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Wind Riemannian spaceforms and Randers–Kropina metrics of constant flag curvature

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Abstract

We provide an application of the recently introduced wind Riemmanian structures to the understanding of Randers metrics of constant flag curvature (CFC). Any wind Riemannian structure is constructed from a Riemannian metric and a vector field with no restriction on its norm. The local and global classifications for wind Riemannian structures of CFC are obtained, generalizing both, the celebrated classification of the CFC Randers case by Bao, Robles and Shen, and its extension to the Kropina case by Yoshikawa, Okubo and Sabau. Remarkably, any incomplete CFC Randers metric can be extended globally to a complete wind Riemannian structure of CFC, providing then a neat interpretation of the Randers global classification.

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Notes

  1. Recall that this link appears already at the level of classical relativistic spacetimes, even if no Finslerian modification of these spacetimes (say, in the spirit of Asanov [1]) is introduced. However, the viewpoint of WRS’s and, with more generality, wind Finslerian structures (also developed in [9]) becomes applicable to Lorentz–Finsler spacetimes, see [17].

  2. It is easy to check that and W are then univocally determined. In [9], general wind Finslerian structures are introduced in a more abstract way, and wind Riemannian ones are then regarded as a particular case of that definition. However, both approaches are clearly equivalent (see [9, Proposition 2.13]).

  3. The critical region is just a closed subset of M. So, more properly, is a Kropina norm at each point p and F becomes a classical Kropina metric in the interior of the critical region.

  4. This property does not hold for the general wind Finslerian structures studied in [9].

  5. So, sometimes a different representative of the conformal class of the SSTK may be preferred. For example, the normalization \(\Lambda \equiv 1\) was chosen in the case of Randers metrics and standard stationary spacetimes studied in [8]. (This is the reason why we preferred to write the metric \(g_0\) in (7) even if it is taken later).

  6. Recall that the equality \(F(\dot{\gamma })= F_l(\dot{\gamma })=c>0\) close to is compatible with because of the Kropina character of \(\Sigma \) at .

  7. Notice that they are affine, and all the affine vector fields of \(\mathbb {R}^n\) have affine natural coordinates. Thus, their completeness follows by direct integration (or by applying general results such as [28, Theorem 1]).

  8. An alternative way of producing this type of examples appears considering the spacetime viewpoint. Take any SSTK splitting with a Cauchy hypersurface which inherits an incomplete Riemannian metric (a Cauchy hypersurface S satisfying this property can be easily constructed in Lorentz–Minkowski spacetime, and moving S with the flow of the natural timelike parallel vector field \(K=\partial _t\) one obtains the required SSTK splitting). The completeness for the induced Randers metric [obtained from the incomplete and the wind W in (7) and (8)] is a consequence of the fact that S is Cauchy (see [8, Theorem 4.4] or the more general [9, Theorem 5.11 (iv)]).

  9. This proof follows the spirit of Berger’s one [6] to prove that any Killing vector field on an even dimensional compact manifold of positive curvature must have a zero.

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Acknowledgements

The authors warmly acknowledge the help and support they have received from Professor D. Bao (San Francisco State University). We especially thank him for confirming our assumption that the computations behind [2, 3] with remain valid for the strict inequality , a crucial step in the proof of Theorem 3.8. Comments by the referees are also acknowledged.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100, Espinardo, Murcia, Spain

    Miguel Angel Javaloyes

  2. Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071, Granada, Spain

    Miguel Sánchez

Authors
  1. Miguel Angel Javaloyes
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  2. Miguel Sánchez
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Correspondence to Miguel Sánchez.

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The authors are partially supported by Spanish MINECO projects with ERDF funds, references MTM2015-65430-P, (MAJ) and MTM2013-47828-C2-1-P (MS).

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Javaloyes, M.A., Sánchez, M. Wind Riemannian spaceforms and Randers–Kropina metrics of constant flag curvature. European Journal of Mathematics 3, 1225–1244 (2017). https://doi.org/10.1007/s40879-017-0186-9

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  • DOI: https://doi.org/10.1007/s40879-017-0186-9

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