Some Curvature Characterizations for Hessian Manifolds with Constant Hessian Sectional Curvature

Year 2017, Volume: 7 Issue: 1, 259 - 265, 31.03.2017

Münevver Yıldırım Yılmaz

Abstract

Let N be a compact hypersurface in the Hessian manifold . We frstly introduce
Hessian manifold concept then using second fundamental tensor, we obtain some curvature characterizations which
are new for Hessian manifolds with constant Hessian sectional curvature

Keywords

Hessian manifold, sectional curvature, second fundamental form

Sabit Hessian Kesit Eğrilikli Hessian Manifoldları için Bazı Eğrilik Karakterizasyonları

Abstract

N , Hessian manifoldunun kompakt bir hiperyüzeyi olsun. Önce Hessian manifold konsepti
hakkında bilgi vereceğiz sonra ikinci temel tensörü kullanarak sabit Hessian kesit eğrilikli Hessian manifoldları
için bazı eğrilik karakterizasyonaları elde edeceğiz

Keywords

Hessian manifoldu, kesit eğriliği, ikinci temel form.

References

• Bektas M¸ Yıldırım M, 2006. Integral inequalities for submanifolds of Hessian manifolds with constant Hessian sectional curvature. Irania. Jour. of Scien. and Techno. Transact. A, 30(2):235-239.
• Camargoa FEC, Chavesa RVB, Sousa Jr LAM, 2008. Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space. Differential Geometry and its Applications. 26(6): 592-599. Caminha A, 2006. A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds. Diff. Geom. and its Applic.. 24(6):652—659.
• Colares AG, De Lima HF, 2002. Some rigidity theorems in semi-Riemannian warped products, Kodai Mathemat. J. 35(2): 268-282.
• Ho PT, 2015. Differentiable rigidity of hypersurface in space forms. Manuscripta mathem. 146: 463-472.
• Kobayashi, S, Nomizu K, 1969. Foundations of differential geometry, Vols. I, II, Inter science, N. York.
• Li, H, 1997. Global rigidity theorems for hypersurfaces. Arkiv Math. 35: 327-351.
• Shima H, 1986. Vanishing theorems for compact Hessian manifolds. Annals Institute. Fourier Grenob.. 36(3):183-205.
• Shima H, 1995. Hessian manifolds of constant Hessian sectional curvature. Journal of Mathematical Society of Japan. 47(4): 735-753.
• Shima H, 2007. The Geometry of Hessian structures. World Scien. Publ., First edition Singapore, Japan. 246p.
• Shima H, Yagi, K,. 1997. Geometry of Hessian manifolds, Differential Geometry and its Applications. 7(3): 277–290.
• Yilmaz MY, Bektas¸M, 2008. A survey on curvatures of Hessian manifolds. Chaos, Solit.& Fract. 38(3): 620-630.