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Book 8A

Smooth Manifolds

The machinery Ricci flow is written in

Charts, tangent spaces, Lie groups, flows and Lie derivatives, tensors and index notation, forms and Stokes, and the curvature of surfaces.

  • Main companionIntroduction to Smooth Manifolds · John M. Lee
  • OptionalDifferential Geometry of Curves and Surfaces · Manfredo do Carmo
In preparation

Contents

  1. 1
    Smooth Structures

    Charts, atlases and smooth maps, and why dimension 3 has only one smooth structure.

    In preparation

  2. 2
    Partitions of Unity

    Gluing local constructions, and why every manifold carries a metric.

    In preparation

  3. 3
    Tangent Vectors and Bundles

    Velocities, differentials, and the tangent and cotangent bundles.

    In preparation

  4. 4
    Submanifolds

    The rank theorem, regular level sets and configuration spaces.

    In preparation

  5. 5
    Lie Groups and Group Actions

    Rotations, quaternions and the groups that carry Thurston’s geometries.

    In preparation

  6. 6
    Flows and the Lie Derivative

    Vector fields, their flows, brackets, and the diffeomorphism invariance behind solitons and DeTurck’s trick.

    In preparation

  7. 7
    Tensors and Index Notation

    Stress, diffusion and the notation every Ricci flow paper uses.

    In preparation

  8. 8
    Differential Forms and Stokes’ Theorem

    Integration on manifolds and integration by parts, the computation behind every monotonicity formula.

    In preparation

  9. 9
    The Curvature of Surfaces

    Principal curvatures, Gauss’s Theorema Egregium and Gauss–Bonnet.

    In preparation

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