Immersions of positively curved manifolds into manifolds with curvature bounded above
HTML articles powered by AMS MathViewer
- by Nadine L. Menninga PDF
- Trans. Amer. Math. Soc. 318 (1990), 809-821 Request permission
Abstract:
Let $M$ be a compact, connected, orientable Riemannian manifold of dimension $n - 1 \geqslant 2$, and let $x$ be an isometric immersion of $M$ into an $n$-dimensional Riemannian manifold $N$. Let $K$ denote sectional curvature and $i$ denote the injectivity radius. Assume, for some constant positive constant $c$, that $K(N) \leqslant 1/(4{c^2}),\quad 1/{c^2} \leqslant K(M)$, and $\pi c \leqslant i(N)$. Then the radius of the smallest $N$-ball containing $x(M)$ is less than $\tfrac {1} {2}\pi c$ and $x$ is in fact an imbedding of $M$ into $N$, whose image bounds a convex body.References
- S. Alexander, Locally convex hypersurfaces of negatively curved spaces, Proc. Amer. Math. Soc. 64 (1977), no. 2, 321–325. MR 448262, DOI 10.1090/S0002-9939-1977-0448262-6
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335 M. P. do Carmo and E. Lima, Immersons of manifolds with nonnegative sectional curvatures, Bol. Soc. Brasil. Mat. 2 (1972), 9-22.
- M. P. do Carmo and F. W. Warner, Rigidity and convexity of hypersurfaces in spheres, J. Differential Geometry 4 (1970), 133–144. MR 266105
- Shiing-shen Chern and Richard K. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306–318. MR 84811, DOI 10.2307/2372684 J. Hadamard, Sur certaines proprietes des trajectories en dynamique, J. Math. Pures Appl. 3 (1897), 331-387.
- John Van Heijenoort, On locally convex manifolds, Comm. Pure Appl. Math. 5 (1952), 223–242. MR 52131, DOI 10.1002/cpa.3160050302
- H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), no. 5, 509–541. MR 442975, DOI 10.1002/cpa.3160300502
- Richard Sacksteder, On hypersurfaces with no negative sectional curvatures, Amer. J. Math. 82 (1960), 609–630. MR 116292, DOI 10.2307/2372973 M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Berkeley, Calif., 1979.
- Joel Spruck, On the radius of the smallest ball containing a compact manifold of positive curvature, J. Differential Geometry 8 (1973), 257–258. MR 339005
- J. J. Stoker, Über die Gestalt der positiv gekrümmten offenen Flächen im dreidimensionalen Raume, Compositio Math. 3 (1936), 55–88 (German). MR 1556933 V. A. Toponogov, Riemannian spaces having their curvature bounded below by a positive number, Amer. Math. Soc. Transl. 37 (1964), 291-336.
- Ivan A. Tribuzy, Convex immersions into positively-curved manifolds, Bol. Soc. Brasil. Mat. 17 (1986), no. 1, 21–39. MR 869706, DOI 10.1007/BF02585474
- F. W. Warner, Extensions of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc. 122 (1966), 341–356. MR 200873, DOI 10.1090/S0002-9947-1966-0200873-6
- H. Wu, The spherical images of convex hypersurfaces, J. Differential Geometry 9 (1974), 279–290. MR 348685
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 318 (1990), 809-821
- MSC: Primary 53C42; Secondary 53C40
- DOI: https://doi.org/10.1090/S0002-9947-1990-0962285-0
- MathSciNet review: 962285