An Introduction to Riemannian Geometry With Applications to Mechanics and Relativity Presented by Mehdi Nadjafikhah © URL: webpages.iust.ac.ir/m_nadjfikhah Email: m_Nadjafikhah@iust.ac.ir

Contents Chapter 1 - Differentiable Manifolds 1.1 Topological Manifolds 1.2 Differentiable Manifolds 1.3 Differentiable Maps 1.4 Tangent Space 1.5 Immersions and Embeddings 1.6 Vector Fields 1.7 Lie Groups 1.8 Orientability 1.9 Manifolds with Boundary

Contents Chapter 2 - Differential Forms 2.1 Tensors 2.2 Tensor Fields 2.3 Differential Forms 2.4 Integration on Manifolds 2.5 Stokes Theorem 2.6 Orientation and Volume Forms

Contents Chapter 3 - Riemannian Manifolds 3.1 Riemannian Manifolds 3.2 Affine Connections 3.3 Levi-Civita Connection 3.4 Minimizing Properties of Geodesics 3.5 Hopf-Rinow Theorem

Contents Chapter 4 – Curvature 4.1 Curvature 4.2 Cartan Structure Equations 4.3 Gauss–Bonnet Theorem 4.4 Manifolds of Constant Curvature 4.5 Isometric Immersions

Contents Chapter 5 - Geometric Mechanics 5.1 Mechanical Systems 5.2 Holonomic Constraints 5.3 Rigid Body 5.4 Non-holonomic Constraints 5.5 Lagrangian Mechanics 5.6 Hamiltonian Mechanics 5.7 Completely Integrable Systems 5.8 Symmetry and Reduction

Contents Chapter 6 – Relativity 6.1 Galileo Spacetime 6.2 Special Relativity 6.3 The Cartan Connection 6.4 Relativity 6.5 Galileo Spacetime 6.6 Special Relativity 6.7 The Cartan Connection