We introduce a new constant for the surfaces of revolution homeomorphic to the 2-sphere. We prove a sphere theorem for radial curvature, assuming an inequality in the constant and the ratio of the difference of the maximal distance to the base point from the diameter of the reference surface and the injectivity radius of the base point. Namely, if a compact pointed Riemannian $n$-manifold which is referred to a surface of revolution satisfies the inequality, then it is topologically an $n$-sphere.
References
K.Grove and K.Shiohama, A generalized sphere theorem, Ann. of Math. 106, 201–211(1977). MR500705 10.2307/1971164 K.Grove and K.Shiohama, A generalized sphere theorem, Ann. of Math. 106, 201–211(1977). MR500705 10.2307/1971164
N. Innami, K. Shiohama and Y. Uneme, The Alexandrov–Toponogov comparison theorem for radial curvature, Nihonkai Math. J., 24, 57–91, (2013). MR3178500 euclid.nihmj/1393273815
N. Innami, K. Shiohama and Y. Uneme, The Alexandrov–Toponogov comparison theorem for radial curvature, Nihonkai Math. J., 24, 57–91, (2013). MR3178500 euclid.nihmj/1393273815
Y. Itokawa, Y. Machigashira and K. Shiohama, Maximal diameter theorems for manifolds with restricted radial curvature, Tohoku Math. Publ., 20, 61–68, (2001). MR1864887Y. Itokawa, Y. Machigashira and K. Shiohama, Maximal diameter theorems for manifolds with restricted radial curvature, Tohoku Math. Publ., 20, 61–68, (2001). MR1864887
Y.Itokawa, Y.Machigashira and K.Shiohama, Generalized Toponogov's theorem for manifolds with radial curvature bounded below, Exploration in complex and Riemannian geometry, Contemporary Math. 332, 121–130(2003). MR2018335 10.1090/conm/332/05932 Y.Itokawa, Y.Machigashira and K.Shiohama, Generalized Toponogov's theorem for manifolds with radial curvature bounded below, Exploration in complex and Riemannian geometry, Contemporary Math. 332, 121–130(2003). MR2018335 10.1090/conm/332/05932
W.Klingenberg, Manifolds with restricted conjugate locus, Ann. of Math. 78, 527–547, (1963). MR159289 10.2307/1970539 W.Klingenberg, Manifolds with restricted conjugate locus, Ann. of Math. 78, 527–547, (1963). MR159289 10.2307/1970539
K.Kondo and S.Ohta, Topology of complete manifolds with radial curvature bounded from below, GAFA 17, 1237–1247(2007). MR2373016 10.1007/s00039-007-0625-8 K.Kondo and S.Ohta, Topology of complete manifolds with radial curvature bounded from below, GAFA 17, 1237–1247(2007). MR2373016 10.1007/s00039-007-0625-8
K.Kondo and M.Tanaka, Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below II, Trans. Amer. Math. Soc., 362, 6293–6324, (2010). MR2678975 10.1090/S0002-9947-2010-05031-7 K.Kondo and M.Tanaka, Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below II, Trans. Amer. Math. Soc., 362, 6293–6324, (2010). MR2678975 10.1090/S0002-9947-2010-05031-7
H.Lee, Generalized Alexandrov-Toponogov theorem for radially curved manifolds and its application. Kyushu J. Math. 59, 1–9(2005). MR2188599 10.2206/kyushujm.59.365 H.Lee, Generalized Alexandrov-Toponogov theorem for radially curved manifolds and its application. Kyushu J. Math. 59, 1–9(2005). MR2188599 10.2206/kyushujm.59.365
Y.Machigashira, Manifolds with pinched radial curvature, Proc. Amer. Math. Soc., 118, 979–985, (1993). MR1136236 10.1090/S0002-9939-1993-1136236-0 Y.Machigashira, Manifolds with pinched radial curvature, Proc. Amer. Math. Soc., 118, 979–985, (1993). MR1136236 10.1090/S0002-9939-1993-1136236-0
Y.Mashiko and K.Shiohama, Comparison geometry referred to warped product models, Tohoku Math. J. 58, 461–473(2006). MR2297194 10.2748/tmj/1170347684 euclid.tmj/1170347684
Y.Mashiko and K.Shiohama, Comparison geometry referred to warped product models, Tohoku Math. J. 58, 461–473(2006). MR2297194 10.2748/tmj/1170347684 euclid.tmj/1170347684
V. B. Marenich, Topology of complete manifolds with bounded radial curvature, Trans. Amer. Math. Soc., 352, 4451–4468, (2000). MR1779483 10.1090/S0002-9947-00-02634-9V. B. Marenich, Topology of complete manifolds with bounded radial curvature, Trans. Amer. Math. Soc., 352, 4451–4468, (2000). MR1779483 10.1090/S0002-9947-00-02634-9
V. B. Marenich and S. J. X. Mendonca, Manifolds with minimal radial curvature bounded from below and big radius, Indiana Univ. Math. J., 48, 249–274, (1999). MR1722200 10.1512/iumj.1999.48.1627V. B. Marenich and S. J. X. Mendonca, Manifolds with minimal radial curvature bounded from below and big radius, Indiana Univ. Math. J., 48, 249–274, (1999). MR1722200 10.1512/iumj.1999.48.1627
R.Sinclair and M.Tanaka, The cut locus of a two-sphere of revolution and Toponogov's comparison theorem, Tohoku Math. J. 59, 379–399(2007). MR2365347 10.2748/tmj/1192117984 euclid.tmj/1192117984
R.Sinclair and M.Tanaka, The cut locus of a two-sphere of revolution and Toponogov's comparison theorem, Tohoku Math. J. 59, 379–399(2007). MR2365347 10.2748/tmj/1192117984 euclid.tmj/1192117984
M.Tanaka, On the cut loci of a von Mangoldt's surface of revolution, J. Math. Soc. Japan 44, 631–641(1992). MR1180440 10.2969/jmsj/04440631 euclid.jmsj/1227106000
M.Tanaka, On the cut loci of a von Mangoldt's surface of revolution, J. Math. Soc. Japan 44, 631–641(1992). MR1180440 10.2969/jmsj/04440631 euclid.jmsj/1227106000
