show/hide this revision's text 2 Corrected description of second variation

I don't particularly understand the point of trying to characterize curvature as the critical point or minimum of some functional, so let me answer the question differently.

Curvature arises naturally as the second derivative of an energy functional evaluated at a critical point as follows:

  • Fix two points on a Riemannian manifold and consider the following (standard) energy functional for curves joining the two points:

$E[\gamma] = \int_0^1 |\gamma'(t)|^2\,dt$

Note that the Riemannian structure is used to define the norm of the velocity vector.

  • It is well known that the critical points of $E$ are constant speed geodesics

  • It is also well known that if $\gamma$ is a critical point of $E$, E$ and $\gamma$ is deformed using a parallel vector field along $\gamma$, then the second variation of $E$ is simply the integral along $\gamma$ of the sectional curvature evaluated on the $2$-plane spanned by $\gamma'$ and the variation of $\gamma$.

So sectional curvature measures in a very precise way how geodesics behave when varied infinitesimally. This for me is the most concrete, direct, and useful way to understand what curvature is.

EDIT: Corrected description of second variation
show/hide this revision's text 1

I don't particularly understand the point of trying to characterize curvature as the critical point or minimum of some functional, so let me answer the question differently.

Curvature arises naturally as the second derivative of an energy functional evaluated at a critical point as follows:

  • Fix two points on a Riemannian manifold and consider the following (standard) energy functional for curves joining the two points:

$E[\gamma] = \int_0^1 |\gamma'(t)|^2\,dt$

Note that the Riemannian structure is used to define the norm of the velocity vector.

  • It is well known that the critical points of $E$ are constant speed geodesics

  • It is also well known that if $\gamma$ is a critical point of $E$, then the second variation of $E$ is simply the integral along $\gamma$ of the sectional curvature evaluated on the $2$-plane spanned by $\gamma'$ and the variation of $\gamma$.

So sectional curvature measures in a very precise way how geodesics behave when varied infinitesimally. This for me is the most concrete, direct, and useful way to understand what curvature is.