Affine functions and Busemann functions on complete Finsler manifolds
HTML articles powered by AMS MathViewer
- by Nobuhiro Innami, Yoe Itokawa, Tetsuya Nagano and Katsuhiro Shiohama PDF
- Proc. Amer. Math. Soc. 149 (2021), 1723-1732 Request permission
Abstract:
We clarify the relation between an affine function and a Busemann function in a geodesically complete Finsler manifold. As an application, we give the characterization of a Minkowski space by means of the dimension of the vector space consisting of all affine functions on a Finsler manifold.References
- D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675, DOI 10.1007/978-1-4612-1268-3
- Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
- H. Busemann and B. B. Phadke, Peakless and monotone functions on $G$-spaces, Tsukuba J. Math. 7 (1983), no. 1, 105–135. MR 703671, DOI 10.21099/tkbjm/1496159666
- R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: topological structure, Invent. Math. 63 (1981), no. 1, 129–157. MR 608531, DOI 10.1007/BF01389196
- R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: differentiable structure, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 4, 357–367 (1982). MR 654202, DOI 10.24033/asens.1410
- Nobuhiro Innami, Differentiability of Busemann functions and total excess, Math. Z. 180 (1982), no. 2, 235–247. MR 661700, DOI 10.1007/BF01318907
- Nobuhiro Innami, Splitting theorems of Riemannian manifolds, Compositio Math. 47 (1982), no. 3, 237–247. MR 681608
- Nobuhiro Innami, A classification of Busemann $G$-surfaces which possess convex functions, Acta Math. 148 (1982), 15–29. MR 666106, DOI 10.1007/BF02392724
- Nobuhiro Innami, Yoe Itokawa, Tetsuya Nagano, and Katsuhiro Shiohama, Blaschke Finsler manifolds and actions of projective Randers changes on cut loci, Trans. Amer. Math. Soc. 371 (2019), no. 10, 7433–7450. MR 3939582, DOI 10.1090/tran/7603
- Yukihiro Mashiko, Affine functions on Alexandrov surfaces, Osaka J. Math. 36 (1999), no. 4, 853–859. MR 1745653
- Yukihiro Mashiko, A splitting theorem for Alexandrov spaces, Pacific J. Math. 204 (2002), no. 2, 445–458. MR 1907900, DOI 10.2140/pjm.2002.204.445
- Sorin V. Sabau and Katsuhiro Shiohama, Topology of complete Finsler manifolds admitting convex functions, Pacific J. Math. 276 (2015), no. 2, 459–481. MR 3374069, DOI 10.2140/pjm.2015.276.459
- Zhongmin Shen, Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001. MR 1845637, DOI 10.1142/9789812811622
- Yoe Itokawa, Katsuhiro Shiohama, and Bankteshwar Tiwari, Strictly convex functions on complete Finsler manifolds, Proc. Indian Acad. Sci. Math. Sci. 126 (2016), no. 4, 623–627. MR 3568254, DOI 10.1007/s12044-016-0307-2
Additional Information
- Nobuhiro Innami
- Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan
- MR Author ID: 199776
- Yoe Itokawa
- Affiliation: Department of Communication and Information Science, Fukuoka Institute of Technology, Fukuoka, 811–0295, Japan
- MR Author ID: 261071
- Tetsuya Nagano
- Affiliation: Department of Information Science, University of Nagasaki, Nagasaki, Japan
- MR Author ID: 254796
- Katsuhiro Shiohama
- Affiliation: Institute of Information Sciences, Fukuoka Institute of Technology, Fukuoka, 811–0295, Japan
- MR Author ID: 160870
- Received by editor(s): June 13, 2020
- Received by editor(s) in revised form: August 4, 2020
- Published electronically: February 11, 2021
- Additional Notes: Research was partially supported by the JSPS KAKENHI Grant Number JP18K03314.
- Communicated by: Jiaping Wang
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 1723-1732
- MSC (2020): Primary 53C20; Secondary 53C22
- DOI: https://doi.org/10.1090/proc/15304
- MathSciNet review: 4242326