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Book 9A

Metrics, Connections and Curvature

Riemannian geometry, part one

Riemannian metrics and model spaces, the Levi-Civita connection, geodesics, the curvature tensors and what they mean, the Laplacian and Bochner formula, Jacobi fields and minimal surfaces.

  • Main companionIntroduction to Riemannian Manifolds · John M. Lee
  • Second voiceRiemannian Geometry · Peter Petersen
In preparation

Contents

  1. 1
    Riemannian Metrics and Model Spaces

    Spheres, hyperbolic space, warped products and the shape of the Earth.

    In preparation

  2. 2
    Connections

    How to differentiate a vector field, parallel transport, and the Foucault pendulum.

    In preparation

  3. 3
    Geodesics and the Exponential Map

    Shortest paths, normal coordinates, cut loci and the injectivity radius.

    In preparation

  4. 4
    Curvature and What It Means

    Riemann, sectional, Ricci and scalar curvature, and why tides are curvature.

    In preparation

  5. 5
    Computing Curvature

    Model spaces, necks, Lie groups and the curvature operator.

    In preparation

  6. 6
    The Laplacian and the Bochner Formula

    Calculus on Riemannian manifolds, spherical harmonics and commuting derivatives.

    In preparation

  7. 7
    Jacobi Fields and Curvature versus Topology

    Variations of length, gravitational lensing, Myers and Cartan–Hadamard.

    In preparation

  8. 8
    Submanifolds and Minimal Surfaces

    The second fundamental form, first and second variation of area, and soap films.

    In preparation

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