Invariants[f] allows one to calculate invariants of a mapping between manifolds (see Mapping) |

- f - mapping identifier.

- If mapping f is the
**embedding of a curve**then the**curve's normalized moving frame**and the**curve's curvatures**are calculated.

- If mapping f is an
**embedding**or**immersion**then the**second fundamental form**and**mean curvature vector**are calculated.

- If mapping f is a
**submersion**then the**A**and**T**invariants are calculated. In that case, some additional calculations are performed: the**mean curvature vector**of corresponding fibers, the**integrability obstruction**of corresponding horizontal distribution and the**riemannian obstruction**(if the submersion is not a riemannian one).

- The corresponding rules are as follows:

- Let mapping
**F:MN**be declared by functions (see Mapping):

- where
**m=dim(M), n=dim(N); {**x_{1}, x_{2}, ..x_{m}**}**are local coordinates on M and**{**y_{1}, y_{2}, ..y_{n}**}**are local coordinates on N.