关于共形平坦(α,β)-度量的两个刚性结果
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摘要
研究了共形平坦(α,β)-度量的刚性性质.首先,在β是关于α的共形1-形式且为闭的条件下,证明了共形平坦(α,β)-度量一定是局部Minkowski度量.其次,根据射影Ricci平坦Randers度量的特性,证明了共形平坦且射影Ricci平坦的Randers度量一定是局部Minkowski度量.
In this paper, we study the rigidity properties of conformally flat(α, β)-metrics. Firstly, under the conditions that β is a closed and conformal 1-form with respect to α, we prove that conformally flat(α, β)-metrics must be Minkowskian. Further, by the properties of the conformally flat(α, β)-metrics and the characterization of projective Ricci flat Randers metrics, we prove that conformally flat and projective Ricci flat Randers metrics must be Minkowskian.
In this paper, we study the rigidity properties of conformally flat(α, β)-metrics. Firstly, under the conditions that β is a closed and conformal 1-form with respect to α, we prove that conformally flat(α, β)-metrics must be Minkowskian. Further, by the properties of the conformally flat(α, β)-metrics and the characterization of projective Ricci flat Randers metrics, we prove that conformally flat and projective Ricci flat Randers metrics must be Minkowskian.
引文
[1] BACSO S, CHENG X Y. Finsler Conformal Transformations and the Curvature Invariances [J]. Publ Math Debrecen, 2007, 70(1/2): 221-231.
[2] RUND H. The Differential Geometry of Finsler Spaces [M]. Berlin: Springer-Verlag, 1959.
[3] ANTONELLI P L, INGARDEN R S, MATSUMOTO M. The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology [M]. Dordrecht: Kluwer Academic Publishers, 1993.
[4] MATSUMOTO M. Finsler Geometry in the 20th-Century [M] //ANTONELLI P L. Handbook of Finsler Geometry. Berlin: Springer-Science, 2003: 1437.
[5] 康琳. 共形平坦的Randers度量 [J]. 中国科学, 2011, 41(5): 439-446.
[6] 穆风, 程新跃, 杨慧章. 共形平坦的(α, β)-度量 [J]. 数学杂志, 2013, 33(2): 275-282.
[7] DOUGLAS J. The General Geometry of Paths [J]. Ann Math, 1927, 29(1-4): 143-168.
[8] BACSO S, PAPP I. A Note on a Generalized Douglas Space [J]. Periodica Mathematicae Hungarica Vol, 2004, 48(1/2): 181-184.
[9] NAJAFI B, SHEN Z M, TAYEBI A. On a Projective Class of Finsler Metrics [J]. Publicationes Mathematicae Debrecen, 2007, 70: 211-219.
[10] SHEN Z M. Differential Geometry of Spray and Finsler Spaces [M]. Dordrecht: Kluwer Academic Publisher, 2001.
[11] 程新跃, 马小玉, 沈玉玲. 射影Ricci曲率及其射影不变性 [J]. 西南大学学报(自然科学版), 2015, 37(8): 92-96.
[12] CHENG X Y, SHEN Y L, MA X Y. On a Class of Projective Ricci Flat Finsler Metrics [J]. Publ Math Debrecen, 2017, 90(1/2): 169-180.
[13] CHENG X Y, MA X Y, SHEN Y L. On Projective RFlat Kropina Metrics [J]. Journal of Mathematics, 2017, 37(4): 705-713.
[14] 程新跃, 李婷婷, 殷丽. 关于射影Ricci曲率的比较定理与共形不变性 [J]. 西南大学学报(自然科学版), 2019, 41(2): 52-59.
[15] CHERN S S, SHEN Z M. Riemann-Finsler Geometry [M]. London: World Scientific, 2005.
[16] CHENG X Y, SHEN Z M. A Class of Finsler Metrics with Isotropic S-Curvature [J]. Israel Journal of Mathematics, 2009, 169(1): 317-340.
[2] RUND H. The Differential Geometry of Finsler Spaces [M]. Berlin: Springer-Verlag, 1959.
[3] ANTONELLI P L, INGARDEN R S, MATSUMOTO M. The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology [M]. Dordrecht: Kluwer Academic Publishers, 1993.
[4] MATSUMOTO M. Finsler Geometry in the 20th-Century [M] //ANTONELLI P L. Handbook of Finsler Geometry. Berlin: Springer-Science, 2003: 1437.
[5] 康琳. 共形平坦的Randers度量 [J]. 中国科学, 2011, 41(5): 439-446.
[6] 穆风, 程新跃, 杨慧章. 共形平坦的(α, β)-度量 [J]. 数学杂志, 2013, 33(2): 275-282.
[7] DOUGLAS J. The General Geometry of Paths [J]. Ann Math, 1927, 29(1-4): 143-168.
[8] BACSO S, PAPP I. A Note on a Generalized Douglas Space [J]. Periodica Mathematicae Hungarica Vol, 2004, 48(1/2): 181-184.
[9] NAJAFI B, SHEN Z M, TAYEBI A. On a Projective Class of Finsler Metrics [J]. Publicationes Mathematicae Debrecen, 2007, 70: 211-219.
[10] SHEN Z M. Differential Geometry of Spray and Finsler Spaces [M]. Dordrecht: Kluwer Academic Publisher, 2001.
[11] 程新跃, 马小玉, 沈玉玲. 射影Ricci曲率及其射影不变性 [J]. 西南大学学报(自然科学版), 2015, 37(8): 92-96.
[12] CHENG X Y, SHEN Y L, MA X Y. On a Class of Projective Ricci Flat Finsler Metrics [J]. Publ Math Debrecen, 2017, 90(1/2): 169-180.
[13] CHENG X Y, MA X Y, SHEN Y L. On Projective RFlat Kropina Metrics [J]. Journal of Mathematics, 2017, 37(4): 705-713.
[14] 程新跃, 李婷婷, 殷丽. 关于射影Ricci曲率的比较定理与共形不变性 [J]. 西南大学学报(自然科学版), 2019, 41(2): 52-59.
[15] CHERN S S, SHEN Z M. Riemann-Finsler Geometry [M]. London: World Scientific, 2005.
[16] CHENG X Y, SHEN Z M. A Class of Finsler Metrics with Isotropic S-Curvature [J]. Israel Journal of Mathematics, 2009, 169(1): 317-340.