By Peter Petersen (auth.)
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Extra resources for Riemannian Geometry
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.