×

On symmetric Finsler spaces of \(H_ p\)-scalar curvature and scalar curvature. (English) Zbl 0607.53016

A Finsler space \(F_ n\) is said to be of Hp-scalar curvature if \(p\cdot H_{\ell ijr}=k(h_{\ell j} h_{ir}-h_{\ell r} h_{ij})\), where \(H_{\ell ijr}\) is the Berwald h-curvature tensor, p is an operator projecting on the indicatrix, \(h_{ij}\) is the angular metric tensor, and k is the curvature scalar. It is proved that for any Berwald space of Hp-scalar curvature \(p\circ H_ i^{\ell}{}_{jr| m}=0\) (\(|\) denoting the h-covariant derivative for the Cartan connection). Some other theorems concerning symmetric \(F_ n\) of Hp-scalar curvature resp. of scalar curvature are given.
Reviewer: L.Tamássy

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. Izumi andT. N. Srivastava, On R3-like Finsler spaces,Tensor (N. S.) 32 (1978), 339–349.MR 80a: 53030 · Zbl 0413.53012
[2] H. Izumi andM. Yoshida, Remarks on Finsler spaces of perpendicular scalar curvature and the propertyH, Tensor (N. S.) 40 (1983), 215–220.Zbl 538: 53033 · Zbl 0538.53033
[3] M. Matsumoto, On the indicatrices of a Finsler space,Period. Math. Hungar. 8 (1977), 185–191.MR 58: 18236 · Zbl 0355.53010 · doi:10.1007/BF02018504
[4] M. Matsumoto,Foundation of Finsler geometry and special Finsler spaces. (To appear) · Zbl 0594.53001
[5] M. Matsumoto andL. Tamássy, Scalar and gradient vector fields of Finsler spaces and holonomy groups of non-linear connections,Demonstratio Math. 13 (1980), 551–564.MR 82c: 53046 · Zbl 0444.53045
[6] R. B. Misra, A symmetric Finsler space,Tensor (N. S.) 24 (1972), 346–350.MR 48: 12436 · Zbl 0232.53034
[7] T. Sakaguchi, On Finsler spaces of scalar curvature,Tensor (N. S.) 38 (1982), 211–219.Zbl 505: 53008 · Zbl 0505.53008
[8] M. Yoshida, On Finsler spaces ofHp-scalar curvature,Tensor (N. S.) 38 (1982), 205–210.Zbl 505: 53007 · Zbl 0505.53007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.