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For any prime number p and any positive integer n, the p^n-rank r_(p^n)(G) of a finitely generated Abelian group G is the number of copies of the cyclic group Z_(p^n) ...

The set of rules for manipulating and calculating with tensors.

C=tauT+kappaB, where tau is the torsion, kappa is the curvature, T is the tangent vector, and B is the binormal vector.

The indices of a contravariant tensor A^j can be lowered, turning it into a covariant tensor A_i, by multiplication by a so-called metric tensor g_(ij), e.g., g_(ij)A^j=A_i.

The indices of a covariant tensor A_j can be raised, forming a contravariant tensor A^i, by multiplication by a so-called metric tensor g^(ij), e.g., g^(ij)A_j=A^i

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

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

If two single-valued continuous functions kappa(s) (curvature) and tau(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for ...

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.

For a smooth harmonic map u:M->N, where del is the gradient, Ric is the Ricci curvature tensor, and Riem is the Riemann tensor.