hypersurfaces in the general inner product spaces

Let a be a symmetric positive definite (n+ 1)×(n+ 1) real matrix for n ≥ 1 and s ∈ r n+1 be a hypersurface. we are supposed to determine the tangent space tps in an arbitrary point p isin; s in the case that the whole space r n+1 admits the inner product with matrix a. among other things, some maximum and minimum properties for the vector fields perpendicular to tangent spaces of hypersurfaces, the compatibility of the image or inverse image of a hypersurface and its tangent space under an embedding, an isometry, and a submersion are also pointed out.