Abstract
One knows how to attach to a compact riemannian manifold (M, g) three real numbers: its volume υ(g), its injectivity radius i(g) and its convexity radius c(g). The present article studies the following problems: do there exist universal constants λ(n), µ(n) such that υ(g)≥λ(n)i n(g), υ(g)≥µ(n)c n(g) for every riemannian manifold of dimension n? An affirmative answer is given to the second problem for any n but with an unsharp constant; an affirmative answer is given to the first problem only when n = 2 but with the sharp bound λ(2) = 4/π.
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References
Bishop, R. L. and Crittenden, R. J., Geometry of Manifolds, Academic Press.
Berger, M., ‘Lectures on Geodesics in Riemannian Geometry’, Tata Institute, Bombay.
Berger, M., Cours de Géométrie, Éditions CEDIC.
Blatter, C., ‘Über Extremallängen auf geschlossenen Flächen’, Comment. Math. Hely. 35 (1961), 153–168.
Federer, H., Geometric Measure Theory, Springer.
Gromoll, D., Klingenberg, W., and Meyer, W., ‘Riemannsche Geometrie im Grossen’, Springer Lectures Notes in Mathematics no. 55.
Green, L. W., ‘Auf Wiedersehenflächen’, Ann. of Math. 78 (1963), 289–299.
Green, L. W., ‘A theorem of E. Hopf’, Michigan Math. J. 5, (1958), 31–34.
Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities,Cambridge University Press.
Lekerkerker, C. G., Geometry of Numbers,Wolters-Noordhoff.
Weinstein, A., ‘On the Volume of Manifolds All of Whose Geodesics Are Closed’, J. of Differential Geometry 9 (1974), 513–517.
Wolf, J. A., Spaces of Constant Curvature,2nd edition, J. Wolf.
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© 1976 D. Reidel Publishing Company, Dordrecht, Holland
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Berger, M. (1976). Some Relations between Volume, Injectivity Radius, and Convexity Radius in Riemannian Manifolds. In: Cahen, M., Flato, M. (eds) Differential Geometry and Relativity. Mathematical Physics and Applied Mathematics, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1508-0_5
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DOI: https://doi.org/10.1007/978-94-010-1508-0_5
Publisher Name: Springer, Dordrecht
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