# zbMATH — the first resource for mathematics

Structure of geodesics in weakly symmetric Finsler metrics on H-type groups. (English) Zbl 07285964
Summary: Structure of geodesic graphs in special families of invariant weakly symmetric Finsler metrics on modified H-type groups is investigated. Geodesic graphs on modified H-type groups with the center of dimension 1 or 2 are constructed. The new patterns of algebraic complexity of geodesic graphs are observed.
##### MSC:
 53C22 Geodesics in global differential geometry 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 53C30 Differential geometry of homogeneous manifolds
Full Text:
##### References:
 [1] Alekseevsky, D.; Arvanitoyeorgos, A., Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc. 359 (2007), 3769-3789 [2] Bao, D.; Chern, S.-S.; Shen, Z., An Introduction to Riemann-Finsler Geometry, Springer Science+Business Media, New York, 2000 [3] Berndt, J.; Kowalski, O.; Vanhecke, L., Geodesics in weakly symmetric spaces, Ann. Glob. Anal. Geom. 15 (1997), 153-156 [4] Berndt, J.; Tricerri, F.; Vanhecke, L., Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Math., vol. 1598, Springer-Verlag, Berlin-Heidelberg-New York, 1995 [5] Deng, S., Homogeneous Finsler Spaces, Springer Science+Business Media, New York, 2012 [6] Dušek, Z., Explicit geodesic graphs on some H-type groups, Rend. Circ. Mat. Palermo, Serie II, Suppl. 69 (2002), 77-88 [7] Dušek, Z., Structure of geodesics in the flag manifold $${\rm SO}(7)/{\rm U}(3)$$, Differential Geometry and its Applications, Proc. 10th Int. Conf. (Kowalski, O., Krupka, D., Krupková, O., Slovák, J., eds.), World Scientific, 2008, pp. 89-98 [8] Dušek, Z., Homogeneous geodesics and g.o. manifolds, Note Mat. 38 (2018), 1-15 [9] Dušek, Z., Geodesic graphs in Randers g.o. spaces, Comment. Math. Univ. Carolin. 61 (2) (2020), 195-211 [10] Dušek, Z.; Kowalski, O., Geodesic graphs on the $$13$$-dimensional group of Heisenberg type, Math. Nachr. 254-255 (2003), 87-96 [11] Gordon, C. S.; Nikonorov, Yu. G., Geodesic orbit Riemannian structures on $$\mathbb{R}^n$$, J. Geom. Phys. 134 (2018), 235-243 [12] Kowalski, O.; Nikčević, S., On geodesic graphs of Riemannian g.o. spaces, Arch. Math. 73 (1999), 223-234, Appendix: Arch. Math. 79 (2002), 158-160 [13] Kowalski, O.; Vanhecke, L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Math. Ital. B(7) 5 (1991), 189-246 [14] Latifi, D., Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phys. 57 (2007), 1421-1433 [15] Lauret, J., Modified H-type groups and symmetric-like Riemannian spaces, Differential Geom. Appl. 10 (1999), 121-143 [16] Riehm, C., Explicit spin representations and Lie algebras of Heisenberg type, J. London Math. Soc. (2) 32 (1985), 265-271 [17] Szenthe, J., Sur la connection naturelle à torsion nulle, Acta Sci. Math. (Szeged) 38 (1976), 383-398 [18] Yan, Z.; Deng, S., Finsler spaces whose geodesics are orbits, Differential Geom. Appl. 36 (2014), 1-23
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.