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The curvature and torsion functions along a space curve determine it up to an orientation-preserving isometry.

On a Riemannian manifold, there is a unique connection which is torsion-free and compatible with the metric. This connection is called the Levi-Civita connection.

Given a geodesic triangle (a triangle formed by the arcs of three geodesics on a smooth surface), int_(ABC)Kda=A+B+C-pi. Given the Euler characteristic chi, intintKda=2pichi, ...

A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers. An extrinsic curvature, on the other hand, is not ...

A necessary and sufficient condition for a curve to be a helix is that the ratio of curvature to torsion be constant.

A function giving the distribution of the interpoint distances of a curve. It is defined by p(r)=1/Nsum_(ij)delta_(r_(ij)=r).

A level set in three dimensions.

The plane spanned by the normal vector N and the binormal vector B.

The plane spanned by the three points x(t), x(t+h_1), and x(t+h_2) on a curve as h_1,h_2->0. Let z be a point on the osculating plane, then [(z-x),x^',x^('')]=0, where ...

If del xF=0 (i.e., F(x) is an irrotational field) in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x), ...