Spherically symmetric Finsler metrics with constant Ricci and flag curvature. (English) Zbl 1363.53067
Spherically symmetric Finsler metrics are first studied by L. Zhou [Publ. Math. 80, No. 1–2, 67–77 (2012; Zbl 1299.53156)]. Recently X. Mo and L. Zhou [“The curvatures of spherically symmetric Finsler metrics in \(R^n\)”, Preprint, arXiv:1202.4543] and X. Mo et al. [“On spherically symmetric Finsler metrics of constant curvature”, Preprint] found three equations that characterize spherically symmetric Finsler metrics of constant curvature and found some new locally projectively flat metrics of constant flag curvature. In this paper the authors show that these three equations can be reduced to two equations.
Suppose that \(\phi=\phi(r,s)\) is a smooth function defined on \([0,\rho)\times (-\rho,\rho)\). A spherically symmetric metric is locally expressed on a ball \(B^n(\rho)\subset R^n\) in the following form: \[ F=| y| \phi(r,s),\quad r=| x|, \quad s=\frac{\langle x,y\rangle}{| y|},\quad y\in T_xB^n(\rho)\equiv R^n. \] The main results of this paper are the following.
Theorem 1.1. Let \(F=| y| \phi(r,s)\) be a spherically symmetric Finsler metric on an open ball \(B^n(\rho)\subset R^n\), \(n\geq 3\). Then \(F\) is of constant flag curvature \(K\) if and only if \[ R_1=K\phi^2,\qquad R_2=0. \]
Here \(R_1, R_2\) and \(R_3\) are defined as expressions of \(\phi(r,s)\) and its derivatives, firstly introduced by Mo-Zhou and Mo-Zhou-Zhu. The authors find that they are related mysteriously as follows.
Theorem 1.2. Let \(F=| y| \phi(r,s)\) be a spherically symmetric Finsler metric on an open ball \(B^n(\rho)\subset R^n(n\geq 3)\). Then \(F\) is of constant flag curvature \(K\) if and only if \[ R_2=0,\qquad R_3=0. \]
Theorem 1.4. Let \(F=| y| \phi(r,s)\) be a spherically symmetric Finsler metric on an open ball \(B^n(\rho)\subset R^n(n\geq 3)\). Then \(\mathrm{Ric}_{ij}=Kg_{ij}\), with \(K\) a constant, if and only if \(\phi\) satisfies \[ (n-1)K\phi^2=(n-1)R_1+(r^2-s^2)R_2,\qquad (n+1)R_3+(r^2-s^2)[R_2]_s=0. \]
The first condition was discussed by Mo-Zhou for the characterization of spherically symmetric Finsler metrics of constant Ricci curvature.
Suppose that \(\phi=\phi(r,s)\) is a smooth function defined on \([0,\rho)\times (-\rho,\rho)\). A spherically symmetric metric is locally expressed on a ball \(B^n(\rho)\subset R^n\) in the following form: \[ F=| y| \phi(r,s),\quad r=| x|, \quad s=\frac{\langle x,y\rangle}{| y|},\quad y\in T_xB^n(\rho)\equiv R^n. \] The main results of this paper are the following.
Theorem 1.1. Let \(F=| y| \phi(r,s)\) be a spherically symmetric Finsler metric on an open ball \(B^n(\rho)\subset R^n\), \(n\geq 3\). Then \(F\) is of constant flag curvature \(K\) if and only if \[ R_1=K\phi^2,\qquad R_2=0. \]
Here \(R_1, R_2\) and \(R_3\) are defined as expressions of \(\phi(r,s)\) and its derivatives, firstly introduced by Mo-Zhou and Mo-Zhou-Zhu. The authors find that they are related mysteriously as follows.
Theorem 1.2. Let \(F=| y| \phi(r,s)\) be a spherically symmetric Finsler metric on an open ball \(B^n(\rho)\subset R^n(n\geq 3)\). Then \(F\) is of constant flag curvature \(K\) if and only if \[ R_2=0,\qquad R_3=0. \]
Theorem 1.4. Let \(F=| y| \phi(r,s)\) be a spherically symmetric Finsler metric on an open ball \(B^n(\rho)\subset R^n(n\geq 3)\). Then \(\mathrm{Ric}_{ij}=Kg_{ij}\), with \(K\) a constant, if and only if \(\phi\) satisfies \[ (n-1)K\phi^2=(n-1)R_1+(r^2-s^2)R_2,\qquad (n+1)R_3+(r^2-s^2)[R_2]_s=0. \]
The first condition was discussed by Mo-Zhou for the characterization of spherically symmetric Finsler metrics of constant Ricci curvature.
Reviewer: Weihuan Chen (Beijing)
MSC:
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
| 53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
