285G, Lecture 1: Flows on Riemannian manifolds

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In the first lecture, we introduce flows $latex t mapsto (M(t), g(t))$ on Riemannian manifolds $latex (M,g)$, which are recipes for describing smooth deformations of such manifolds over time, and derive the basic first variation formulae for how various structures on such manifolds (e.g. curvature, length, volume) change by such flows. (One can view these formulae as describing the relationship between two “infinitesimally close” Riemannian manifolds.) We then specialise to the case of Ricci flow (together with some close relatives of this flow, such as renormalised Ricci flow, or Ricci flow composed with a diffeomorphism flow). We also discuss the “de Turck trick” that modifies the Ricci flow into a nonlinear parabolic equation, for the purposes of establishing local existence and uniqueness of that flow.

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