Skip to main content
SpringerLink
Log in

Inequalities for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11712))

Included in the following conference series:

Abstract

In the present paper, we study Hessian and Einstein-Hessian manifolds with some examples. We establish optimizations of the intrinsic invariant (normalized scalar curvature) for a new extrinsic invariant (generalized normalized Casorati curvatures) on statistical submanifolds in a Hessian manifold of constant Hessian curvature by using algebraic technique. We consider the equality case of the derived inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Amari, S.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics, vol. 28. Springer, New York (1985). https://doi.org/10.1007/978-1-4612-5056-2

    Book  MATH  Google Scholar 

  2. Casorati, F.: Mesure de la courbure des surfaces suivant l’idee commune. Acta Math. 14(1), 95–110 (1890)

    Article  MathSciNet  Google Scholar 

  3. Decu, S., Haesen, S., Verstraelen, L.: Optimal inequalities characterising quasi-umbilical submanifolds. J. Inequal. Pure Appl. Math. 9(3), 1–7 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Furuhata, H.: Hypersurfaces in statistical manifolds. Differ. Geom. Appl. 27, 420–429 (2009)

    Article  MathSciNet  Google Scholar 

  5. Furuhata, H.: Statistical hypersurfaces in the space of Hessian curvature zero. Differ. Geom. Appl. 29, 586–590 (2011)

    Article  MathSciNet  Google Scholar 

  6. Lee, C.W., Yoon, D.W., Lee, J.W.: A pinching theorem for statistical manifolds with Casorati curvatures. J. Nonlinear Sci. Appl. 10, 4908–4914 (2017)

    Article  MathSciNet  Google Scholar 

  7. Mihai, A., Mihai, I.: Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature. Mathematics 44(6) (2018)

    Google Scholar 

  8. Opozda, B.: Bochner’s technique for statistical structures. Ann. Glob. Anal. Geom. 48(4), 357–395 (2015)

    Article  MathSciNet  Google Scholar 

  9. Opozda, B.: A sectional curvature for statistical structures. Linear Algebra Appl. 497, 134–161 (2016)

    Article  MathSciNet  Google Scholar 

  10. Siddiqui, A.N. and Shahid, M.H.: Optimizations on statistical hypersurfaces with Casorati curvatures. Accepted for the publication in Kragujevac Journal of Mathematics (2019)

    Google Scholar 

  11. Shima, H.: Hessian manifolds of constant Hessian sectional curvature. J. Math. Soc. Jpn. 47(4), 735–753 (1995)

    Article  MathSciNet  Google Scholar 

  12. Shima, H.: The Geometry of Hessian Structures. World Scientific, Singapore (2007)

    Book  Google Scholar 

  13. Tripathi, M.M.: Inequalities for algebraic Casorati curvatures and their applications. Note Mat. 37, 161–186 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

The authors would like to thank to anonymous referees for their comments to improve the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India

    Aliya Naaz Siddiqui

  2. M.M. Engineering College, Maharishi Markandeshwar (Deemed to be) University, Mullana, Ambala, 133207, India

    Kamran Ahmad

  3. Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Cenap Özel

Authors
  1. Aliya Naaz Siddiqui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kamran Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cenap Özel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aliya Naaz Siddiqui .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories, Inc., Tokyo, Japan

    Frank Nielsen

  2. Thales, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Siddiqui, A.N., Ahmad, K., Özel, C. (2019). Inequalities for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26980-7_40

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26979-1

  • Online ISBN: 978-3-030-26980-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation