Ngô Quốc Anh

November 26, 2009

R-G: Sectional curvature

Filed under: Riemannian geometry — Ngô Quốc Anh @ 14:40

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

Definition. The sectional curvature of the plane spanned by the (linearly independent) tangent vectors $X, Y \in T_xM$ of the Riemannian manifold $M$ is

$\displaystyle K\left( {X,Y} \right) = \frac{{\left\langle {R\left( {X,Y} \right)Y,X} \right\rangle }}{{\left\langle {X,X} \right\rangle \left\langle {Y,Y} \right\rangle - {{\left\langle {X,Y} \right\rangle }^2}}}$.

In local coordinates, if

$\displaystyle X = {X^i}\frac{\partial }{{\partial {x^i}}}, \quad Y = {Y^j}\frac{\partial }{{\partial {x^j}}}$

we then have

$\displaystyle R\left( {X,Y} \right)Y = {X^i}{Y^j}{Y^k}R\left( {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^j}}}} \right)\frac{\partial }{{\partial {x^k}}} = {X^i}{Y^j}{Y^k}R_{kij}^l\frac{\partial }{{\partial {x^l}}}$

which implies

$\displaystyle\begin{gathered} \left\langle {R\left( {X,Y} \right)Y,X} \right\rangle = {X^i}{Y^j}{Y^k}R_{kij}^l\left\langle {\frac{\partial }{{\partial {x^l}}},{X^m}\frac{\partial }{{\partial {x^m}}}} \right\rangle \hfill \\ \qquad= {X^i}{Y^j}{X^m}{Y^k}R_{kij}^l{g_{lm}} \hfill \\ \qquad= {R_{mkij}}{X^i}{Y^j}{X^m}{Y^k} \hfill \\ \qquad = {R_{ijmk}}{X^i}{Y^j}{X^m}{Y^k}. \hfill \\ \end{gathered}$

Besides

$\displaystyle\begin{gathered} \left\langle {X,X} \right\rangle \left\langle {Y,Y} \right\rangle - {\left\langle {X,Y} \right\rangle ^2} = {X^i}{X^m}{g_{im}}{Y^j}{Y^k}{g_{jk}} - {\left( {{X^\alpha }{Y^\beta }{g_{\alpha \beta }}} \right)^2} \hfill \\ \qquad= {X^i}{X^m}{g_{im}}{Y^j}{Y^k}{g_{jk}} - {X^\alpha }{Y^\beta }{g_{\alpha \beta }}{X^\gamma }{Y^\delta }{g_{\gamma \delta }} \hfill \\ \qquad= \left( {{g_{im}}{g_{jk}} - {g_{ij}}{g_{mk}}} \right){X^i}{X^m}{Y^j}{Y^k}. \hfill \\\end{gathered}$

Thus

$\displaystyle K\left( {X,Y} \right) = \frac{{{R_{ijmk}}{X^i}{Y^j}{X^m}{Y^k}}}{{\left( {{g_{im}}{g_{jk}} - {g_{ij}}{g_{mk}}} \right){X^i}{X^m}{Y^j}{Y^k}}}$.

To be exact, without using Einstein summation convention, one reads the above identity as following

$\displaystyle K\left( {X,Y} \right) = \frac{{\sum\limits_{ijmk} {{R_{ijmk}}{X^i}{Y^j}{X^m}{Y^k}} }}{{\sum\limits_{ijmk} {\left( {{g_{im}}{g_{jk}} - {g_{ij}}{g_{mk}}} \right){X^i}{X^m}{Y^j}{Y^k}} }}$.

We refer the reader to this topic for examples. In addition, if we choose

$\displaystyle {g_{ij}} = {\left( {\displaystyle\frac{2}{{1 - {{\left| y \right|}^2}}}} \right)^2}{\delta _{ij}}$

then the sectional curvature of $g$ is $-1$.