By Bang-yen Chen

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DUP) Z = 0. Bejancu informed me that he also obtained this proposition independently. In Bejancu, Kon and Yano (1980), they proved that if 17P= 0 and N is anti- By an anti-holomorphic submanifold, we holomorphic, then N is a CR-product. mean a CR-submanifold with Jy-( = T 1N. 2. 3) r< 4h2 + 4h + p2 - p The equality holds if and only if (a) N Zies in a totaZZy geodesic C Ph+p+hp of CFJ, (b) ZocaZZy, N is the Riemannian product of a totaZZy geodesic, complex submanifold C Ph and a totaZZy geodesic, totaZZy real submanifold R PP and (c) the immersion in induced from the Segre imbedding.

M+(p) is a totally geodesic submanifold of the symmetric space M= G/H. 2 The normal space Tp M+ (p) of M (p) at p in M is the tangent space of a complete connected totaZZy geodesic submanifold M_(p). It is well-known that every complete totally geodesic submanifold of a symmetric space is a symmetric space. For a symmetric space M the dimension of a maximal flat totally geodesic submanifold of M is called the rank of M, denoted by rk M. From the equation of Gauss, it follows that rk B< rk M if B is totally geodesic in M.

15). 5 (Chen 1979a). 16). 2. Lawson (1970) showed that for any positive integer g, there is a closed minimal surface m in the unit 3-sphere S3 such that the genus of M is g and the area of M is <87r. From this fact, we know that for any closed oriented surface M, there exists an immersion from M into E3 whose total mean curvature is < 87r. 6. Let M be a closed surface which is not homeomorphic to 29 22 S , RP or S1 xS1. 3. The relations between total mean curvature and the theory of variations have been studied in Chen (1972e, 1973a), Willmore and Jhaveri (1972).

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Geometry of submanifolds and its applications by Bang-yen Chen


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