대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제37권3호
  • /
  • Pages.851-864
  • /
  • 2022
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

CHEN INVARIANTS AND STATISTICAL SUBMANIFOLDS

  • 투고 : 2021.05.21
  • 심사 : 2021.10.14
  • 발행 : 2022.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We define a kind of sectional curvature and 𝛿-invariants for statistical manifolds. For statistical submanifolds the sum of the squared mean curvature and the squared dual mean curvature is bounded below by using the 𝛿-invariant. This inequality can be considered as a generalization of the so-called Chen inequality for Riemannian submanifolds.

키워드

과제정보

The authors wish to express their gratitude to Yukihiko Okuyama and Kimitake Sato for their kind help.

참고문헌

  1. M. E. Aydin, A. Mihai, and I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat 29 (2015), no. 3, 465-476. https://doi.org/10.2298/FIL1503465A
  2. B.-Y. Chen, A general optimal inequality for arbitrary Riemannian submanifolds, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 3, Article 77, 10 pp.
  3. B.-Y. Chen, A. Mihai, and I. Mihai, A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature, Results Math. 74 (2019), no. 4, Paper No. 165, 11 pp. https://doi.org/10.1007/s00025-019-1091-y
  4. H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical manifolds, in Geometry of Cauchy-Riemann submanifolds, 179-215, Springer, Singapore, 2016.
  5. H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato, and M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys. 117 (2017), 179-186. https://doi.org/10.1016/j.geomphys.2017.03.010
  6. C. W. Lee, D. W. Yoon, and J. W. Lee, A pinching theorem for statistical manifolds with Casorati curvatures, J. Nonlinear Sci. Appl. 10 (2017), no. 9, 4908-4914. https://doi.org/10.22436/jnsa.010.09.31
  7. A. Mihai and I. Mihai, The δ(2, 2)-invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature, Entropy 22 (2020), no. 2, Paper No. 164, 8 pp. https://doi.org/10.3390/e22020164
  8. B. Opozda, A sectional curvature for statistical structures, Linear Algebra Appl. 497 (2016), 134-161. https://doi.org/10.1016/j.laa.2016.02.021
  9. N. Satoh, H. Furuhata, I. Hasegawa, T. Nakane, Y. Okuyama, K. Sato, M. H. Shahid, and A. N. Siddiqui, Statistical submanifolds from a viewpoint of the Euler inequality, Inf. Geom. 4 (2021), no. 1, 189-213. https://doi.org/10.1007/s41884-020-00032-4
  10. H. Shima, The geometry of Hessian structures, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. https://doi.org/10.1142/9789812707536