Compactness criteria for Riemannian manifolds
HTML articles powered by AMS MathViewer
- by Gregory J. Galloway
- Proc. Amer. Math. Soc. 84 (1982), 106-110
- DOI: https://doi.org/10.1090/S0002-9939-1982-0633289-3
- PDF | Request permission
Abstract:
Ambrose, Calabi and others have obtained Ricci curvature conditions (weaker than Myers’ condition) which ensure the compactness of a complete Riemannian manifold. Using standard index form techniques we relate the problem of finding such Ricci curvature criteria to that of establishing the conjugacy of the scalar Jacobi equation. Using this relationship we obtain a Ricci curvature condition for compactness which is weaker than that of Ambrose and, in fact, which is best among a certain class of conditions.References
- W. Ambrose, A theorem of Myers, Duke Math. J. 24 (1957), 345–348. MR 89464
- Eugenio Calabi, On Ricci curvature and geodesics, Duke Math. J. 34 (1967), 667–676. MR 216429
- 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
- Theodore Frankel and Gregory J. Galloway, Energy density and spatial curvature in general relativity, J. Math. Phys. 22 (1981), no. 4, 813–817. MR 617327, DOI 10.1063/1.524961
- Detlef Gromoll and Wolfgang Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75–90. MR 247590, DOI 10.2307/1970682
- Einar Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234–252. MR 27925, DOI 10.1090/S0002-9947-1948-0027925-7
- Richard A. Moore, The behavior of solutions of a linear differential equation of second order, Pacific J. Math. 5 (1955), 125–145. MR 68690
- S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404. MR 4518
- Zeev Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428–445. MR 87816, DOI 10.1090/S0002-9947-1957-0087816-8
- Rolf Schneider, Konvexe Flächen mit langsam abnehmender Krümmung, Arch. Math. (Basel) 23 (1972), 650–654 (German). MR 317234, DOI 10.1007/BF01304947
- C. A. Swanson, Comparison and oscillation theory of linear differential equations, Mathematics in Science and Engineering, Vol. 48, Academic Press, New York-London, 1968. MR 0463570
- Aurel Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115–117. MR 28499, DOI 10.1090/S0033-569X-1949-28499-6
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 106-110
- MSC: Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1982-0633289-3
- MathSciNet review: 633289