Abstract
The statistical structure on a manifold \(\mathfrak {M}\) is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection \(\nabla \) on the \(T\mathfrak {M}\), such that \(\nabla g\) is totally symmetric, forming, by definition, a “Codazzi pair” \(\{\nabla , g\}\). In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on \(T\mathfrak {M}\)), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of \(\nabla \) with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.
Keywords
- Conformal Transformation
- General Transformation
- Projective Transformation
- Invertible Operator
- Leibniz Rule
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Amari, S., Nagaoka, H.: Method of Information Geometry. Oxford University Press, AMS Monograph, Providence, RI (2000)
Masayuki, H., Matsuzoe, H.: Geometry of precontrast functions and nonconservative estimating functions. In: International Workshop on Complex Structures, Integrability and Vector Fields, vol. 1340(1), AIP Publishing (2011)
Ivanov, S.: On dual-projectively flat affine connections. J. Geom. 53(1–2), 89–99 (1995)
Kurose, T.: On the divergences of 1-conformally flat statistical manifolds. Tohoku Math. J. Second Ser. 46(3), 427–433 (1994)
Kurose, T.: Conformal-projective geometry of statistical manifolds. Interdisc. Inf. Sci. 8(1), 89–100 (2002)
Lauritzen, S.: Statistical manifolds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, vol. 10, pp. 163–216. IMS Lecture Notes, Hayward, CA (1987)
Liu, H.L., Simon, U., Wang, C.P.: Codazzi tensors and the topology of surfaces. Ann. Glob. Anal. Geom. 16(2), 189–202 (1998)
Liu, H.L., Simon, U., Wang, C.P.: Higher order Codazzi tensors on conformally flat spaces. Contrib. Algebra Geom. 39(2), 329–348 (1998)
Matsuzoe, H.: On realization of conformally-projectively flat statistical manifolds and the divergences. Hokkaido Math. J. 27, 409–421 (1998)
Nomizu, K., Sasaki, T.: Affine Differential Geometry - Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)
Pinkall, U., Schwenk-Schellschmidt, A., Simon, U.: Geometric methods for solving Codazzi and Monge-Ampere equations. Math. Ann. 298(1), 89–100 (1994)
Schwenk-Schellschmidt, Angela, Udo Simon, and Martin Wiehe.: Generating higher order Codazzi tensors by functions. TU, Fachbereich Mathematik (3), 1998
Schwenk-Schellschmidt, A., Simon, U.: Codazzi-equivalent affine connections. RM 56(1–4), 211–229 (2009)
Simon, U.: Codazzi tensors. In: Ferus, D., Ktihnel, W., Simon, U., Wegner, B.: Global Differential Geometry and Global Analysis, vol. 838, pp. 289–296. Springer, Berlin Heidelberg (1981)
Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes Science University of Tokyo, Tokyo (1991)
Simon, U.: Transformation techniques for partial differential equations on projectively flat manifolds. RM 27(1–2), 160–187 (1995)
Simon, U.: Affine differential geometry. In: Dillen, F., Verstraelen, L. (eds.) Handbook of Differential Geometry, vol. 1, pp. 905–961. Elsevier Science (2000)
Simon, U.: Affine hypersurface theory revisited: gauge invariant structures. Russian Mathematics c/c of Izvstiia-Vysshie Uchebnye Zavedeniia Matematika 48(11), 48 (2004)
Uohashi, K.: On \(\alpha \)-conformal equivalence of statistical manifolds. J. Geom. 75, 179–184 (2002)
Zhang, J.: Divergence function, duality, and convex analysis. Neural Comput. 16, 159–195 (2004)
Zhang, J.: Nonparametric information geometry: from divergence function to referential-representational biduality on statistical manifolds. Entropy 15, 5384–5418 (2013)
Acknowledgment
This work was completed while the second author (J.Z.) was on sabbatical visit at the Center for Mathematical Sciences and Applications at Harvard University under the auspices and support of Prof. S.T. Yau.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Tao, J., Zhang, J. (2015). Transformations and Coupling Relations for Affine Connections. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_36
Download citation
DOI: https://doi.org/10.1007/978-3-319-25040-3_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25039-7
Online ISBN: 978-3-319-25040-3
eBook Packages: Computer ScienceComputer Science (R0)