Intrinsic geometry and analysis of Finsler structures. (English) Zbl 1484.53049
Summary: In this short note, we prove that if \(F\) is a weak upper semicontinuous admissible Finsler structure on a domain in \(\mathbb {R}^n\), \(n\geq 2\), then the intrinsic distance and differential structures coincide.
MSC:
| 53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
| 46E99 | Linear function spaces and their duals |
References:
| [1] | Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405-1490 (2014) · Zbl 1304.35310 · doi:10.1215/00127094-2681605 |
| [2] | Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry, vol. 200. Graduate Texts in Mathematics. Springer, New York (2000) · Zbl 0954.53001 |
| [3] | Briani, A., Davini, A.: Monge solutions for discontinuous Hamiltonians. ESAIM Control Optim. Calc. Var. 11(2), 229-251 (2005) · Zbl 1087.35023 · doi:10.1051/cocv:2005004 |
| [4] | Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, vol. 33. Graduate Studies in Mathematics. American Mathematical Society, Providence (2001) · Zbl 0981.51016 |
| [5] | Davini, A.: Smooth approximation of weak Finsler metrics. Differ. Integral Equ. 18(5), 509-530 (2005) · Zbl 1212.41092 |
| [6] | De Cecco, G., Palmieri, G.: Intrinsic distance on a LIP Finslerian manifold. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 17, 129-151 (1993) · Zbl 0852.53050 |
| [7] | De Cecco, G., Palmieri, G.: LIP manifolds: from metric to Finslerian structure. Math. Z. 218(2), 223-237 (1995) · Zbl 0819.53014 · doi:10.1007/BF02571901 |
| [8] | Garroni, A., Ponsiglione, M., Prinari, F.: From 1-homogeneous supremal functionals to difference quotients: relaxation and \[\Gamma\] Γ-convergence. Calc. Var. Partial Differ. Equ. 27(4), 397-420 (2006) · Zbl 1105.47056 · doi:10.1007/s00526-005-0354-5 |
| [9] | Guo, C.-Y., Xiang, C.-L., Yang, D.: The \[L^\infty\] L∞-variational problems associated to measurable Finsler structures. Nonlinear Anal. 132, 126-140 (2016) · Zbl 1329.49004 · doi:10.1016/j.na.2015.10.025 |
| [10] | Koskela, P., Shanmugalingam, N., Zhou, Y.: Intrinsic geometry and analysis of diffusion processes and \[L^\infty\] L∞-variational problems. Arch. Ration. Mech. Anal. 214(1), 99-142 (2014) · Zbl 1302.60113 · doi:10.1007/s00205-014-0755-8 |
| [11] | Koskela, P., Zhou, Y.: Geometry and analysis of Dirichlet forms. Adv. Math. 231(5), 2755-2801 (2012) · Zbl 1253.53035 · doi:10.1016/j.aim.2012.08.004 |
| [12] | Sturm, K.T.: Is a diffusion process determined by its intrinsic metric? Chaos Solitons Fractals 8(11), 1855-1860 (1997) · Zbl 0939.58032 · doi:10.1016/S0960-0779(97)00030-1 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
