Geometry of null curves. (English) Zbl 0956.53015

From the introduction: The theory of space curves of a Riemannian manifold is fully developed and its local and global geometries are well-known. In case \(M\) is proper semi-Riemannian there are three categories of curves: spacelike, timelike, and null, depending on their causal character. Since the induced metric on a null curve is degenerate, this case is much more complicated and also different from the nondegenerate ones. The subject of this paper is to study null curves in semi-Riemannian manifolds of index two. We show that it is possible to construct two types of Frenet frames, each invariant under any causal change. This is then followed by a deviation of the general Frenet equations.


53B25 Local submanifolds
53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics