Differential Geometry Dominic Leung 梁树培 Lecture 13

The following observations will be helpful in showing that the metric (3.39) has constant curvature K. Define r(u) = {∑ (ui)2}1/2 and σ(r) = -ln(1 + (K/4) r2) . Then { e- σ(∂/∂ui} is a local orthonormal frame field, and the corresponding local orthonormal coframe field is { 𝜃 𝑖 } , where 𝜃 𝑖 = e σ 𝑑𝑢 𝑖 . Now d 𝜃 𝑖 = e σ d σ Ʌ 𝑑𝑢 𝑖 = 𝑗 𝜃 𝑗 Ʌ (∂σ /∂uj) 𝑑𝑢 𝑖 = 𝑗 𝜃 𝑗 Ʌ ((∂σ /∂uj) 𝑑𝑢 𝑖 - (∂σ /∂ui) 𝑑𝑢 𝑗 ) Let 𝜋 𝑗 𝑖 denotes the term in the parenthesis. Then we have 𝜋 𝑗𝑖 = 𝜋 𝑗 𝑘  𝑖𝑘 = - 𝜋 𝑖𝑗 . Therefore by Theorem 1.5 the connection 1-form ω 𝑖𝑗 with respect to the local coframe 𝑑𝑢 𝑖 is given by 𝜋 𝑖𝑗 = (∂σ /∂ui) 𝑑𝑢 𝑗 - (∂σ /∂uj) 𝑑𝑢 𝑖

According to Theorem 1.5 the curvature form Ωij and the curvature tensors can be computer from Ωij = d 𝜋 𝑖𝑗 + ∑ 𝑙 𝜋 𝑖𝑙 Ʌ 𝜋 𝑗𝑙 A direct computations shows that d 𝜋 𝑖𝑗 + ∑ 𝑙 𝜋 𝑖𝑙 Ʌ 𝜋 𝑗𝑙 = - K e σ 𝑑𝑢 𝑖 Ʌ e σ 𝑑𝑢 𝑗 = - K 𝜃 𝑖 Ʌ 𝜃 𝑗 The following partial derivatives of σ will be very helpful in the computations ∂σ /∂uj = - (K/2) e σ 𝑢 𝑖 ∂2σ/∂ 𝑢 𝑘 ∂ 𝑢 𝑙 = (K/2)2 e 2σ 𝑢 𝑘 𝑢 𝑙 for k ≠ 𝑙 ∂2σ/(∂ 𝑢 𝑘 )2 = - (K/2) e σ(1 – (K/2) e σ ( 𝑢 𝑘 )2)

Riemann’s classical metric tensor (3 Riemann’s classical metric tensor (3.39) for a Riemannian manifold has been generalized to generalized Riemannian manifold as follows. Suppose 𝑑𝑠 2 = ϵ1𝑑𝑢1 2 + … + ϵ𝑚𝑑𝑢𝑚 2 1 + 𝐾 4 ϵ𝑖 𝑢𝑖 2 2 ϵi = ±1 (3.40) where K is a real number. Then the generalized Riemannian space with (3.40) as metric tensor has constant sectional curvature K. The computation to show that the sectional curvature for the metric (3.40) is in fact K is similar to that for the Riemannian case with the obvious modifications.

2.8 The metric space structure of a Riemannian manifold We note that if x(t) is a solution of (2.1) of §5-2 in [1], so is x(αt) for any constant α ϵ R. Denoting the geodesic in Theorem 3.1 with c(0) = p and ċ(0) = v by cv . We obtain cv(t) = cv(t /) for   0, t  [0, ϵ]. In particular, cv is defined on [0, ϵ/]. Since cv depends smoothly on v and { v  TpM : ||v|| = 1} is compact, there exists ϵ0  0 with the property that, for ||v|| = 1, cv is defined at least on [0, ϵ0]. Therefore, for any w ϵ Mp with ||w||  ϵ0, cw is defined at least on [0,1]. Definition 2.2. Let M be a Riemannian manifold, p  M, Vp := {v ϵ TpM : cv is defined in [0,1]}, let expp: Vp  M be by defined expp (v) = cv (1). The map expp is called the exponential map of M at p. A connection is complete if every maximal geodesic is complete (has the form (t), - < t <  ).

Theorem. (Hopf-Rinow). Let M be a connected Riemannian manifold Theorem. (Hopf-Rinow) . Let M be a connected Riemannian manifold. Then the following conditions are equivalent. (i) M is a complete metric space. (ii) The Levi-Civita connection on M is complete. (iii) For some x ϵ M, expx is defined on all of TxM (iv) Every closed metric ball in M is compact. A proof of this fundamental theorem in Riemannian Geometry can be found in many books in differential geometry, like Lectures on Differential Geometry By S.S. Chern and others. In this book, the theorem is actually proved for a Finsler manifold.

Differential Geometry Dominic Leung 梁树培 Lecture 14