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Riemannian Geometry Cambridge University Press 0521450543 - The Geometry of Total Curvature on Complete Open Surfaces Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Excerpt More information 1 Riemannian geometry We introduce the basic tools of Riemannian geometry that we shall need as- suming that readers already know basic facts about manifolds. Only smooth manifolds are discussed unless otherwise mentioned. The use of a local chart (local coordinates) will be convenient for readers at the beginning. Tensor cal- culus is used to introduce geodesics, parallelism, covariant derivative and the Riemannian curvature tensor. In Sections 1.1 to 1.3 the Einstein convention will be adopted without further mention. However, it is not convenient for later discussion, for example, the second variation formula or Jacobi fields. To avoid confusion we shall use vector fields and connection forms to discuss such matters as Jacobi fields and conjugate points. 1.1 The Riemannian metric Let M be an n-dimensional, connected and smooth manifold and (U, x) local coordinates around a point p ∈ M. A point q ∈ U is expressed as x(q) = 1 ,..., n ∈ ⊂ n (x (q) x (q)) x(U) R . The tangent space to M at p is denoted by = Tp M or Mp and TM : p∈M Tp M denotes the tangent bundle over M.We denote by π : TM → M the projection map. Let X (M) and X (U) be the spaces of all smooth vector fields over M and U respectively and C∞(M) the space of all smooth functions over M. A positive definite smooth symmetric bilinear form g : X (M) × X (M) → C∞(M) is by definition a Riemannian metric over M. The metric g is locally expressed in (U, x) as follows. Let Xi ∈ X (U) be the ith basis-vector field tangent to the ith coordinate curve, i.e., −1 i i n Xi := d(x )(∂/∂x ), where ∂/∂x is the canonical vector field in x(U) ⊂ R n parallel to the ith coordinate axis of R . Then Tq M for every q ∈ U is spanned by X1(q),...,Xn(q). If X, Y : U → TU are local vector fields expressed as 1 © Cambridge University Press www.cambridge.org Cambridge University Press 0521450543 - The Geometry of Total Curvature on Complete Open Surfaces Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Excerpt More information 2 1 Riemannian geometry = φi = ψ j = , X i Xi and Y j X j and if gij : g(Xi X j ) then n i j g(X, Y ) = φ ψ gij, (1.1.1) i, j=1 = where gij g ji and the gij are smooth functions onU. Thus the length (or norm) v v ∈ v = v,v 1/2 = vi v j 1/2 of a vector Tp M is defined by : gp( ) ( gij(p) ) , i where v := v Xi (p). The angle (u,v) between two vectors u and v in Tp M is thus defined by u,v cos (u,v):= . (1.1.2) u v Here the angle (u,v) takes values in [0,π] and u,v=gp(u,v). The volume of a parallel n-cube in Tp M spanned by X1(p),..., Xn(p)is 1/2 given by |X1(p) ∧···∧Xn(p)|=(det gij(p)) . Thus the volume element dM of M is expressed as follows: 1/2 1 n dM = (det gij) dx ∧···∧dx . (1.1.3) The manifold M equipped with a Riemannian metric g is called a Riemannian manifold and denoted by (M, g) or simply by M. By a smooth curve c :[α, β] → M we always mean that there exists an open interval I ⊃ [α, β] such that c is defined over I and is regular at all points on I . A piecewise-smooth curve c :[α, β] → M is a continuous map consisting of finitely many smooth curves. Namely, there are finitely many points t0 = α< < ··· < = β | = ,..., − t1 tk such that c [ti ,ti+1], for every i 0 k 1, is a smooth curve. A piecewise-smooth vector field X along a curve c :[α, β] → M is by α, β → π ◦ = definition a piecewise-smooth mapX :[ ] Tc M such that X(t) c(t) ∈ α, β = α, β → for all t [ ], where Tc M : t∈[α,β] Tc(t) M. A curve c :[ ] U in a coordinate neighborhood is expressed by x ◦ c(t) = (x1(t),...,xn(t)), for t ∈ [α, β]. Its velocity vector field is n dxi c˙(t) = Xi ◦ c(t). i=1 dt The length L(c) of a curve c :[α, β] → U is given by β β n i j dx dx L(c) = c˙ dt = gij(c(t)) dt. (1.1.4) α α i, j=1 dt dt If s(t) for t ∈ [α, β] is the length of the subarc c|[α,t] of c then ds(t)/dt = c˙(t) > 0, and hence s(t) has an inverse, t = t(s). Thus c can be parameterized © Cambridge University Press www.cambridge.org Cambridge University Press 0521450543 - The Geometry of Total Curvature on Complete Open Surfaces Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Excerpt More information 1.2 Geodesics 3 by its arc length s ∈ [0, L], where L = L(c) is the total length of c. From the relation ds(t) = c˙ dt we derive a quadratic differential form n 2 i j ds = gijdx dx , i, j=1 which we call the line element of M. 1.2 Geodesics From now on the Einstein convention is used. Once the length of a curve in M has been established, we discuss a curve with the special property of having a local minimum in length among the neighboring curves with the same endpoints. This is referred to as the locally minimizing property. Such a curve, a geodesic, is obtained as the solution of a nonlinear second-order ordinary differential equation, (1.2.1) below, whose coefficients depend only on the gij and their partial derivatives. Thus geodesics are defined as the solutions of (1.2.1). The set of all solutions with a fixed starting point p corresponds to a domain in Tp M that is star-shaped with respect to the origin. Thus the exponential map and its injectivity radius at that point is introduced here. Definition 1.2.1. A unit-speed curve c :[α, β] → M is said to have the locally minimizing property iff there exists for every s ∈ [α, β] a positive number δ with [s − δ, s + δ] ⊂ I and a neighborhood N around c([s − δ, s + δ]) such that c([s − δ, s + δ]) has the minimum length among all the curves in N joining c(s − δ)toc(s + δ). We define the Christoffel symbols i 1 i = ∂ + ∂ − ∂ , jk : 2 g ( j g k k g j g jk) j ij where ∂ j gk := ∂gk /∂x and (g ) is the inverse matrix of (gij), i.e., i = δi . g gk : k δi δi = = δi = = Here j is the Kronecker delta, i.e., k 1 for i k and k 0 for i k. With this notation we prove Theorem 1.2.1. If a unit-speed curve c :[α, β] → M has the locally minimiz- ing property, then the local expression x ◦ c(s) = (x1(s),..., xn(s)) © Cambridge University Press www.cambridge.org Cambridge University Press 0521450543 - The Geometry of Total Curvature on Complete Open Surfaces Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Excerpt More information 4 1 Riemannian geometry of c in a coordinate neighborhood U satisfies d2xi dx j dxk + i = 0. (1.2.1) ds2 jk ds ds Proof. Since the discussion is local, we may restrict ourselves to the case where c([α, β]) is contained entirely in a coordinate neighborhood U and write x ◦ c(s) = (x1(s),..., xn(s)). A variation along c :[α, β] → U is by definition a (piecewise-smooth) map V :(−ε0,ε0) × [α, β] → U such that V (0, s) = c(s) for all s ∈ [α, β] and Vε(s):= V (ε, s) for every ε ∈ (−ε0,ε0) is a curve Vε :[α, β] → U. If x ◦ 1 n V (ε, s) = (x (ε, s),..., x (ε, s)) for (ε, s) ∈ (−ε0,ε0) × [α, β] then the vari- ational vector field Y :[α, β] → Tc M associated with V is given by ∂ ∂xi Y s = V , = , s X ◦ c s . ( ) d (0 s) ∂ε ∂ε (0 ) i ( ) The length L(ε) of each variation curve Vε is given by β ∂xi (ε, s) ∂x j (ε, s) L(ε):= L(Vε) = gij ds. α ∂s ∂s Thus we have, by taking the arc length parameter 0 ≤ s ≤ L =: L(c), L dL ∂ ∂xi (ε, s) ∂x j (ε, s) = gij ds dε ∂ε ∂s ∂s ε=0 0 ε= 0 1 L ∂xk ∂xi ∂x j ∂2xi ∂x j = ∂ g + g s. ( k ij) ∂ε ∂ ∂ 2 ij ∂ ∂ε ∂ d 2 0 s s s s ε=0 Since the second term in the integrand can be expressed as ∂ ∂xi ∂x j ∂ ∂x j ∂xk g − g ∂s ij ∂ε ∂s ∂s kj ∂s ∂ε we have dL(0) ∂xi ∂x j L = g , s ε ij ∂ε ∂ (0 ) d s 0 L j 2 j i j k ∂x ∂x ∂ x 1 ∂x ∂x ∂x − ∂g + g − ∂ g ds. kj ∂ ∂ kj ∂ 2 k ij ∂ ∂ ∂ε 0 s s s 2 s s (0,s) = ∂ + ∂ − ∂ k = By setting 2[ij; k]: i g jk j gik k gij, we see that ijgk [ij; ], and © Cambridge University Press www.cambridge.org Cambridge University Press 0521450543 - The Geometry of Total Curvature on Complete Open Surfaces Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Excerpt More information 1.2 Geodesics 5 then the integrand can be rewritten as 1 dx dx j d2x j ∂xk (∂g + ∂ g − ∂ g ) + g 2 kj j k k j ds ds kj ds2 ∂ε d2x j dx dxm ∂xk = g + j .
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