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A note on harmonic maps of statistical manifolds

Year 2019, Volume: 68 Issue: 2, 1370 - 1376, 01.08.2019
https://doi.org/10.31801/cfsuasmas.530449

Abstract

We show an existence and uniqueness result for a class of maps from a flat statistical manifold into a Riemannian manifold in a given homotopy class when the target Riemannian manifold is of negative sectional curvature under a global topological non-triviality condition. We also show that due to dualistic structure of the domain manifold the result is still valid in dual coordinates.

References

  • Jost, J. and Şimşir, F. M., Affine harmonic maps, Analysis, 29, (2009) 185--197.
  • Jost, J. and Şimşir, F. M., Nondivergence harmonic maps, Harmonic maps and differential geometry, Contemp. Math., 542, 231--238, 2011.
  • Al'ber, Spaces of mappings into a manifold with negative curvature, Sov.Math.Dokl.9, 6--9, 1967.
  • Amari, S. I. and Nagaoka, H., Methods of information geometry, Transl.Math. Monogr. 191, AMS & Oxford Univ. Press, 2000.
  • Chentsov, N.N., Statistical decision rules and optimal inferences, AMS, 1982 (Translation of the Russian version, Nauka, Moscow, 1972)
  • Cheng, S.Y. and Yau, S.T., The real Monge-Ampère equation and affine flat structures, in: S.S.Chern, W.T.Wu (eds.), Differential geometry and differential equations, Proc.Beijing Symp. 1980, (1982) 339--370.
  • Kurose, T., Dual connections and affine geometry, Mathematische Zeitschrift, 203, 115--121, 1990.
  • Lauritzen, S.L., Statistical manifolds, in: Differential Geometry in Statistical Inference, Institute of Mathematical Statistics Lecture Notes, 10, pp. 163-218, Berkeley 1987.
  • Noguchi, M., Geometry of Statistical Manifolds, Differantial Geometry and its Applications, 2, 197--222, 1992.
  • Shima, H., The Geometry of Hessian Structures, 13--32, World Scientific, 2007.
Year 2019, Volume: 68 Issue: 2, 1370 - 1376, 01.08.2019
https://doi.org/10.31801/cfsuasmas.530449

Abstract

References

  • Jost, J. and Şimşir, F. M., Affine harmonic maps, Analysis, 29, (2009) 185--197.
  • Jost, J. and Şimşir, F. M., Nondivergence harmonic maps, Harmonic maps and differential geometry, Contemp. Math., 542, 231--238, 2011.
  • Al'ber, Spaces of mappings into a manifold with negative curvature, Sov.Math.Dokl.9, 6--9, 1967.
  • Amari, S. I. and Nagaoka, H., Methods of information geometry, Transl.Math. Monogr. 191, AMS & Oxford Univ. Press, 2000.
  • Chentsov, N.N., Statistical decision rules and optimal inferences, AMS, 1982 (Translation of the Russian version, Nauka, Moscow, 1972)
  • Cheng, S.Y. and Yau, S.T., The real Monge-Ampère equation and affine flat structures, in: S.S.Chern, W.T.Wu (eds.), Differential geometry and differential equations, Proc.Beijing Symp. 1980, (1982) 339--370.
  • Kurose, T., Dual connections and affine geometry, Mathematische Zeitschrift, 203, 115--121, 1990.
  • Lauritzen, S.L., Statistical manifolds, in: Differential Geometry in Statistical Inference, Institute of Mathematical Statistics Lecture Notes, 10, pp. 163-218, Berkeley 1987.
  • Noguchi, M., Geometry of Statistical Manifolds, Differantial Geometry and its Applications, 2, 197--222, 1992.
  • Shima, H., The Geometry of Hessian Structures, 13--32, World Scientific, 2007.
There are 10 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Fatma Muazzez Şimşir 0000-0001-8205-8668

Publication Date August 1, 2019
Submission Date July 11, 2017
Acceptance Date August 28, 2018
Published in Issue Year 2019 Volume: 68 Issue: 2

Cite

APA Şimşir, F. M. (2019). A note on harmonic maps of statistical manifolds. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1370-1376. https://doi.org/10.31801/cfsuasmas.530449
AMA Şimşir FM. A note on harmonic maps of statistical manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2019;68(2):1370-1376. doi:10.31801/cfsuasmas.530449
Chicago Şimşir, Fatma Muazzez. “A Note on Harmonic Maps of Statistical Manifolds”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 2 (August 2019): 1370-76. https://doi.org/10.31801/cfsuasmas.530449.
EndNote Şimşir FM (August 1, 2019) A note on harmonic maps of statistical manifolds. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 2 1370–1376.
IEEE F. M. Şimşir, “A note on harmonic maps of statistical manifolds”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 2, pp. 1370–1376, 2019, doi: 10.31801/cfsuasmas.530449.
ISNAD Şimşir, Fatma Muazzez. “A Note on Harmonic Maps of Statistical Manifolds”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/2 (August 2019), 1370-1376. https://doi.org/10.31801/cfsuasmas.530449.
JAMA Şimşir FM. A note on harmonic maps of statistical manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1370–1376.
MLA Şimşir, Fatma Muazzez. “A Note on Harmonic Maps of Statistical Manifolds”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, 2019, pp. 1370-6, doi:10.31801/cfsuasmas.530449.
Vancouver Şimşir FM. A note on harmonic maps of statistical manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(2):1370-6.

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Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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