The sectional curvature at a point 
of a Riemannian manifold 
assigns a real
number 
to each subspace 
of dimension 2 in the tangent space 
. If 
and 
are linearly independent
and span 
,
then
 |
where 
is the Riemann tensor and the angle brackets denote
the Riemannian metric. The denominator
is the squared area of the parallelogram
spanned by 
and 
,
so the value depends only on the two-plane 
and not on the choice of 
and 
.
On a Riemannian manifold of dimension 2, there is only one tangent two-plane at each point, and its sectional curvature is the Gaussian curvature. In higher dimensions, the values over all tangent two-planes determine the Riemann tensor. They also control the relative acceleration of nearby geodesics through the geodesic-deviation equation.
The Hopf conjecture gives two prominent global restrictions on sectional curvature. Its sign form predicts that a compact manifold of even dimension equipped with a Riemannian metric of positive sectional curvature has positive Euler characteristic. Its product form predicts that the direct product of two closed manifolds of positive dimension admits no Riemannian metric with positive sectional curvature.
