chapter  5
20 Pages

Submanifolds in Riemannian Space

Let us be given an n-dimensional Riemannian manifold R 11 endowed with the metric dl 2 = gudx; dxi .

We say that the Riemannian manifold is isometrically immersed into V 111 as a submanifold F 11 if the length of any curve in F" being evaluated with the metric d/2 is equal to the length of the same curve being evaluated with the metric ds 2 of ambient space. This means that for any infinitesimal motion, the differential dx; and corresponding differentials dya are related by

d "'d (3-d id ,j aa(j Y Y - g ij X X . Since y"' are the functions of xi, then dyet = ~: dxi . Comparing the coefficients of two forms we obtain

As ya are the functions on F 11 , their usual derivatives are covariant one. Hence the latter equation can be rewritten as

&". {&"} As dya = ~; dx', the values ~; with i fixed form the components of vectors on vm tangent to coordinate curves x' in F". Their linear combinations form the tangent space Tx. Let ~/3 be the components of a vector normal to F", i.e.