HOMOGENEOUS GEODESICS IN HOMOGENEOUS SUB-FINSLER MANIFOLDS
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In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of $\ell^{p}$-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.
Abstract
In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of $\ell^{p}$-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.
참고문헌 (20)
- A. D. Aleksandrov, "A theorem on triangles in a metric space and some of its applications", Trudy Mat. Inst. Steklov. Izdat. Akad. Nauk SSSR, Moscow, 38, pp.5~23 (1951)
- D. Bao, "An Introduction to Riemann-Finsler Geometry", Graduate Texts in Mathematics (2000), Springer New York[DOI]
- V. N. Berestovskii, "Homogeneous manifolds with intrinsic metric. II", Siberian Mathematical Journal, 30(2), pp.180~191 (1989), Springer Science and Business Media LLC[DOI]
- V. Berestovskii, "Riemannian manifolds and homogeneous geodesics, Springer Monographs in Mathematics" (2020), Springer
- Salomon Bochner, "Locally Compact Groups of Differntiable Transformations", The Annals of Mathematics, 47(4), pp.639~ (1946), JSTOR[DOI]
- Dmitri Burago, "A Course in Metric Geometry", Graduate Studies in Mathematics (2001), American Mathematical Society[DOI]
- Luca Capogna, "Smoothness of subRiemannian isometries", American Journal of Mathematics, 138(5), pp.1439~1454 (2016), Project MUSE[DOI]
- Shiing-Shen Chern, "Riemann-Finsler Geometry", Nankai Tracts in Mathematics (2005), WORLD SCIENTIFIC[DOI]
- D. van Dantzig, "Über metrisch homogene räume", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 6(1), pp.367~376 (1928), Springer Science and Business Media LLC[DOI]
- S. Q. Deng, "Homogeneous Finsler spaces, Springer Monographs in Mathematics" (2012), Springer
- Shaoqiang Deng, "The group of isometries of a Finsler space", Pacific Journal of Mathematics, 207(1), pp.149~155 (2002), Mathematical Sciences Publishers[DOI]
- Z. Dusek, "Survey on homogeneous geodesics", Note Mat, 28(suppl. 1), pp.147~168 (2009)
- Zdeněk Dušek, "The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds", Journal of Geometry and Physics, 60(5), pp.687~689 (2010), Elsevier BV[DOI]
- M. Gromov, "Math. Sci. Res. Inst. Publ", pp.8~ (1987), Springer
- Oldřich Kowalski, "On the existence of homogeneous geodesics in homogeneous Riemannian manifolds", Geometriae Dedicata, 81(1, pp.3) (2000), 209~214[DOI]
- O. Kowalski, "Riemannian manifolds with homogeneous geodesics", Boll. Un. Mat. Ital. B, 5(7), pp.189~246 (1991)
- S. B. Myers, "The Group of Isometries of a Riemannian Manifold", The Annals of Mathematics, 40(2), pp.400~ (1939), JSTOR[DOI]
- Alexey Podobryaev, "Homogeneous geodesics in sub-Riemannian geometry", ESAIM: Control, Optimisation and Calculus of Variations, 29, pp.11~ (2023), EDP Sciences[DOI]
- Z. Shen, "Lectures on Finsler Geometry" (2001), World Sci. Publishing
- Zaili Yan, "On the existence of homogeneous geodesic in homogeneous Finsler spaces", Journal of Geometry and Physics, 124, pp.264~267 (2018), Elsevier BV[DOI]