Published online by Cambridge University Press: 22 January 2016
Let M be an n-dimensional manifold immersed in an m-dimensional euclidean space Em and let ∇ and ∇̃ be the covariant differentiations of M and Em, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by
(1.1) ∇̃XY = ∇XY + h(X,Y).
This work was partially supported by NSF Grant GP-36684.
A partial result of this paper was announced in the following article “Some integral inequalities of two geometric invariants” appeared in Bull. Amer. Math. Soc. 81 (1975), 177-178.
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