A pinching theorem for statistical manifolds with Casorati curvatures
-
2449
Downloads
-
4119
Views
Authors
Chul Woo Lee
- Department of Mathematics, Kyungpook National University, Daegu 41566, South Korea.
Dae Won Yoon
- Department of Mathematics Education, Gyeongsang National University and RINS, Jinju 52828, South Korea.
Jae Won Lee
- Department of Mathematics Education, Gyeongsang National University and RINS, Jinju 52828, South Korea.
Abstract
With a pair of conjugate connections \(\overline{\nabla}\) and \(\overline{\nabla}^*\), we derive optimal Casorati inequalities with the normalized scalar curvature on submanifolds of a statistical manifold of constant curvature.
Share and Cite
ISRP Style
Chul Woo Lee, Dae Won Yoon, Jae Won Lee, A pinching theorem for statistical manifolds with Casorati curvatures, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4908--4914
AMA Style
Lee Chul Woo, Yoon Dae Won, Lee Jae Won, A pinching theorem for statistical manifolds with Casorati curvatures. J. Nonlinear Sci. Appl. (2017); 10(9):4908--4914
Chicago/Turabian Style
Lee, Chul Woo, Yoon, Dae Won, Lee, Jae Won. "A pinching theorem for statistical manifolds with Casorati curvatures." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4908--4914
Keywords
- Statistical manifolds
- dual connection
- Casorati curvature
- \(\delta\)-Casorati curvature
- normalized scalar curvature
MSC
References
-
[1]
S.-I. Amari, Differential-Geometrical Methods in Statistics, Springer, New York (1985)
-
[2]
F. Casorati, Mesure de la courbure des surfaces suivant l’Idée commune, Acta Math., 14 (1890), 95–110.
-
[3]
S. Decu, S. Haesen, L. Verstraelen , Optimal inequalities involving Casorati curvatures , Bull. Transilv. Univ. Braşov Ser. B (N.S.), 14 (2007), 85–93.
-
[4]
S. Decu, S. Haesen, L. Verstraelen, Optimal inequalities characterising quasi-umbilical submanifolds, JIPAM. J. Inequal. Pure Appl. Math., 2008 (2008), 7 pages.
-
[5]
H. Furuhata , Hypersurfaces in statistical manifolds, Differential Geom. Appl., 27 (2009), 420–429.
-
[6]
V. Ghişoiu , Inequalities for the Casorati curvatures of slant submanifolds in complex space forms, Riemannian geometry and applications, 2011 (2011), 10 pages.
-
[7]
J. J. Koenderink, Surface shape, the science and the looks in ”Handbook of experimental phenomenology: Visual perception of shape, space and appearance” edited by L. Albertazzi, John Wiley and Sons Ltd. Chichester, (2013), 165–180.
-
[8]
D. Kowalczyk , Casorati curvatures, Bull. Transilv. Univ. Brasov, 1 (2008), 209–213.
-
[9]
C. W. Lee, J. W. Lee, G.-E. Vîlcu , A new proof for some optimal inequalities involving generalized normalized \(\delta\)-Casorati curvatures, J. Inequal. Appl., 2015 (2015), 9 pages.
-
[10]
C. W. Lee, J. W. Lee, G.-E. Vîlcu, D. W. Yoon , Optimal inequalities for the Casorati curvatures of submanifolds of generalized space forms endowed with semi-symmetric metric connections, Bull. Korean Math. Soc., 52 (2015), 1631– 1647.
-
[11]
J. W. Lee, C. W. Lee, D. W. Yoon, Inequalities for generalized \(\delta\)-Casorati curvatures of submanifolds in real space forms endowed with a semi-symmetric metric connection, Rev. Un. Mat. Argentina, 57 (2016), 53–62.
-
[12]
J. Lee, G.-E. Vîlcu , Inequalities for generalized normalized \(\delta\)-Casorati curvatures of slant submanifolds in quaternionic space forms, Taiwanese J. Math., 19 (2015), 691–702.
-
[13]
C. W. Lee, D. W. Yoon, J. W. Lee, Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections, J. Inequal. Appl., 2014 (2014), 9 pages.
-
[14]
B. Ons, P. Verstraelen , some geometrical comments on vision and neurobiology, seeing Gauss and Gabor walking by, when looking through the window of the Parma at Leuven in the company of Casorati, Kragujevac J. Math., 35 (2011), 317–325.
-
[15]
V. Slesar, B. Şahin, G.-E. Vîlcu, Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms, J. Inequal. Appl., 2014 (2014), 10 pages.
-
[16]
M. M. Tripathi, Inequalities for algebraic Casorati curvatures and their applications, Note Mat., 37 (2017), 161–186.
-
[17]
L. Verstraelen , Geometry of submanifolds I. , The first Casorati curvature indicatrices, Kragujevac J. Math., 37 (2013), 5–23.
-
[18]
P.W. Vos, Fundamental equations for statistical submanifolds with applications to the Bartlett correction, Ann. Inst. Statist. Math., 41 (1989), 429–450.
-
[19]
P. Zhang, L. Zhang , Casorati inequalities for submanifolds in a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection, Symmetry, 2016 (2016), 9 pages.
-
[20]
P. Zhang, L. Zhang, Inequalities for Casorati curvatures of submanifolds in real space forms, Adv. Geom., 16 (2016), 329–335.