Pavla Musilová

Differential invariants of immersions of manifolds with metric fields

The problem of finding all rth order differential invariants of immersions of manifolds with metric fields, with values in a left (G 1 m ×G 1 n )-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that rth order differential invariants depending on an immersion f:M → N of smooth manifolds M and N and metric fields on them can be factorized through metrics, curvature tensors and their covariant differentials up to the order (r−2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential of Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.

Natural operators of smooth mappings of manifoldswith metric fields

The concepts of both a natural bundle and a natural operator generalized for the case of the category of cartesian product of two manifolds and product of local diffeomorphisms are introduced. The existence of a bijective correspondence between k-th order natural operators and equivariant mappings of corresponding type fibers is proved. A basis of invariants of arbitrary order is constructed for natural operators of smooth mappings of manifolds endowed with metric fields, with values in a natural bundle of order one.