Abstract
Schouten tensor, which is expressed by the Ricci curvature and scalar curvature is a Codazzi tensor on a Riemannian manifold M(dimM>3)with harmonic Weyl conformal curvature tensor. By using this tensor, an operator r can be induced, which is self-adjoint relative to the L2 - inner product. Using this operator, some equalities and inequalities are obtained. Then by equalities between certain function on a compact local conformally symmetric Riemannie manifold, Einstein manifold and constant sectional curvature space are characterized. Some new theorems are established.
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References
Hertrich-Jeromin, U.: Models in Meobius Differential Geometry, pp.16–22 (2001)
Okumura, M.: Hypersurfaces and a pinching problem on the second fundamental tensor. Amer. J. Math., 207–213 (1974)
Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann., 195–204 (1977)
Chern, S.S., Do Carmo, M., Kobayashi, S.: Minimal submanifold of a sphere with second fundamentalform of constant length. In: Browder, F.E. (ed.) Fuctional Analysis and Related Fields, pp. 59–75. Springer, New York (1970)
Li, H.: Hypersurface with constant scalar curvature in space forms. Math. Ann., 665–672 (1996)
Yau, S.T.: Lectures on Differential Geometry, pp. 231–250. Higher Education Press (2004)
Baek, J.O., Suh, Y.J.: KYUNGPOOK Math. J. 44 Conformally Recurrent Riemannian Manifolds with Harmonic Conformal Curvature Tensor, 47–61 (2004)
Malek, F., Samavaki, M.: On weakly symmetric Riemannian manifolds. Differential Geometry - Dynamical Systems 10, 215–220 (2008)
Stanis law Ewert-Krzemieniewski, Conformally Flat Totally Umbilical Submanifolds in Some Semi-Riemannian Manifolds, KYUNGPOOK Math. J. 48,183-194 (2008)
Murathana, C., O zgur̈, C.: Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions. In: Proceedings of the Estonian Academy of Sciences, pp. 210–216 (2008)
Nan, J., Xinghua, M., Yunwei, X., Yamian, P.: Conformal Flat Manifold and A Pinching Problem on the Schouten Tensor. In: DCABES 2009 Proceedings, pp. 99–100 (2009)
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Ji, N., Luo, Y., Yan, Y. (2010). Applications of Schouten Tensor on Conformally Symmetric Riemannie Manifold. In: Zhu, R., Zhang, Y., Liu, B., Liu, C. (eds) Information Computing and Applications. ICICA 2010. Communications in Computer and Information Science, vol 105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16336-4_16
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DOI: https://doi.org/10.1007/978-3-642-16336-4_16
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