Immersions in ℝn

Suppose f: M → ℝn is a smooth immersion of an m-dimensional manifold M in ℝn, 1 ≤ m ≤ n. If (p1,...,pm) are local coordinates in M, then the vectors f _i = ∂f/∂pi constitute a basis in the tangent m-dimensional plane T to (M, f) at the point p. When n > m we can choose an orthonormed basis (v ₁ ,..., v _n−m ) in the (n − m)-dimensional orthogonal complement N to T. Each unit normal v ∈ N can be assigned the second fundamental form of the m-dimensional surface (M, f) by using the equality II (v) = <d2f, v>. Denote mH _j = trII(v _j ) = k_{1 j} +···k _mj , where tr is the trace of the form; k_{1 j} ,···,k _mj are the principal curvatures of (M, f) with respect to the normal v _j , i.e. the eigenvalues of II(v _j ). The vector 1$$H = {H^j}{v_j}$$ (the sum being taken over j from 1 to n − m) does not depend on the choice of orthonormed basis {v _j } in N. This vector H is said to be the mean curvature vector of the m-dimensional surface (M, f) at the point p ∈ M and its norm 2$$|H| = \frac{1}{m}\sqrt {\sum {_{j = 1}^{n - m}{{({k_{1j}} + \cdots + {k_{mj}})}^2}} } $$ is the absolute mean curvature.