On twisted Riemannian extensions associated with Szabó metrics

Abstract

Let $M$ be an n-dimensional manifold with a torsion free afine connection $\nabla$ and let $T^*M$ be the cotangent bundle. In this paper, we consider some of the geometric aspects of a twisted Riemannian extension which provide a link between the afine geometry of $(M,\nabla)$ and the neutral signature pseudo-Riemannian geometry of $T^*M$. We investigate the spectral geometry of the Szabó operator on $M$ and on $T^*M$.

Keywords

• Afine connection,Cyclic parallel,Szabó manifold,Twisted Riemannian extension

References

1. Brozos-Vázquez, M., García-Rio, E., Gilkey, P., Nikevi¢, S. and Vázquez-Lorenzo, R. The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics 5 (Morgan and Claypool Publishers, 2009).
2. Calvino-Louzao, E, García-Rio, E., Gilkey, P. and Vázquez-Lorenzo, R. The geometry of modied Riemannian extensions, Proc. R. Soc. A. 465, 2023-2040, 2009.
3. Calvino-Louzao, E., García-Rio, E. and Vázquez-Lorenzo, R. Riemann extensions of torsion-free connections with degenerate Ricci tensor, Canad. J. Math. 62 (5), 1037-1057, 2010.
4. Diallo, A. S. and Massamba, F. Ane Szabó connections on smooth manifolds, Rev. Un. Mat. Argentina 58 (1), 37-52, 2017.
5. Fiedler, B. and Gilkey, P. Nilpotent Szabo, Osserman and Ivanov-Petrova pseudo- Riemannian manifolds, Contemp. Math. 337, 53-64, 2003.
6. García-Río, E., Gilkey, P., Nikcevi¢, S. and Vázquez-Lorenzo, R. Applications of Ane and Weyl Geometry, Synthesis Lectures on Mathematics and Statistics 13 (Morgan and Claypool Publishers, 2013).
7. García-Rio, E., Kupeli, D. N., Vázquez-Abal, M. E. and Vázquez-Lorenzo, R. Ane Osserman connections and their Riemannian extensions, Dierential Geom. Appl. 11, 145-153, 1999.
8. Gilkey, P. B., Ivanova, R. and Zhang, T. Szabó Osserman IP Pseudo-Riemannian manifolds, Publ. Math. Debrecen 62, 387-401, 2003.

9. Gilkey, P. and Stavrov, I. Curvature tensors whose Jacobi or Szabó operator is nilpotent on null vectors, Bull. London Math. Soc. 34 (6), 650-658, 2002.
10. Kowalski, O. and Sekizawa, M. The Riemann extensions with cyclic parallel Ricci tensor, Math. Nachr. 287 (8-9), 955-961, 2014.
11. Patterson, E. M. and Walker, A. G. Riemann extensions, Quart. J. Math. Oxford Ser. 3, 19-28, 1952.
12. Szabó, Z. I. A short topological proof for the symmetry of 2 point homogeneous spaces, Invent. Math. 106, 61-64, 1991.
13. Yano, K. and Ishihara, S. Tangent and cotangent bundles: dierential geometry, Pure and Applied Mathematics 16 ( Marcel Dekker, New York, 1973).