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An Extension Theorem for Weighted Ricci Curvature on Finsler Manifolds

Year 2019, Volume: 40 Issue: 4, 867 - 874, 31.12.2019

Yasemin Soylu

https://doi.org/10.17776/csj.618537

Abstract

Let (M,F) be a forward complete and connected Finsler manifold of dimensional n ≥2 . In this study, we extend Wan’s extension theorem in Riemannian manifolds to Finsler manifolds by using the weighted Ricci curvature RicN bounded below. The proof of theorem is obtained by the Laplacian comparison theorem on Finsler manifolds and the excess function.

Keywords

Distance function Finsler manifold Weighted Ricci curvature

References

  • [1] Ohta S., Finsler interpolation inequalities, Calc. Var. Partial Differ. Equ., 36 (2009) 211-249.
  • [2] Wu B., A note on the generalized Myers theorem for Finsler manifolds, Bull. Korean Math. Soc., 50 (2013) 833–837.
  • [3] Yin S., Two compactness theorems on Finsler manifolds with positive weighted Ricci curvature, Results Math., 72 (2017) 319–327.
  • [4] Myers S.B., Riemannian manifolds with positive mean curvature, Duke Math. J., 8-2 (1941) 401-404.
  • [5] Calabi E., On Ricci curvature and geodesics, Duke Math. J., 34 (1967) 667–676.
  • [6] Cheeger J., Gromov M. and Taylor M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., 17-1 (1982) 15-53.
  • [7] Wan J., An extension of Bonnet–Myers theorem, Math. Z., 291 (2019) 195–197.
  • [8] Qui H., Extensions of Bonnet-Myers’ type theorems with the Bakry-Emery Ricci curvature, https://arxiv.org/abs/1905.01452, (2019).
  • [9] Lott J., Some geometric properties of the Bakry-Emery-Ricci Tensor, Comment. Math. Helv., 78 (2003) 865-883.
  • [10] Wei G. and Wylie W., Comparison geometry for the Bakry-Emery-Ricci tensor, J. Differential Geom., 83 (2009) 337-405.
  • [11] Deshmukh S. and Al-Solamy F.R., Conformal vector fields on a Riemannian manifold, Balkan J. Geom. Appl., 19 (2014) 86-93.
  • [12] Eker S., The Bochner Vanishing Theorems on the Conformal Killing Vector Fields, TWMS J. App. Eng. Math., 9-1 Special Issue (2019) 114-120.
  • [13] Soylu Y., Upper Bounds on the Diameter for Finsler Manifolds with Weighted Ricci Curvature, Miskolc Math. Notes, 19-2 (2018) 1173-1184.
  • [14] Wu B. and Xin Y., Comparison theorems in Finsler geometry and their applications, Math. Ann., 337 (2007) 177–196.
  • [15] Shen Z., Lectures on Finsler Geometry, World Scientific, Singapore, 2001.
  • [16] Ohta S. and Sturm K.T., Heat flow on Finsler manifolds, Commun. Pure Appl. Math., 62 (2009) 1386-1433.
  • [17] Ohta S. and Sturm K.T., Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds, Adv. Math., 252 (2014) 429-448.

Year 2019, Volume: 40 Issue: 4, 867 - 874, 31.12.2019

Yasemin Soylu

https://doi.org/10.17776/csj.618537

Abstract

References

  • [1] Ohta S., Finsler interpolation inequalities, Calc. Var. Partial Differ. Equ., 36 (2009) 211-249.
  • [2] Wu B., A note on the generalized Myers theorem for Finsler manifolds, Bull. Korean Math. Soc., 50 (2013) 833–837.
  • [3] Yin S., Two compactness theorems on Finsler manifolds with positive weighted Ricci curvature, Results Math., 72 (2017) 319–327.
  • [4] Myers S.B., Riemannian manifolds with positive mean curvature, Duke Math. J., 8-2 (1941) 401-404.
  • [5] Calabi E., On Ricci curvature and geodesics, Duke Math. J., 34 (1967) 667–676.
  • [6] Cheeger J., Gromov M. and Taylor M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., 17-1 (1982) 15-53.
  • [7] Wan J., An extension of Bonnet–Myers theorem, Math. Z., 291 (2019) 195–197.
  • [8] Qui H., Extensions of Bonnet-Myers’ type theorems with the Bakry-Emery Ricci curvature, https://arxiv.org/abs/1905.01452, (2019).
  • [9] Lott J., Some geometric properties of the Bakry-Emery-Ricci Tensor, Comment. Math. Helv., 78 (2003) 865-883.
  • [10] Wei G. and Wylie W., Comparison geometry for the Bakry-Emery-Ricci tensor, J. Differential Geom., 83 (2009) 337-405.
  • [11] Deshmukh S. and Al-Solamy F.R., Conformal vector fields on a Riemannian manifold, Balkan J. Geom. Appl., 19 (2014) 86-93.
  • [12] Eker S., The Bochner Vanishing Theorems on the Conformal Killing Vector Fields, TWMS J. App. Eng. Math., 9-1 Special Issue (2019) 114-120.
  • [13] Soylu Y., Upper Bounds on the Diameter for Finsler Manifolds with Weighted Ricci Curvature, Miskolc Math. Notes, 19-2 (2018) 1173-1184.
  • [14] Wu B. and Xin Y., Comparison theorems in Finsler geometry and their applications, Math. Ann., 337 (2007) 177–196.
  • [15] Shen Z., Lectures on Finsler Geometry, World Scientific, Singapore, 2001.
  • [16] Ohta S. and Sturm K.T., Heat flow on Finsler manifolds, Commun. Pure Appl. Math., 62 (2009) 1386-1433.
  • [17] Ohta S. and Sturm K.T., Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds, Adv. Math., 252 (2014) 429-448.
There are 17 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Yasemin Soylu 0000-0001-9009-1214

Publication Date December 31, 2019
Submission Date September 11, 2019
Acceptance Date December 19, 2019
Published in Issue Year 2019Volume: 40 Issue: 4

Cite

APA Soylu, Y. (2019). An Extension Theorem for Weighted Ricci Curvature on Finsler Manifolds. Cumhuriyet Science Journal, 40(4), 867-874. https://doi.org/10.17776/csj.618537