Minkowski空间的等价性定理及在Finsler几何的应用
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摘要
首先利用中心仿射几何中的结果建立了Minkowski空间的等价性定理.作为在Finsler几何中的应用,我们证明满足一定条件的Landsberg空间为Berwald空间,这些条件可以是具有闭的Cartan型形式,S曲率为零或平均Berwald曲率为零.
We first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As applications in Finsler geometry, we prove that a Finsler manifold is a Berwald space if it is a Landsberg space and satisfies one of the following conditions: closed Cartan-type form, vanishing S curvature or vanishing mean Berwald curvature.
We first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As applications in Finsler geometry, we prove that a Finsler manifold is a Berwald space if it is a Landsberg space and satisfies one of the following conditions: closed Cartan-type form, vanishing S curvature or vanishing mean Berwald curvature.
引文
[1]Aikou T., Some remarks on Berwald manifolds and Landsberg manifolds, Acta Math. Academiae Paedagogicae Nyiregyháziensis, 2010, 26:39-148.
[2]Alvarez Paiva J. C., Some Problems on Finsler Geometry, Handbook of Differential Geometry, Vol. 2, NorthHolland, Elsevier, 2006.
[3]Bao D., On two curvature-driven problems in Riemann-Finsler geometry, Advanced Studies in Pure Mathematics, Math. Soc. Japan, 2007, 48:19-71.
[4]Bao D., Chern S. S., Shen Z., An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, 2000.
[5]Blaschke W., Vorlesungenüber Differentialgeometrie II, Affine Differentialgeometrie, Springer, Berlin, 1923.
[6]Bryant R. L., Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math., 2002, 28:161-203.
[7]Chern S. S., Local Equivalence and Euclidean Connections in Finsler Spaces, Chern Selected Papers, II,95-121, 1989.
[8]Chern S. S., Shen Z., Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol. 6, World Scientific,Singapore, 2005.
[9]Feng H., Li M., Adiabatic limit and connections in Finsler geometry, Communications in Analysis and Geometry, 2013, 21:607-624.
[10]Hildebrand R., Centro-affine hypersurface immersions with parallel cubic form, Beitrage zur Algebra und Geometrie, 2015, 56:593-640.
[11]IchijyōY., Finsler manifolds modeled on a Minkowski space, J. Math. Kyoto Univ., 1976, 16:639-652.
[12]IchijyōY., On special Finsler connections with the vanishing hv-curvature tensor, Tensor(N.S.), 1978, 32:149-155.
[13]Laugwitz D., Differentialgeometrie in Vektorraumen, Friedr. Vieweg&Sohn, Braunschweig, 1965.
[14]Laugwitz D., Zur Differentialgeometrie der Hyperfl(a|¨)chen in Vektorraumen und zur affingeometrischen Deutung der Theorie der Finsler-Raume, Math. Z., 1957, 67:63 74.
[15]Laugwitz D., Eine Beziehung zwischen affiner und Minkowskischer Differentialgeometrie, Publ. Math. Debrecen, 1957, 5:72-76.
[16]Li A., Simon U., Zhao G., Global Affine Differential Geometry of Hypersurfaces, W. de Gruyter, Berlin-New York, 1993.
[17]Mo X., An Introduction to Finsler Geometry, Peking University Series in Math., Vol. 1, World Scientific,Singapore, 2006.
[18]Mo X., Huang L., On characterizations of Randers norms in a Minkowski space, International Journal of Mathematics, 2010, 21:523-535.
[19]Nomizu K., Sasaki T., Affine Differential Geometry, Cambridge University Press, Cambridge, 1994.
[20]Schneider R., Zur affinen Differentialgeometrie in Groβen, I, Math. Z., 1967, 101:375-406.
[21]Schneider R., Zur affinen Differentialgeometrie im Groβen,Ⅱ, Math. Z., 1967, 102:1-8.
[22]Simon U., Schwenk-Schellschmidt A., Viesel H., Introduction to the Affine Differential Geometry of Hypersurfaces, Science University Tokyo, Tokyo, 1991.
[23]Shen Y., Shen Z., Introduction to Modern Finsler Geometry, World Scientific, Singapore, 2016.
[24]Shen Z., Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
[25]Shen Z., Lectures on Finsler Geometry, World Scientific, Singapore, 2001.
[26]Shen Z., Landsberg Curvature, S-curvature and Riemann Curvature, Riemann-Finsler Goemetry, MSRI Publications, 2004, 50:303-355.
[27]Song P., Li M., On some properties of the Cartan type one form of a Finsler manifold, Journal of Southwest University(Natural Science Edition), 2014, 36(10):119-123.
[28]Szilasi J., Vincze C., A new look at Finsler connections and special Finsler manifolds. Acta Mathematica Academiae Paedagogicae Nyiregyháziensis, 2000, 16:33-63.
[29]Zhang W., Lectures on Chern-Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, Vol.4, World Scientific, Singapore, 2001.
[2]Alvarez Paiva J. C., Some Problems on Finsler Geometry, Handbook of Differential Geometry, Vol. 2, NorthHolland, Elsevier, 2006.
[3]Bao D., On two curvature-driven problems in Riemann-Finsler geometry, Advanced Studies in Pure Mathematics, Math. Soc. Japan, 2007, 48:19-71.
[4]Bao D., Chern S. S., Shen Z., An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, 2000.
[5]Blaschke W., Vorlesungenüber Differentialgeometrie II, Affine Differentialgeometrie, Springer, Berlin, 1923.
[6]Bryant R. L., Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math., 2002, 28:161-203.
[7]Chern S. S., Local Equivalence and Euclidean Connections in Finsler Spaces, Chern Selected Papers, II,95-121, 1989.
[8]Chern S. S., Shen Z., Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol. 6, World Scientific,Singapore, 2005.
[9]Feng H., Li M., Adiabatic limit and connections in Finsler geometry, Communications in Analysis and Geometry, 2013, 21:607-624.
[10]Hildebrand R., Centro-affine hypersurface immersions with parallel cubic form, Beitrage zur Algebra und Geometrie, 2015, 56:593-640.
[11]IchijyōY., Finsler manifolds modeled on a Minkowski space, J. Math. Kyoto Univ., 1976, 16:639-652.
[12]IchijyōY., On special Finsler connections with the vanishing hv-curvature tensor, Tensor(N.S.), 1978, 32:149-155.
[13]Laugwitz D., Differentialgeometrie in Vektorraumen, Friedr. Vieweg&Sohn, Braunschweig, 1965.
[14]Laugwitz D., Zur Differentialgeometrie der Hyperfl(a|¨)chen in Vektorraumen und zur affingeometrischen Deutung der Theorie der Finsler-Raume, Math. Z., 1957, 67:63 74.
[15]Laugwitz D., Eine Beziehung zwischen affiner und Minkowskischer Differentialgeometrie, Publ. Math. Debrecen, 1957, 5:72-76.
[16]Li A., Simon U., Zhao G., Global Affine Differential Geometry of Hypersurfaces, W. de Gruyter, Berlin-New York, 1993.
[17]Mo X., An Introduction to Finsler Geometry, Peking University Series in Math., Vol. 1, World Scientific,Singapore, 2006.
[18]Mo X., Huang L., On characterizations of Randers norms in a Minkowski space, International Journal of Mathematics, 2010, 21:523-535.
[19]Nomizu K., Sasaki T., Affine Differential Geometry, Cambridge University Press, Cambridge, 1994.
[20]Schneider R., Zur affinen Differentialgeometrie in Groβen, I, Math. Z., 1967, 101:375-406.
[21]Schneider R., Zur affinen Differentialgeometrie im Groβen,Ⅱ, Math. Z., 1967, 102:1-8.
[22]Simon U., Schwenk-Schellschmidt A., Viesel H., Introduction to the Affine Differential Geometry of Hypersurfaces, Science University Tokyo, Tokyo, 1991.
[23]Shen Y., Shen Z., Introduction to Modern Finsler Geometry, World Scientific, Singapore, 2016.
[24]Shen Z., Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
[25]Shen Z., Lectures on Finsler Geometry, World Scientific, Singapore, 2001.
[26]Shen Z., Landsberg Curvature, S-curvature and Riemann Curvature, Riemann-Finsler Goemetry, MSRI Publications, 2004, 50:303-355.
[27]Song P., Li M., On some properties of the Cartan type one form of a Finsler manifold, Journal of Southwest University(Natural Science Edition), 2014, 36(10):119-123.
[28]Szilasi J., Vincze C., A new look at Finsler connections and special Finsler manifolds. Acta Mathematica Academiae Paedagogicae Nyiregyháziensis, 2000, 16:33-63.
[29]Zhang W., Lectures on Chern-Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, Vol.4, World Scientific, Singapore, 2001.