]> The Curvature of the Torus

The Curvature of the Torus

Let's get bent

We begin with the parameterized surface:

[torus with coordinate axes]
x u v = x = c + a cos v cos u y = c + a cos v sin u z = a sin v

Take the partial derivatives of this parameterization and compute inner products to find the coefficients of the first fundamental form: E = c + a cos v 2 , F = 0 , G = a 2 . This gives us the line element ds 2 = c + a cos v 2 du 2 + a 2 dv 2 , from which we read off the metric:

g ij = c + a cos v 2 0 0 a 2

Some straightforward and boring computation yields the nonzero Christoffel symbols of the second kind:

Γ uv u = Γ vu u = a sin v c + a cos v Γ uu v = 1 a sin v c + a cos v

Another two pages of index juggling and basic algebra gives the nonzero components of the Riemann tensor:

R     vuv u = R     vvu u = a cos v c + a cos v R     uvu v = R     uuv v = 1 a cos v c + a cos v

Contract to get the Ricci tensor:

R ij = R     imj m = 1 a cos v c + a cos v 0 0 a cos v c + a cos v

Finally, contract with the upper form of the metric to get the Ricci scalar (a.k.a. the curvature scalar):

R = g ij R ij = 2 cos v a c + a cos v

The result is twice the Gaussian curvature, as expected.

What does the Gaussian curvature tell us about the torus? Since c > a the denominator is always positive, so the sign of the curvature is determined only by cos v. The illustration shows regions of different curvature: on the outside of the torus curvature is positive (blue), on the inside it's negative (red), and at the top and bottom circles it's zero (grey).

[torus showing regions of different curvature]

Understanding the torus's curvature will help us in our search for the torus's geodesics.

Last updated 25 April 2005
All contents released into the public domain by Mark L. Irons