STM Article Repository

So far, higher order Gaussian curvatures of hypersurfaces have been introduced by using principal curvatures. The principal curvatures are the eigenvalues of the matrix of the second fundamental form in an orthonormal basis of the tangent space. The principal directions are the corresponding eigenvectors. Normal curvatures, principal curvatures for hypersurface and the relevant higher order Gaussian curvatures are invariants independent of the choice of coordinates.

In this paper we introduce new definitions that are higher order Gaussian curvatures of submanifold and higher order Gaussian curvatures of parallel submanifold. (n −1)− dimensional submanifolds of Riemannian n −manifold are hypersurfaces. (n −1)− dimensional hypersurface of n − dimensional manifold has only one normal space and so has one shape operator. When higher order Gaussian curvatures of hypersurfaces are calculated, the calculations are made out of a single shape operator. But when calculating higher order Gaussian curvatures of k −dimensional submanifold in n −dimensional manifold, the calculations are made out of (n − k) shape operators. We define them by taking into account and using their principal curvatures. We present relationship between higher order Gaussian curvatures of submanifold and provide higher order Gaussian curvatures of parallel submanifold.