curvature

Tags: #geomtop/Riemannian-geometry #physics #geomtop/differential-geometry Refs: connection curvature Lie algebra valued form

curvature

Definition

Given a connection \(\nabla\) on \(E\searrow M\) a Riemannian manifold the curvature form is a 2-form \(F_\nabla\) with values in \({ \operatorname{End} }(E) \cong E {}^{ \vee }\otimes E\), i.e. \begin{align*} F_\nabla \in \Omega^2_{M}({ \operatorname{End} }E) \cong {{\Gamma}\qty{ { {\bigwedge}^{\scriptscriptstyle \bullet}} ^2 {{\mathbf{T}}M} \otimes{ \operatorname{End} }E} } \qquad F_\nabla(X, Y)({-}) = \qty{ [\nabla_X, \nabla_Y] -\nabla_{[X, Y]}} ({-}) \end{align*}

The covariant exterior derivative satisifes \(d_\nabla^2 s = F_\nabla \wedge s\) for \(s\in { { {\Omega}^{\scriptscriptstyle \bullet}} }_M(E)\) an \(E{\hbox{-}}\)valued form, and thus \(d^2=0\) when the curvature form vanishes. This yields a flat bundle.

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Cartan’s Structure Equation

If \(\nabla = A\) locally as a matrix of 1-forms, then \(F_\nabla = dA + A{ {}^{ \scriptscriptstyle\wedge^{2} } }\) or \(F_\nabla = dA + {1\over 2}[A, A]\).

Other types

Types of curvature:

  • Riemannian curvature
  • Gaussian curvature
    • Sectional curvature: defined in terms of the Riemannian curvature, and determines the Riemannian curvature completely. Essentially the Gaussian curvature of the geodesic surface with the same tangent at \(p\), i.e. the image of \(\exp_p\).

How these are related: recover Ricci curvature by contracting Unsorted/Riemannian curvature, and you get scalar curvature by taking the trace of Ricci curvature.

Misc

  • Can define parallel vector fields as \(\nabla X = 0\), a PDE.
    • Don’t generally exist, this is an overdetermined equation. The integrability condition for this equation is equivalent to \(\mathop{\mathrm{Curv}}(\nabla) = 0\).
    • If curvature vanishes, parallel transport along every curve can be used to define parallel vector fields on \(M\).

p Curvature

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#geomtop/Riemannian-geometry #physics #geomtop/differential-geometry