Abstract
Bianchit has generalized the study of a surface in eucidean 3-space by considering a hypersurface, V^, immersed in an arbitrary enveloping space of one more dimension. Associated with such a hypersurface there are two differential quadratic forms, which in this paper we denote by ga#duadu# and Q,a#duadu#, where the u's (a, ,B =1, 2, * * *, n) are coordinates in V,. These forms, known as the first and second fundamental forms of the hypersurface respectively, are defined precisely as are the two fundamental forms of a surface in ordinary 3-space; the first in the case of a positive definite form gives the square of the element of the arc,
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