Riemannian Geometry

The following sections are included:

• Tangent Space

  • Tangent vectors and inner product

  • Riemannian metric

  • Examples of metric tensor

• Covariant Derivative (Connection)

  • Definition

  • Time-dependent case

• Riemannian Connection

  • Definition

  • Christoffel symbol

• Covariant Derivative along a Curve

  • Derivative along a parameterized curve

  • Parallel translation

  • Dynamical system of an invariant metric

• Structure Equations

  • Structure equations and connection forms

  • Two-dimensional surface M²

  • Example: Poincaré surface (I)

• Geodesic Equation

  • Local coordinate representation

  • Group-theoretic representation

  • Example: Poincaré surface (II)

• Covariant Derivative and Parallel Translation

  • Parallel translation again

  • Covariant derivative again

  • A formula of covariant derivative

• Arc-Length

• Curvature Tensor and Curvature Transformation

  • Curvature transformation

  • Curvature tensor

  • Sectional curvature

  • Ricci tensor and scalar curvature

• Jacobi Equation

  • Derivation

  • Initial behavior of Jacobi field

  • Time-dependent problem

  • Two-dimensional problem

  • Isotropic space

• Differentiation of Tensors

  • Lie derivatives of 1-form and metric tensor

  • Riemannian connection ∇

  • Covariant derivative of tensors

• Killing Fields

  • Killing vector field X

  • Isometry

  • Positive curvature and simplified Jacobi equation

  • Conservation of 〈X, T〉 along γ(s)

  • Killing tensor field

• Induced Connection and Second Fundamental Form