On the projective Ricci curvature
- PDF / 163,826 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 43 Downloads / 197 Views
https://doi.org/10.1007/s11425-020-1705-x
. ARTICLES .
On the projective Ricci curvature In Memory of Professor Zhengguo Bai (1916–2015)
Zhongmin Shen1 & Liling Sun2,†,∗ 1Department
of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA; 2Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China Email: [email protected], [email protected] Received February 23, 2020; accepted May 25, 2020
Abstract
The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a
manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics. Keywords MSC(2010)
spray, Finsler metric, Randers metric, projective Ricci curvature 53B40, 53C60
Citation: Shen Z M, Sun L L. On the projective Ricci curvature. Sci China Math, 2021, 64, https://doi.org/ 10.1007/s11425-020-1705-x
1
Introduction
In Finsler geometry, there are many important Riemannian quantities such as the Riemann curvature and the Ricci curvature, etc., and non-Riemannian quantities such as the Berwald curvature and the S-curvature, etc. The Ricci curvature is defined as the trace of the Riemann curvature. Together with the S-curvature, the Ricci curvature plays an important role in Finsler geometry. The volume of geodesic balls can be controlled by the lower bounds of the Ricci curvature and the S-curvature (see [4, 6]). The volume of geodesic balls can be also controlled by a single bound of the N-Ricci curvature which is the combination of the Ricci curvature and the S-curvature (see [3]). Let G be a spray on an n-dimensional manifold M . Given a volume form dV on M , we can construct a new spray by ˆ := G + 2S Y. G n+1 ˆ The spray G is called the projective spray of (G, dV ). The projective Ricci curvature of (G, dV ) is ˆ namely, defined as the Ricci curvature of G, PRic(G,dV ) := RicGˆ .
(1.1)
† Current address: School of Science, Jimei University, Xiamen 361021, China * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
2
Shen Z M et al.
Sci China Math
By a simple computation, we have the following formula for the projective Ricci curvature: { PRic(G,dV ) = Ric + (n − 1)
[ ]2 } S|0 S + , n+1 n+1
(1.2)
where Ric = RicG is the Ricci curvature of the spray G, S = S(G,dV ) is the S-curvature of (G, dV ) and ˆ remains unchanged under a S|0 is the covariant derivative of S along a geodesic of G. It is known that G projective change of G with dV fixed, and thus PRic(G,dV ) = RicG ˆ is a projective invariant of (G, dV ) (see [5]). We make the following definition. Definition 1.1. A spray G on an n-dimensional manifold M is said to be proj
Data Loading...
