In order to correlate the

tangent space with the Grassmann manifold, we review a theorem about geodesics and horizontal lifts [1].

for any v [member of] TxS, and hence grad(f) = d[f.sup.#] is the projection of grad(F) = d[F.sup.#] on the

tangent space [T.sub.x]S.

q [member of] G is an element of the Lie group, and its

tangent space is [T.sub.q]G.

For the manifold [S.sub.n], the horizontal

tangent space at x [member of] [S.sub.n] is [H.sub.x][S.sub.n] = [span.sub.R]{Re Z(x), Im Z(x)}.

where X*, Y* [member of] [chi](M*) form an orthogonal basis {X*([alpha]*(t*)), Y *([alpha]*(t*)) of a

tangent space at each point [alpha]*(t*) of M*.

Let {[e.sub.i]}, i = 1, 2, ..., n be an orthonormal basis of the

tangent space at any point.

Another important concept is the

tangent space to a manifold in a certain point, which is basically a first-order (vector space) approximation of the manifold at this point.

Let [M.sup.n] be a complete hypersurface and {[e.sub.1],...,[e.sub.n]} an orthonormal basis of the

tangent space [T.sub.x](M) at a point x [member of] M such that A[e.sub.i] = [[lambda].sub.i][e.sub.i], 1 [??] i [??] n.

For any vector field X of [??], we define a mapping J from the

tangent space of [??] onto itself is given by JX = [J.sub.m]PX + [J.sub.n]QX, then we see that

The components of the metric tensor g ([x.sub.N]) = [[eta].sub.ik]d[X.sup.i] [cross product] d[X.sup.k] describing the locally flat

tangent space [T.sub.x](M) of rigid frames at a point [x.sub.N] = [x.sub.N] ([x.sup.a]) are given by

The normal map image is usually blue because the normals are captured in

tangent space. Later on, using a DOT3 texture operation we are able to render the low resolution model and make it look like the high quality one.

By finite type m we mean that the

tangent space of M at the origin is spanned by commutators of length m of sections of [T.sup.1,0]M [symmetry] [T.sup.0,1]M and it is not spanned by commutators of length at most m - 1.