By Bibhutibhushan Datta

ISBN-10: 8170204607

ISBN-13: 9788170204602

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4). Let M 2 be a complete non-flat Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then the only entire solutions to the maximal surface equation (2) are the constant functions. 2 is obtained as a consequence of the following result, (we refer the reader to Ref. 2, Section 4 for a detailed proof). 3 (Ref. 1). Let M 2 be a (non necessarily complete) Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then any maximal surface Σ2 in M 2 × R1 which is complete with respect to the metric induced from the Riemannian product M 2 × R is totally geodesic.

Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann. 210 (1974) 163–170. 6. K. Sakamoto, Planer geodesic immersions, Tˆ ohoku Math. J. 29 (1977) 25–56. 7. T. Sugiyama, Totally geodesic K¨ ahler immersions in veiw of curves of order two, to appear in Geom. Dedicata. 8. T. Sugiyama and T. Adachi, Totally umbilic isometric immersions and curves of order 2, Monatsh. Math. 150 (2007) 73–81. Differential Geometry and its Applications Proc. , in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp.

We have a decomposition of p into a sum of Ad(K)-invariant and Ad(K)irreducible submodules: p = p1 ⊕ p2 ⊕ p(1,2) ⊕ p(1,3) ⊕ p(2,3) . 1 (Ref. 13). Assume that k1 , k2 , k3 ≥ 2, and at most one of k1 , k2 is equal to 2. Then there are no pairwise Ad(K)-isomorphic submodules among p1 , p2 and p(i,j) (1 ≤ i < j ≤ 3). 1 are satisﬁed, then we have a complete description of all Ad(K)-invariant metrics on G/H. Let ρ be any Ad(K)-invariant metric on G/H with corresponding Ad(K)-invariant inner product (·, ·) on p.

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