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초록· 키워드
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Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.
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참고문헌 (18)
참고문헌 신청
H. Akbar-Zadeh / 1988 / Sur les espaces de Finsler a courbures sectionnelles constantes / Acad. Roy. Belg. Bull. Cl. Sci. 74(10):281~322
H. Akbar-Zadeh / 1995 / Generalized Einstein manifolds / J. Geom. Phys 17(4):342~380
D. Bao / 2000 / Graduate Texts in Mathematics, 200 / Springer-Verlag
D. Bao / 2004 / Ricci and ag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry / Math. Sci. Res. Inst. Publ. 50:197~259
B. Bidabad / 2014 / A classication of complete Finsler manifolds through the conformal theory of curves / Dierential Geom. Appl 35:350~360
H. Akbar-Zadeh / 1995 / Generalized Einstein manifolds / J. Geom. Phys 17(4):342~380
D. Bao / 2000 / Graduate Texts in Mathematics, 200 / Springer-Verlag
D. Bao / 2004 / Ricci and ag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry / Math. Sci. Res. Inst. Publ. 50:197~259
B. Bidabad / 2014 / A classication of complete Finsler manifolds through the conformal theory of curves / Dierential Geom. Appl 35:350~360
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