COMPOSITION OF CONFORMAL AND PROJECTIVE MAPPINGS OF GENERALIZED RIEMANNIAN SPACES IN EISENHART’S SENSE PRESERVING CERTAIN TENSORS

Miloš Z. Petrović

DOI Number
https://doi.org/10.22190/FUMI240905059P
First page
873
Last page
889

Abstract


The composition of conformal and projective mappings between Riemannian spaces that were at the same time harmonic had been studied by S. E. Stepanov,
I. G. Shandra in 2003 and further developed in I. Hinterleitner’s Ph.D. thesis in 2009. Conformal and projective mappings of Riemannian spaces preserving certain tensors were studied by O. Chepurna in the 2012 Ph.D. thesis. We consider conformal and projective mappings of generalized Riemannian spaces in Eisenhart’s sense and find necessary and sufficient conditions for these mappings to preserve curvature, Ricci and traceless Ricci tensors and some of their linear combinations. Particularly, as an additional contribution to related results collected in the Ph.D. thesis by O. Chepurna, we find that the following result holds in the case of Riemannian spaces: if a conformal mapping f1 : M → M^ is preserving the traceless Ricci tensor and a projective mapping f2 : M^ → M is preserving the traceless Ricci tensor then the Yano tensor of concircular curvature is invariant with respect to the composition f3 = f1 ◦ f2 : M → M.

Keywords

conformal mapping, geodesic mapping, generalized Riemannian space, Riemannian curvature tensor, traceless Ricci tensor, Weyl’s tensor of projective curvature, Weyl’s conformal curvature tensor, Yano’s tensor of concircular curvature.

Full Text:

PDF

References


V. E. Berezovski, S. Bácsó and J. Mikeš: Diffeomorphism of affine connected spaces which preserved Riemannian and Ricci curvature tensors. Miskolc Math. Notes 18 (2017), 117–124.

O. Chepurna: Diffeomorphisms of Riemannian spaces with preserved Einstein tensor. Ph. D. Thesis, Palacký University, Olomouc (2012).

L. P. Eisenhart: Generalized Riemannian spaces I. Proc. Natl. Acad. Sci. USA 37 (1951), 311–315.

I. Hinterleitner: Special mappings of equidistant spaces. Ph. D. Thesis, Brno University of Technology, Brno (2009).

S. M. Minčić: Independent curvature tensors and pseudotensors of spaces with nonsymmetric affine connexion. In: Differential geometry. (Colloquium on Differential Geometry held in Budapest from 3 to 7 September 1979, organized by the Janos Bolyai Mathematical Society) (Gy. Soos, J. Szenthe, eds.), North-Holland Publishing

Company, Amsterdam (1982), 445–460.

S. M. Minčić, M. S. Stanković and Lj. S. Velimirović: Generalized Riemannian spaces and spaces of non-symmetric affine connection (1st ed.). University of Niš, Faculty of sciences and mathematics, Niš (2013).

J. Mikeš: Differential geometry of special mappings (1st ed.). Palacký Univ. Press, Faculty of Science, Olomouc (2015).

M. Z. Petrović: On composition of geodesic and conformal mappings between generalized Riemannian spaces preserving certain tensors. Presented at the 8th European Congress of Mathematics, University of Primorska, Portorož (2021).

M. Z. Petrović, M. S. Stanković and P. Peška: On conformal and concircular diffeomorphisms of Eisenhart’s generalized Riemannian spaces. Mathematics 7 626, (2019).

M. Z. Petrović and A. M. Velimirović: Projective curvature tensors of some special manifolds with non-symmetric linear connection. Mediterr. J. Math. 18 124, (2020).




DOI: https://doi.org/10.22190/FUMI240905059P

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)