The partial positivity of the curvature in Riemannian symmetric spaces

Abstract

In this paper, the partial positivity (resp., negativity) of the curvature of all irreducible Riemannian symmetric spaces is determined. From the classifications of abstract root systems and maximal subsystems, the author gives the calculations for symmetric spaces both in classical types and in exceptional types.

Keywords

Partial positivity / Symmetric space / Semi-simple Lie algebra

References

[1] Freudenthal H., de Vris H.. Linear Lie groups, 1969, New York: Academic Press

[2] Frankel T.. Manifolds with positive curvature. Pacific Jour. Math., 1961, 11: 165-174

[3] Frankel T.. On the fundmental group of minimal submanifolds. Ann. of Math., 1966, 83: 68-73

[4] Kenmotsu K., Xia C. Y.. Intersections of minimal submanifolds in Riemannian manifold with partially positive curvature. Kodai Math. J., 1995, 18: 242-249

[5] Kenmotsu K., Xia C. Y.. Hadamard-Frankel type theorems for manifolds with partially positive curvature. Pacific Journal Math., 1996, 176: 129-139

[6] Knapp A. W.. Lie Groups, Beyond an Introduction, 2002 Second Edition Boston: Birkhauser

[7] Wallach N. R.. On maximal subsystems of root systems. Canad. J. Math., 1968, 20: 555-574

[8] Helgason S.. Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, 2001, Providence, RI: A. M. S.

[9] Lee N.. On the lower bound of the mean curvature of constant mean curvature hypersurface in non-compact symmetric spaces and manifolds with a pole. Tohoku Math. Jour., 1995, 47(2): 499-508

[10] Lee N.. Determination of the partial positivity of the curvature in symmetric spaces. Ann. di Matim. Pura ed Appl., 1996, 171: 107-129

[11] Samelson H.. Notes on Lie Algebras, 1989, New York: Universitext, Springer-Verlag

[12] Shen Z.. On complete manifolds of nonnegative kth-Ricci curvature. Trans. Amer. Math. Soc., 1993, 338: 289-310

[13] Wu H.. Manifolds of partially positive curvature. Indiana Univ. Math. J., 1987, 36: 525-548