Minimizing properties of geodesics and convex neighorhoods Seminar Riemannian Geometry Summer Semester 2015 Prof. Dr. Anna Wienhard, Dr. Gye-Seon Lee Hyukjin Bang July 10, 2015 1. Normal spheres In the sequal, we understand the tangent space to T P M at v T p M as T P M itself. Lemma 1 (Gauss). Let p M, v T p M s. t. exp p v is defined. Let w T v T p M. Then (dexp p ) v (v), (dexp p ) v (w) = v, w. Proof. Recall that (dexp p ) v (v): T p M T v exp p M, whereas, T v B ε (0) T v T p M T P M. We shall prove this lemma in three steps. (i) (dexp p ) v (v), (dexp p ) v (v) = v, v. (ii) (dexp p ) v (v), (dexp p ) v (w N ) = 0, where w = w T + w N. (iii) t f t,f s =0. (i) (dexp p ) v (v) = v. Let u e a curve in T p M, u : R I T p M, such that u(0) = v T v T p M T P M. Choose u(t) = tv among its equivalent classes, namely v (t) = v(t + α), where α R. With this construction, (dexp p ) v (v), (dexp p ) v (v) = d dt (exp p(v)), d t=0 dt (exp p(v)) t=0 = d dt (γ(t, p, v)), d t=0 dt (γ(t, p, v)) = v, v. t=0 We now separate w into its tangential, and normal component to v.
(ii) (dexp p ) v (v), (dexp p ) v (w N ) = 0 We define the curve u as u: [ ε, ε] [0, 1] T p M, (s, t) tv + tsw N. It follows, u(0,1) = v, u t (s, t) = v + sw N, Furthermore, we consider the parametrized surface Under this construction; u s (0, t) = tw N. f: [ ε, ε] [0, 1] M, (s, t) exp p (tv + tsw N ). (dexp p ) v (v) = (dexp p ) u(0,1) ( u t (0,1)) = t exp p u(s, t) t=1,s=0 = f t (0,1), (iii) (dexp p ) v (w N ) = (dexp p ) u(0,1) ( u s (0,1)) = s exp p u(s, t) t=1,s=0 = f s (0,1), Hence, t f t, f s (0,t) = 0 (dexp p ) v (v), (dexp p ) v (w N ) = f t, f s (0,1). We are verifying that the aove scalar product is actually independent on the variale t, so that we can conclude that f (0,1) = 0 through the property lim f t 0 s, f t s (0, t) = lim t 0 (dexp p) tv (tw N ) = 0. Since the map t f(s, t) is geodesic for all s and t, D t f, f + t s f, D f = t t s f, D f = 1 t s t 2 s f, f. t t t f t, f s = And since the map t f(s, t) is geodesic, the function t f, f is t t constant. Thus s f, f = v + sw t t s=0 s N, v + sw N = 2 v, w N = 0. s=0 (i),(ii), and (iii) yield us to complete the proof of the lemma with the ilinearity of the scalar product.
It is said that U = exp p V is a normal neighorhood of p, if exp p is a diffeomorphism on a neighorhood V of the origin in T p M. B ε (p) = exp p B ε (0), a normal all (or a geodesic all) if the closure of epsilon all is in such V. Moreover, the Gauss lemma gives us that the normal sphere S ε (p) = B ε (p) is a hypersurface in the manifold M, which is orthogonal to the geodesics starting from p. Further, the geodesics in B ε (p) with starting point p are called radial geodesics. 2. The minimizing property of geodesics. In this section we will oserve that geodesics locally minimize the arc length. Namely, Proposition 2. Let p M, U = exp p V, B = B ε (p). Let γ: [0,1] B e a geodesic segment with γ(0) = p, and c: [0,1] M e any piecewise differentiale curve with c(0) = γ(0), c(1) = γ(1). Then it holds for γ and c, l(γ) l(c). The equality holds if and only if c([0,1]) = γ([0,1]). Proof. w.l.o.g. suppose that c([0,1]) B. Otherwise cut c into parts along the oundary of B and e parametrized y [0,1]. To apply the Gauss lemma we shall find a parametrized surface f, such that f(γ(t), t) = exp p (γ(t) v(t)), where t v(t) a curve in T p M, v(t) = 1, and γ: (0,1] R is a positive piecewise differentiale function. Simple calculus gives us, except for some singularities, Applying the Gauss lemma yields, f as, ε 1 dc dt = f dγ γ dt + f t., f γ t = 0. For f = 1, γ dc dt 2 = dγ dt 2 + f t 2 dγ dt 2, ε 1 dc dt dt dγ dt dt dγ dt dt γ(1) γ(ε). Taking ε 0 shows what it was claimed. If the equality holds for some c, then f γ = 0, which let v(t) e constant, meaning that the curve is rescaled form of γ. We have now reached that there is a piecewise differentiale curve which locally minimizes ε 1
the arc length. We shall further prove that this curve is a geodesic y showing the existence of normal neighorhoods. Theorem 3. For any p in M, there exist a neighorhood W of p, which is a normal neighorhood of each of its points. In other words, p M, W M, δ > 0, such that q W, exp q is a diffeomorphism on B δ (0) T q M and exp q (B δ (0)) W. Proof. We shall prove this theorem in following manner. (i) (ii) Construct a local diffeomorphism Applying the inverse function theorem. For proof of this theorem, see the script of Analysis 2, Prof. Knüpfer, SS2014. Let ε > 0, V a neighorhood of p M, Ω = {(q, v) TM; q V, v T q M, v < ε}. Define F: Ω M M, (q, v) (q, exp q v). F is then a local diffeomorphism around (p,0). Indeed, (dexp p ) 0 = I, and the matrix df (p,0) is ( I I ). Thus we have a 0 I local diffeomorphism. From the inverse function theorem, there exists a neighorhood Ω Ω of (p, 0), and a neighorhood W of (p, p), so that F: Ω W is ijective and differentiale. By choosing Ω = {(q, v) TM; q V V, v T q M, v < δ} and W such that W W W, we get the assertion. Remark: It is said that W is a totally normal neighorhood of p M. As we have seen that there is a unique minimizing geodesic γ with l(γ) < δ, for given two points of W. By having this theorem we can conclude that there is a unique v in T σ M, for (σ, τ) in W, such that γ (0) = v. Corollary 4. A piecewise differentiale curve γ: [a, ] M with parameter proportional to arc length is a geodesic, if for any other piecewise differentiale curve c: [a, ] M, c(a ) = γ(a) and c( ) = γ(), holds l(γ) l(c). Proof. We shall prove with the compactness argument. There exists a pathwise connected compactum K, with K. Let W e a totally normal neighorhood of γ(t), t K. Then the inclusion γ(k) W holds. Consider the restriction of γ K : K W. Since γ K (K) is in a normal all, it has equal length of a radial geodesic in W. From the minimizing property, γ K is a geodesic. Furthermore, since it holds for all t [a, ] that d dt dγ = 0, which implies, dγ dt dt C dγ, dγ = const. Thus, if C is non-zero, γ must e a regular curve. (If C is equal to dt dt zero, γ degenerates to a point in M.)
In following example, we are using this corollary aove, to determine the geodesics in so called the Loatschevski plane, whilst the geodesics is eing mapped to geodesics y isometries of a Riemannian manifold. Example: Given that G is the upper half-plane in two dimensional Euclid space with the Riemannian metric g 11 = g 22 = 1 y 2, g 12 = g 21 = 0. We are claiming that the segment γ: [a, ] G, a > 0, γ(t) = (0, t) is the image of a geodesic under an isometry. [Indeed, for any c: [a, ] G, t (x(t), y(t)) with c(a) = (0, a) and c() = (0, ), it holds that l(c) = dc dt = (dx dt dt )2 + ( dx dt )2 dt y dy dt dy dt dy = l(γ). a a a dt y a dt y a y Hence γ is the minimizer of arc length, y the corollary it is a geodesic.] To this purpose, we take the mo ius transformations Φ acd : C R 2 C as our isometries, given y z az+, ad c = 1. For details, see the script of Funktionentheorie, Prof. cz+d Kohnen. These transformations give us rays or half circles, which vary on the values of a,,c and d. 3. Convex neighorhoods We have oserved that every point in a Riemannian manifold has a totally normal neighorhood. Still, there are cases that our geodesics are not completely lying in a certain totally normal neighorhood. In this section we will see that a totally normal neighorhood W can e chosen on a Riemannian manifold, so that W is ecoming strongly convex. Definition 5. A suset S M is strongly convex if for any two points q 1, q 2 S there exists unique minimizing geodesic γ such that (γ([q 1, q 2 ])) S. Lemma 6. For any p M, there is an upper oundary c for radius r, such that any geodesic in M that is tangent at q M to S r (p) = B r (p) stays out of the B r (p) = exp p B r (0). Proof. Let W e a totally normal neighorhood of p M. Since all geodesics in W can e considered to have the velocity one through the natural parametrization, it is enough to prove with the unit tangent undle T 1 W = {(q, v); q W, v T q M, v = 1}. Consider the differentiale mapping γ: ( ε, ε) T 1 W M, where t γ(t, q, v) e the geodesic with the initial condition at t = 0, γ passes q = γ(0, q, v), v = 1. Let u(t, q, v) = exp p 1 (γ(t, q, v)) and define F: ( ε, ε) T 1 W R, F(t, q, v) = u(t, q, v) 2. F From the constructions follows = 2 t u, u, 2 F = 2 u t t 2 2, u + 2 u t2 t 2. Further, we choose our radius r > 0, so that B r (p) = exp p B r (0) W. Oserve that if γ is tangent to S r (p) =
B r (p), at q = γ(0, q, v), then F = t u, u = 0, from the Gauss lemma. We are (0,q,v) t (0,q,v) showing that F has a strict minimum at (0, q, v), for some enough small radius r. For starters we oserve for q = p, we have u(t, p, v) = exp 1 p (γ(t, p, v)) = tv, as well as 2 F t 2 (0,p,v) = 2 t (tv) 2 =2 v 2 = 2 > 0. Thus there exist a neighorhood Ω W of p such that 2 F t 2 (0,q,v) > 0, for all q Ω, v T q M, v = 1. Now choose an upper oundary c s.t. exp p B c (0) Ω. Since a strict minimum gives us the minimizer for distance etween p and to a point p γ that is moving along the geodesic γ, with minimizing property there is another geodesic in B c (p) = exp p B c (0), joining p and p γ, which is tangent to S r (p) = B r (p) at q = γ(0, q, v), as claimed. Proposition 7. (Convex neighorhoods). For any p M, there is a numer β > 0 such that the normal all B β (p) is strongly convex. Proof. Let c e the upper oundary given in the aove lemma for a p M. Choose c 2 > δ > 0 and W s. t. q W, exp q is a diffeomorphism on B δ (0) T q M and exp q (B δ (0)) W. We shall prove that B β (p) is strongly convex for β < δ B β (p) W. Given that two random points q 1, q 2 B β (p), let γ e the geodesic with l(γ) < 2δ < c, joining q 1 and q 2. Assume that (γ([q 1, q 2 ])) B β (p) i.e. B β (p) is not strongly convex. Then there is a point x in (γ([q 1, q 2 ])), with the maximum distance r from q. Hence for enough small ε > 0, B ε (x) B r (p), which is contradiction to the lemma 6.