# Riemannian Geometry by Peter Petersen (auth.) By Peter Petersen (auth.)

Best differential geometry books

Gradient flows in metric spaces and in the space of probability measures

This booklet is dedicated to a thought of gradient flows in areas which aren't unavoidably endowed with a usual linear or differentiable constitution. It involves components, the 1st one referring to gradient flows in metric areas and the second dedicated to gradient flows within the area of chance measures on a separable Hilbert house, endowed with the Kantorovich-Rubinstein-Wasserstein distance.

Geometry from Dynamics, Classical and Quantum

This booklet describes, through the use of ordinary concepts, how a few geometrical buildings typical this present day in lots of components of physics, like symplectic, Poisson, Lagrangian, Hermitian, and so forth. , emerge from dynamics. it really is assumed that what will be accessed in real studies whilst learning a given approach is simply its dynamical habit that's defined by utilizing a relatives of variables ("observables" of the system).

Diffeology

Diffeology is the 1st textbook at the topic. it truly is aimed to graduate scholars and researchers  who paintings in differential geometry or in mathematical physics

Degenerate Complex Monge–Ampère Equations

Complicated Monge–Ampère equations were some of the most robust instruments in Kähler geometry seeing that Aubin and Yau’s classical works, culminating in Yau’s technique to the Calabi conjecture. A amazing program is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. in recent times degenerate complicated Monge–Ampère equations were intensively studied, requiring extra complex instruments.

Extra resources for Riemannian Geometry

Sample text

5 Curvature and the Riemannian Metric Let us list some important properties that we can easily derive. These properties will form the basis for many of our developments in comparison geometry. Comparison geometry is the study of how curvature inequalities influence the geometry and topology of the underlying manifold. In all of these results we assume that a smooth distance function is given on some open subset of a Riemannian manifold. Moreover, we start at some point p in this domain and consider what happens to the metric or Hessian along the integral curve for the gradient through this point.

The Hessian of a generic function cannot, of course, exhibit such predictable behavior (namely, being a solution to a PDE). It is only geometrically relevant functions that behave so nicely. n, can) we have arrived at a "new" result, that is, one that is not part of standard multivariable calculus. n, can), they all satisfy this same equation. This will become an important point later on. The second and third fundamental equations are also known as the Gauss equations and Codazzi-Mainardi equations, respectively.

On IRn the gradient is defined as v f = 8ij ai(f)aj = L;'=l ai (J)ai. But this formula depends on the fact that we used Cartesian coordinates. ) ae, because after change of coordinates, this does not equal ax (f) ax + oy (f) oy. , coordinate-independent description). One rule of thumb for items that are invariantly defined is that they should satisfY the Einstein summation convention, where one sums over identical super- and subscripts. 1 Connections 21 V f = o;(f)o; is not. The metric g = gijdxidxj and gradient V f = gijo;(f)aj are invariant expressions that also depend on our choice of metric.