
Algebraic Geometry Santa Cruz 1995, Part 2 by Kollar J., Lazarsfeld R., Morrison D. (eds.)