Geodesic Flows       Gabriel P. Paternain

第一章 Introduction

1. Geodesic flow of a complete Riemannian manifold
   這裡有用E-L equation導出geodesic方程)。
   geodesic是曲線C 滿足能量E的Euler-Lagrange quation 因此測地線是energy functional 的critical point。[12 Geometric Analysis]
   一維的極小流形即測地線。
   所謂geodesic flow : a type of (parabolic)PDE for a geometric object such as a Riemann metric or an embedding。
   帶電黑洞中的測地流線 就像Ricci flow解決Poincare 猜想。
2. Symplectic manifolds and contact manifolds
3. The geometry of the tangent bundle  (Sasaki metric) [symplectic form] [contact form]
4. The cotangent bundle
5. Jacobi fields  and the differential of the geodesic flow
6. The asymptotic cycle and the stable norm
7. [Geodesic Flows] --- (1)2022夏令營  到 [Richard S. Hamilton]可以download Sir的Major publications

第二章 The Geodsic Flow acting on Lagrangian subspaces

1. Twist properties 這裡有幾個習作
2. Riccati equations
3. The Grassmannian bundle of Lagrangian subspaces
4. The Maslov index
5. The geodesic flow acting at the level Of Lagrangian subspaces
6. Continuous invariant Lagrangian subbundles in SM
7. Birkhoff's second theorem of geodesic flows

第三章 Geodsic arcs , Counting Functions and Topological Entropy

1. The counting functions
2. Entropies and Yomdin's theorem
3. Geodesic arcs and topological entropy
4. Manning's inequality
5. A uniform version of Yomdin's theorem

第四章 Mane's Formula for Geodesic Flows and Convex Billards

1. Time shifts that avoid the vertical
2. Mane's formula for Geodesic flows
3. Manifold without conjugate points
4. A formula for the topological entropy for manifolds of positive sectional curvature
5. Mane's theorem for conex billiards
6. Further results and problems on the subject

第五章 Topological Entropy and Loop Space Homology

1. Rationally elliptic and rationally hyperbolic manifolds
2. Morse theory of the loop space
3. Topological conditions that ensure positive entropy
4. Entropies of manifolds
5. Further results and problems on the subject

1. Riemannian manifolds with integrable geodesic flows      Andrew Miller