# Article

Full entry | PDF   (0.5 MB)
Keywords:
Finsler metric; unitary invariance; isometries; Riemannian metric
Summary:
In this paper we study isometry-invariant Finsler metrics on inner product spaces over \$\mathbb{R}\$ or \$\mathbb{C}\$, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results for more specific classes of metrics, including the class of Riemannian metrics. Our main result states that for an isometry-invariant Finsler metric the only possible linear maps under which the metric is invariant are scalar multiples of isometries. Furthermore, we characterize the metrics invariant with respect to all linear maps of this type.
References:
[1] Abate, M., Patrizio, G.: Finsler metrics – a global approach. Lecture Notes in Mathematics, vol. 1591, Springer Verlag, Berlin, 1994, With applications to geometric function theory. DOI 10.1007/BFb0073980 | MR 1323428 | Zbl 0837.53001
[2] Arcozzi, N., Rochberg, R., Sawyer, E., Wick, B.D.: Distance functions for reproducing kernel Hilbert spaces. Function spaces in modern analysis, Contemp. Math., vol. 547, Amer. Math. Soc., Providence, RI, 2011, pp. 25–53. MR 2856478 | Zbl 1236.46023
[3] Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Math., vol. 33, American Mathematical Society, Providence, RI, 2001, pp. xiv+415. MR 1835418 | Zbl 0981.51016
[4] Kobayashi, S.: Geometry of bounded domains. Trans. Amer. Math. Soc. 92 (1959), 267–290. DOI 10.1090/S0002-9947-1959-0112162-5 | MR 0112162 | Zbl 0136.07102
[5] McCarthy, P.J., Rutz, S.F.: The general four-dimensional spherically symmetric Finsler space. Gen. Relativity Gravitation 25 (6) (1993), 589–602. DOI 10.1007/BF00757070 | MR 1218065 | Zbl 0806.53022
[6] Xia, H., Zhong, Ch.: A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature. Differential Geom. Appl. 43 (2015), 1–20. DOI 10.1016/j.difgeo.2015.08.001 | MR 3421873 | Zbl 1328.53031
[7] Zhong, Ch.: On unitary invariant strongly pseudoconvex complex Finsler metrics. Differential Geom. Appl. 40 (2015), 159–186. DOI 10.1016/j.difgeo.2015.02.002 | MR 3333101 | Zbl 1320.53095
[8] Zhou, L.: Spherically symmetric Finsler metrics in \$R^n\$. Publ. Math. Debrecen 80 (1–2) (2012), 67–77. MR 2920216

Partner of