对偶平坦(α,β)-度量的共形不变性
The Conformal Invariances of the Dually Flat (α, β)-metrics
本文主要研究了两个(α,β)-度量之间的共形变换.证明了:若F是一个局部对偶平坦的正则(α,β)-度量且与度量F共形相关,即F=eσ(x)F,那么度量F也是一个局部对偶平坦的(α,β)-度量当且仅当共形变换是一个位似.进一步,在度量具有奇异性的情形,我们证明了两个局部对偶平坦广义Kropina度量之间的任一共形变换必然是一个位似.
We study the conformal transformations between two (α, β)-metrics. We prove that, if F is a locally dually flat regular (α, β)-metric and is conformally related to F, that is, F=eσ(x)F, then F is also a locally dually flat (α, β)-metric if and only if the conformal transformation is a homothety. Further, in the case with singularity, we prove that any conformal transformation between two locally dually flat general Kropina metrics must be a homothety.
共变换 / 形局部对偶平坦芬斯勒度量 / (&alpha / &beta / )-度量 / 广义Kropina度量 {{custom_keyword}} /
conformal transformation / locally dually flat Finsler metric / (α, β)-metric / general Kropina metric {{custom_keyword}} /
[1] Amari S. I., Differential-Geometrical Metholds in Statistics, Springer Lecture Notes in Statistics, Vol. 28, Springer-Verlag, Berlin, 1985.
[2] Amari S. I., Nagaoka H., Methods of Information Geometry, AMS Translation of Math. Monographs, Vol. 191, Oxford University Press, 2000.
[3] Antonelli P. L., Ingarden R. S., Matsumoto M., The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993.
[4] Bácsó S., Cheng X., Finsler conformal transformations and the curvature invariances, Publicationes Mathematicae Debrecen, 2007, 70(1/2):221-231.
[5] Cheng X., On (α, β)-metrics of scalar flag curvature with constant S-curvature, Acta Mathematica Sinica, 2010, 26(9):1701-1708.
[6] Chen G., Cheng X., Zou Y., On conformal transformations between two (α, β)-metrics, Differential Geometry and Its Applications, 2013, 31(2):300-307.
[7] Cheng X., Shen Z., Zou Y., On locally dually flat Finsler metrics, International Journal of Mathematics, 2010, 21(11):1531-1543.
[8] Chern S. S., Shen Z., Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol.6, World Scientific, Singapore, 2005.
[9] Fefferman C., Graham C. R., The Ambient Metric, Princeton University Press, Princeton and Oxford, 2012.
[10] Li B., On dually flat Finsler metrics, Differential Geometry and Its Applications, 2013, 31(6):718-724.
[11] Matsumoto M., Finsler Geometry in the 20th-Century, Handbook of Finsler Geometry, Edited by P. L. Antonelli, Kluwer Academic Publishers, Dordrecht, 2004.
[12] Matsumoto M., Hōjō S. I., A conclusive theorem on C-reducible Finsler spaces, Tensor, N. S., 1978, 32:225-230.
[13] Rund H., The Differential Geometry of Finsler Spaces, Springer-Verlag, Berlin, 1959.
[14] Shen Z., Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
[15] Shen Z., Riemann-Finsler geometry with applications to information geometry, Chin. Ann. Math., 2006, 27(1):73-94.
[16] Shen B., Tian Y., Dually flat Kropina metrics, Adv. Math. China, 2017, 46:453-462.
[17] Xia Q., On locally dually flat (α, β)-metrics, Differential Geometry and Its Applications, 2011, 29(2):233-243.
[18] Yu C., On dually flat Randers metrics, Nonlinear Analysis, 2014, 95:146-155.
国家自然科学基金资助项目(11871126,11371386);重庆师范大学科学基金(17XLB022)
/
〈 | 〉 |