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M2 is an oriented surface in E2t. If M2 is pseudoumbilical, the Lipschitz-Killing curvature takes maximum in the direction of mean curvature vector. The converse is also investigated. Furthermore assuming that M2 is closed, pseudo-umbilical and its Gaussian curvature has some nonnegative lower bound, M2 is completely determined by the M-index of M2. 1. Let M2 be an oriented Riemannian surface with an isometric immersion x:M2-*E2+N in a euclidean space E2?1N. Let F(M2) and F(E2+N) be the bundles of orthonormal frames of M2 and E2+AN respectively. Throughout this paper we assume that the mean curvature vector H of M2 is nowhere zero. Let B be the set of elements b= (p, ei, e2> * * *, e2+N) such that (p, el, e2) E F(M2), e3=HIIHI and that (x(p), el, e2, e3, e2+N) e F(E2+N) whose orientation is coherent with that of E2+N, identifying ei with dx(ei), i= 1, 2. Let x: B--+F(E2+N ) be the mapping naturally defined by x(b)=(x(p), el, * *, e21N) We have the differential forms wi, wci, (,ia w?ap (1? i,]j2, 3_o, i3?2+N) on B derived from the basic forms and the connection forms on F(E2+-N) through x as follows. 2+N dx = woel + c2e2, deA = 2 coBeB, (DAB = -.')B,l B==1 (A, B = 1, 2, *,2 + N); 2 0 = E Aaijco)i AXij = Aajij=l In the following, for the summation notations >S, 'S' and we mean ,2 andN 2+?N for the indices i, a, /, r, t we mean _

Surfaces with maximal Lipschitz-Killing curvature in the direction of mean curvature vector