Sectional curvature: Difference between revisions

Revision as of 04:36, 7 March 2007

This article defines a notion of curvature for a differential manifold equipped with a Riemannian metric

Definition

Given data

A Riemannian manifold $(M, g)$ viz a differential manifold $M$ equipped with a Riemannian metric $g$ .

Definition part

Let $Π$ be a tangent plane to $M$ at a point $p \in M$ . Then, the sectional curvature of $Π$ at $p$ is defined as follows: take two linearly independent vectors $X$ and $Y$ in $Π$ , and calculate:

$< R (X, Y) Y, X >$

viz the inner product of $R (X, Y) Y$ and $X$ with respect to $g$ .

Divide this by the square of the area of the parallelogram formed by $X$ and $Y$ . This ratio defines the sectional curvature of $Π$ , denoted as $K (Π)$ .

Related notions

Related notions of curvature

Ricci curvature associates a real number to every tangent direction at every point
Scalar curvature assocaites a real number to every point

Related metric properties

@@ Line 13: / Line 13: @@
 <math><R(X,Y)Y,X></math>
-viz the inner product of <math>R(X,Y)Y</math> and <math>X</math> with respect to <math>g</math>. This defines the sectional curvature.
+viz the inner product of <math>R(X,Y)Y</math> and <math>X</math> with respect to <math>g</math>.
-''This doesn't sound right. Check it''
+Divide this by the square of the area of the parallelogram formed by <math>X</math> and <math>Y</math>. This ratio defines the sectional curvature of <math>\Pi</math>, denoted as <math>K(\Pi)</math>.
 ==Related notions==