Glossary of Riemannian and Metric Geometry - S

S

Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe the shape operator of a hypersurface,

It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.

Shape operator for a hypersurface M is a linear operator on tangent spaces, S_p: T_pM→T_pM. If n is a unit normal field to M and v is a tangent vector then

(there is no standard agreement whether to use + or − in the definition).

Short map is a distance non increasing map.

Smooth manifold

Sol manifold is a factor of a connected solvable Lie group by a lattice.

Submetry a short map f between metric spaces is called a submetry if for any point x and radius r we have that image of metric r-ball is an r-ball, i.e.

Sub-Riemannian manifold

Systole. The k-systole of M, is the minimal volume of k-cycle nonhomologous to zero.