Computations of framing invariants of multi-dimensional bands on submanifolds \(F^ m\) of spaces \(V^ n(K)\), \(n>m\), of constant curvature \(K\) (Q1357936)

Let \(V^n(K)\) be a Riemannian manifold of constant curvature \(K\), \(F^m\) a smooth submanifold, and \(P^\perp(F^m)\) the bundle of orthogonal frames in vector subspaces normal to \(F^m\). A curve in \(F^m\) together with a section of \(P^\perp(F^m)\) on it is called a band on \(F^m\). This band is complemented to the section of \(P(F^m)\oplus P^\perp(F^m)\) by the tangent, principal normal, binormal, etc. unit vectors of this curve (with respect to the Levi-Civita connection \(\nabla\) of \(F^m\)). The corresponding Frenet formulas are deduced. The main results give the expression of the invariants for a band under changes of the section of \(P^\perp(F^m)\). In these expressions the normal components in these Frenet formulas (called normal curvature vectors) and their covariant derivatives in the normal connection are used.