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# An introduction to tensor analysis by Leonard Lovering Barrett

By Leonard Lovering Barrett

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We have the following relation between total mean curvature and knot number p. 4 (Wintgen 1978). surface into E4. Let f:M Then we have a2dV > 4Trp . For the proof see Wintgen (1978). 96. 1. , the product surface of two plane circles of the same radius, into Em has total mean curvature >27r2. For arbitrary flat surfaces we have the following best possible result. 1 (Chen 1981). Let M be a closed flat surface. 1) a2dV > 2Tr2. if and only if M is imbedded in an affine 4-space of Em The equality holds as a Clifford torus.

8. A compact symmetric space M is an equal-antipodal-pair space if and only if M is either a rank one symmetric space or one of the following spaces G2, GI, and EN . Any isometric totally geodesic imbedding f: B ---M gives rise to a mapping P(f) : P(B) -}P(M) induced by the mapping carring (o,p,B+(p),B_(p)) into (f(o),f(p), M+(f(p)),M_(f(p))). 5 that f(B+(p))C M+(f(p)) totally geodesic submanifolds. 9. It is easy f(B_(p))C M (f(p)) The later one follows from (ii-c). this is an important fact, we express totally geodesic immersion.

Throughout this chapter, N is assumed to be a CR-submanifold of a Kaehler manifold M unless mentioned otherwise. 1. 3) AJEX = -AEJX. and This follows from the indentities, JVUZ + J h (U, Z) = -A JZU + DUfZ and = . For a CR-submanifold N, PR°1= {0}. 1, we obtain the following fundamental result for CR-submanifolds. 2. The totally real distribution r1 of a CR-submanifold of a Kaehler manifold is always integrable. This theorem was generalized to CR-submanifolds in a locally conformal almost Kaehler manifold by Blair and Chen (1979).