Geometry, Algebra and Applications: From Mechanics to Cryptography by Marco Castrillón López Luis Hernández Encinas Pedro Martínez Gadea & Mª Eugenia Rosado María

Author:Marco Castrillón López, Luis Hernández Encinas, Pedro Martínez Gadea & Mª Eugenia Rosado María
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where and . The Weingarten operator A is a -bilinear map and the normal connection is a connection in the normal bundle .

Moreover, if are two normal vector fields, then

which shows that the normal connection is metric for the fibre metric in the normal bundle .

A normal vector field is said to be parallel respect to if .

Let be a vector bundle with a connection D, compatible with a metric on E. The parallel transport induced by D defines an isometry between any two different fibres of (see [18] for details). Thus, the norm of a parallel section remains constant and the angle between two parallel sections also remains constant. In the present case of having a submanifold M of a Riemannian manifold , the vector bundle we are considering is the normal bundle, and the metric is the restriction of to normal vectors.

Many results about curves in Riemannian manifolds have been obtained in the past. We would like to point out that generalizations of Frenet frames have been obtained in [13] and [9], in [16] for the case of spaces of constant curvature, and in [17] for the case of the Minkowski space. Besides in [4] some results about the total curvature of a curve in a Riemannian manifold are also obtained.

The aim of the present note is not to show Frenet frames in Riemannian manifolds, but RM frames. Nevertheless, I would like to comment the very beautiful main results of the paper [16]: (1) a Frenet theorem: two curves in the Euclidean, Spherical or Hyperbolic space are congruent if and only if their curvatures are equal, and (2) the converse: Frenet’s theorem holds for curves in a connected Riemannian manifold (M, g) if and only if (M, g) is of constant curvature. Thus, one cannot expect to extend this Theorem to other Riemannian manifolds.

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