常曲率统计流形中子流形的广义标准δ-Casorati曲率不等式
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摘要
考虑常曲率统计流形中的子流形,利用Oprea最优化方法得到了关于广义标准δ-Casorati曲率的一些几何不等式,并分别给出子流形标准数量曲率的上界和下界以及等号成立时子流形的性态.
We considered submanifolds in statistical manifolds of constant curvatures by using Oprea's optimization method,and obtained some geometric inequalities involving the generalized normalized δ-Casorati curvatures.We gave the upper bound and the lower bound of the normalized scalar curvature of the submanifolds,respectively,and the properties of submanifolds satisfying the equality cases.
We considered submanifolds in statistical manifolds of constant curvatures by using Oprea's optimization method,and obtained some geometric inequalities involving the generalized normalized δ-Casorati curvatures.We gave the upper bound and the lower bound of the normalized scalar curvature of the submanifolds,respectively,and the properties of submanifolds satisfying the equality cases.
引文
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[8]LEE J W,LEE C W,YOON D W.Inequalities for Generalizedδ-Casorati Curvatures of Submanifolds in Real Space Forms Endowed with a Semi-symmetric Metric Connection[J].Revista de la Unión Matemtica Argentina,2016,57(2):53-62.
[9]PARK K S.Inequalities for the Casorati Curvatures of Real Hypersurfaces in Some Grassmannians[J].Taiwanese Journal of Mathematics,2018,22(1):63-77.
[10]SU Man,ZHANG Liang,ZHANG Pan.Some Inequalities for Submanifolds in a Riemannian Manifold of Nearly Quasi-constant Curvature[J].Filomat,2016,31(8):2467-2475.
[11]TRIPATHI M M.Inequalities for Algebraic Casorati Curvatures and Their Applications[J].Note di Mathematica,2017,37(1):161-186.
[12]ZHANG Pan,ZHANG Liang.Inequalities for the Casorati Curvatures of Submanifolds in Real Space Forms[J].Advances in Geometry,2016,16(3):329-335.
[13]AMARI S.Differential-Geometrical Methods in Statistics[M].Berlin:Springer-Verlag,1985:168-178.
[14]LEE C W,YOON D W,LEE J W.A Pinching Theorem for Statistical Manifolds with Casorati Curvatures[J].Journal Nonlinear Sciences and Applications,2017,4(10):4908-4914.
[15]AYDIN M E,MIHAI A,MIHAI I.Generalized Wintgein Inequality for Statistical Submanifolds in Statistical Manifolds of Constant Curvature[J].Bulletin of Mathematical Sciences,2017,7(1):155-166.
[16]AYDIN M E,MIHAI A,MIHAI I.Some Inequalities on Submanifolds in Statistical Manifolds of Constant Curvature[J].Filomat,2015,29(3):465-477.
[17]VOS P W.Fundamental Equations for Statistical Submanifolds with Applications to the Bartlett Correction[J].Annals of Institute Statistical Mathematics,1989,41(3):429-450.
[2]CASORATI F.Mesure de la Courbure des Surfaces Suivant L’idée Commune[J].Acta Mathematica,1890,14(1):95-110.
[3]DECU S,HAESEN S,VERSTRAELEN L.Optimal Inequalities Involving Casorati Curvatures[J].Transactions of the Indian Institute of Metals,2007,14(5):85-93.
[4]DECU S,HAESEN S,VERSTRAELEN L.Optimal Inequalities Characterising Quasi-umbilical Submanifolds[J].Journal of Inequalities Pure and Applied Mathematics,2008,9(3):1-7.
[5]LEE C W,YOON D W,LEE J W.Optimal Inequalities for the Casorati Curvatures of Submanifolds of Real Space Forms Endowed with Semi-symmetric Metric Connections[J/OL].Journal of Inequalities and Applications,2014-09-01.http://www.journalofinequalitiesandapplications.com/content/2014/1/327.
[6]LEE C W,LEE J W,VLCU G E,et al.Optimal Inequalities for the Casorati Curvatures of Submanifolds of Generalized Space Forms Endowed with Semi-symmetric Metric Connections[J].Bulletin Korean Mathematical Society,2015,52(5):1631-1647.
[7]LEE J,VLCU G E.Inequalities for Generalized Normalizedδ-Casorati Curvatures of Slant Submanifolds in Quaternionic Space Forms[J].Taiwanese Journal of Mathematics,2015,19(3):691-702.
[8]LEE J W,LEE C W,YOON D W.Inequalities for Generalizedδ-Casorati Curvatures of Submanifolds in Real Space Forms Endowed with a Semi-symmetric Metric Connection[J].Revista de la Unión Matemtica Argentina,2016,57(2):53-62.
[9]PARK K S.Inequalities for the Casorati Curvatures of Real Hypersurfaces in Some Grassmannians[J].Taiwanese Journal of Mathematics,2018,22(1):63-77.
[10]SU Man,ZHANG Liang,ZHANG Pan.Some Inequalities for Submanifolds in a Riemannian Manifold of Nearly Quasi-constant Curvature[J].Filomat,2016,31(8):2467-2475.
[11]TRIPATHI M M.Inequalities for Algebraic Casorati Curvatures and Their Applications[J].Note di Mathematica,2017,37(1):161-186.
[12]ZHANG Pan,ZHANG Liang.Inequalities for the Casorati Curvatures of Submanifolds in Real Space Forms[J].Advances in Geometry,2016,16(3):329-335.
[13]AMARI S.Differential-Geometrical Methods in Statistics[M].Berlin:Springer-Verlag,1985:168-178.
[14]LEE C W,YOON D W,LEE J W.A Pinching Theorem for Statistical Manifolds with Casorati Curvatures[J].Journal Nonlinear Sciences and Applications,2017,4(10):4908-4914.
[15]AYDIN M E,MIHAI A,MIHAI I.Generalized Wintgein Inequality for Statistical Submanifolds in Statistical Manifolds of Constant Curvature[J].Bulletin of Mathematical Sciences,2017,7(1):155-166.
[16]AYDIN M E,MIHAI A,MIHAI I.Some Inequalities on Submanifolds in Statistical Manifolds of Constant Curvature[J].Filomat,2015,29(3):465-477.
[17]VOS P W.Fundamental Equations for Statistical Submanifolds with Applications to the Bartlett Correction[J].Annals of Institute Statistical Mathematics,1989,41(3):429-450.