Riemannian DNA, Inequalities and Their Applications

Bang-Yen Chen

doi:10.6180/jase.2000.3.3.01

Riemannian DNA, Inequalities and Their Applications

Mathematics / Statistics

Bang-Yen Chen This email address is being protected from spambots. You need JavaScript enabled to view it.¹

¹Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, USA

Received: March 1, 2000
Accepted: September 1, 2000
Publication Date: September 1, 2000

Download Citation: ||https://doi.org/10.6180/jase.2000.3.3.01

ABSTRACT

The main purpose of this survey article is to present the new type of Riemannian curvature invariants (Riemannian DNA) and the sharp inequalities, involving these invariants and the squared mean curvature, originally introduced and established in [7,8]. These Riemannian DNA affect the behavior in general of the Riemannian manifold and they have several interesting connections to several areas of mathematics. For instance, they give rise to new obstructions to minimal and Lagrangian isometric immersions. Moreover, these invariants relate closely to the first nonzero eigenvalue of the Laplacian on a Riemannian manifold. These invariants together with the sharp inequalities gives rise naturally to the notion of “ideal immersions” or the notion of “the best ways of living”. We also explain the physical meaning of the notion of ideal immersions for Riemannian manifolds in a Riemannian space form based again on the sharp inequalities.

Keywords: Riemannian invariants, squared mean curvature, tension, ideal immersion, best way of living, Riemannian DNA

REFERENCES

[1] Berger, M., “La geometrie metrique des varietes Riemnniennes,” Elie Cartan et les Mathematiques d’Aujourd’Hui, Asterisque, pp. 9-66 (1985).
[2] Blair, D. E., Dillen, F., Verstraelen, L. and Vrancken, L., “Calabi curves as holomorphic Legendre curves and Chen’s inequality," Kyunpook Math. J. Vol. 35, Jun, pp.407-416 (1995).
[3] Chen, B. Y., “On the total curvature of immersed manifolds I,” Amer. J. Math. Vol. 93, pp. 148-162 (1971).
[4] Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publ., Co. River Edge, NJ (1984).
[5] Chen, B. Y.,“Mean curvature and shape operator of isometric immersions in realspace-forms,” Glasgow Math. J. Vol. 38, pp. 87-97 (1996).
[6] Chen, B. Y., “Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension,” Glasgow Math. J. Vol. 41, pp. 33-41 (1999).
[7] Chen, B. Y., “Strings of Riemannian invariants, inequalities, ideal immersions and their applications,” Third Pacific Rim Geom. Conf., Intern. Press, Cambridge, MA, pp. 105-127 (1998).
[8] Chen, B. Y., “Some new obstructions to minimal and Lagrangian isometric immersions,” Japan. J. Math. Vol. 26, pp. 105-127 (2000).
[9] Chen, B. Y., “Ideal Lagrangian immersions in complex space forms,” Math. Proc. Cambridge Phil. Soc. Vol. 128, pp. 511-533 (2000).
[10] Chen, B. Y., “Riemannian submani-folds, ” Handbook of Differential Geometry, Vol. I, (North Holland Publ.) pp. 187-418 (2000). [11] Chen, B. Y., Dillen, F. and Verstraelen, L., “Conformally flat ideal hypersurfaces,” (preprint).
[12] Chen, B. Y., Dillen, F. Verstraelen, L. and Vrancken, L.,“Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces,” Proc. Amer. Math. Soc. Vol. 128, pp. 589-598 (2000).
[13] Chen, B. Y.and Vrancken, L., “CRsubmanifolds of complex hyperbolic spaces satisfying a basic equality,” Israel J. Math. Vol. 110, pp. 341-358 (1999).
[14] Chen, B. Y. and Yang, J., “Elliptic functions, theta function and hypersurfaces satisfying a basic equality,” Math. Proc. Cambridge Phil. Soc. Vol. 125, pp. 463-509 (1999).
[15] Chern, S. S., Minimal Submanifolds in a Riemannian Manifold, Univ. of Kansas, Lawrence, Kansas (1968).
[16] Dajczer, M. and Florit, L. A., “On Chen’s basic equality,” Illinois J. Math. Vol. 42, pp. 97-106 (1998).
[17] Defever, F., Mihai, I. and Verstraelen, L., “B. Y. Chen’s inequality for C-totally real submanifolds in Sasakian space forms,” Boll. Un. Mat. Ital. Ser. B Vol. 11, pp.365-374 (1997).
[18] Dillen, F., Petrovic, M., and Verstraelen, L., “Einstein, conformally flat and semisymmetric submanifolds satisfying Chen’s equality,” Israel J. Math. Vol. 100, pp. 163- 169 (1997).
[19] Dillen, F. and Vrancken, L., “Totally real submanifolds in 6-sphere satisfying Chen’s equality,” Trans. Amer. Math. Soc. Vol. 348, pp. 1633-1646 (1996).
[20] Gromov, M., “A topological technique for the construction of solutions of differential equations and inequalities,” Intern. Congr. Math. Nice 1970, Vol 2, pp. 221-225, (1971).
[21] Nagano, T., “On the minimum eigenvalues of the Laplacians in Riemannian manifolds,” Sci. Papers College Gen. Edu. Univ. Tokyo, Vol 11, pp. 177-182, (1961).
[22] Osserman, R., “Curvature in the eighties,” Amer. Math. Monthly, Vol 97, pp. 731-754 (1990).
[23] Sasahara, T., “CR-submanifolds in complex hyperbolic spaces satisfying an equality of Chen,” Tsukuba J. Math. Vol. 23, pp. 565- 583(1999).