A general version of Beltrami's theorem in Finslerian setting
We present a new proof of a Finslerian version of Beltrami's theorem (1865) which works also in dimension 2.
math.DG↗
arXiv subjects
Georgeta Creţu
Publications and source records attributed to Georgeta Creţu.
A characterisation for Finsler metrics of constant curvature and a Finslerian version of Beltrami Theorem
We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.
math.DG↗
Frobenius integrability and Finsler metrizability for $2$-dimensional sprays
For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous $1$-form from the annihilator of the Berwald distribution. We discuss both the degenerate and non-degenerate cases using the fact that the regularity of the Finsler function is encoded into a regularity condition of a $2$-form, canonically associated to the given spray. The integrability of the Berwald distribution and the regularity of the $2$-form have simple and useful expressions in terms of the Berwald frame.
math.DG↗