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In this paper, we study a class of Finsler metric. First, we find some rigidity results of the dually flat $(\alpha,\beta)$-metric where the underline Riemannian metric $\alpha$ satisfies nonnegative curvature properties. We give a new geometric approach of the Monge-Amp\'ere type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.
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참고문헌 (11)
참고문헌 신청
L. Caffarelli / 2003 / An extension to a theorem of Jorgens, Calabi, and Pogorelov / Comm. Pure Appl. Math 56(5):549~583
E. Calabi / 1958 / Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jorgens / Michigan Math. J 5:105~126
A. Caminha / 2010 / Complete foliations of space forms by hypersurfaces / Bull. Braz. Math. Soc. (N.S.) 41(3):339~353
X. Cheng / 2012 / Finsler Geometry–An approach via Ransders spaces / Springer, Science Press
X. Cheng / 2010 / On locally dually flat Randers metrics / Internat. J. Math 21:1531~1543
E. Calabi / 1958 / Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jorgens / Michigan Math. J 5:105~126
A. Caminha / 2010 / Complete foliations of space forms by hypersurfaces / Bull. Braz. Math. Soc. (N.S.) 41(3):339~353
X. Cheng / 2012 / Finsler Geometry–An approach via Ransders spaces / Springer, Science Press
X. Cheng / 2010 / On locally dually flat Randers metrics / Internat. J. Math 21:1531~1543
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