On characterization of riemannian manifolds
DOI:
https://doi.org/10.4067/S0716-09172008000200001Keywords:
Geodesics, convexity, axiomatic geometry, isosceles triangles, geodésicas, convexidad, geometría axiomática, triángulos isosceles.Abstract
This survey, present some results about characterization of Riemannian manifolds by using notions of convexity. The first part deals with immersed manifolds and the second part gives a characterization for the Euclidean space and for the Euclidean sphere.Downloads
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References
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[9] Green, L.W. - Auf Wiedersehensflchen. Annals of Mathematics, 78 (2), (1963).
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[12] Karcher, H. - Schnittort und konvexe Mengen in vollstndigen Riemannschen Mannigfaltigkeiten. Math. Annalen, 117, pp. 105-121, (1968).
[13] Lima, E. L. - Commuting Vector Fields on S3. Ann. of Math. 81, pp. 70-81, (1965).
[14] Maeda, M. - On the injective radius of noncompact Riemannian Manifold. Proc. Japan Acad. 50, pp. 148-151, (1974).
[15] Nash, J. - The embedding problem for Riemannian manifolds. Ann. of Math. 63, pp. 20-63, (1956).
[16] Rodrigues L. - Geometria das subvariedades. Monografias do IMPA, (1976).
[17] Sacksteder, R. - On hypersurfaces with no negative sectional curvatures. Amer. J. Math. 82, pp. 609-630, (1960).
[18] Spivak M. - A conprehensive introduction to differential geometry, volume IV. Pulblish or Perish - Inc Boston Mass.
[19] Stokes, J. J. - ber die Gestalt der positiv gekrmmten offenen Flche. Compositio Mathematica 3, 55-88, (1936).
[20] Tribuzy, I. A. - A characterization of Rn. Archiv der Mathematik, maio (1978).
[21] Tribuzy, I. A. - Convexidade em Variedades Riemannianas, Tese IMPA, (1978).
[22] Tribuzy, I. A. - Convex immersions into positively-curved manifolds. Bol. Soc. Bras. Mat. Vol. 17 n0 1, pp. 21-39, (1986).
[23] Tribuzy, I. A. - Isosceles triangles in Riemannian geometry - a characterization of the n-sphere. Bol. Soc. Bras. Mat. New Series 38(4), pp. 573-583, (2007).
[24] Whitehead - Convex regions in geometry of paths. Quart. J. Math. Oxford, 3 (1932).
[25] Yang, C. T. - Odd-dimensional wiedersehen manifolds are spheres. J. Diff. Geometry, 15, pp. 91-96, (1980).
[26] Kazdan, A. - An isoperimetric inequality and wiedersehen manifolds - seminar on differential geometry. Annals of Math. Studies, 102, (1982).
[2] Bishop, R. L. - Infinitesimal convexity implies local convexity. Indiana Math. J. 29, pp. 169-172, (1974).
[3] do Carmo, M. P. - Geometria Riemanniana. Rio de Janeiro, (1979).
[4] do Carmo, M. P. and Lima, E. L. - Isometric immersions with semidefinite second quadratic forms. Archiv der Math. 20, pp. 173-175, (1969).
[5] do Carmo M. P. and Warner, F. - Rigidity and convexity of hypersurfaces in spheres. j. Diff. Geom. 4, pp. 133-144, (1970).
[6] Cartan, E. - Lenons sur le geometrie des espaces de Riemann. Paris, (1946).
[7] Cheeger. J. and Ebin, D.G. - Comparison Theorems in Riemannian Geometry. North-Holand Publishing Co., Amsterdam, (1975).
[8] Eschenburg, J. H - Horospheres and the stable part of geodesic flow. Math.Z., (1977).
[9] Green, L.W. - Auf Wiedersehensflchen. Annals of Mathematics, 78 (2), (1963).
[10] Hadamard, J. - Sur certaines proprits des trajectories en dynamique. J. Math. Pures Appl. 3 (1897), 331-387.
[11] Im Hof, H. C. and Ruh, E. A. - An equivariant pinching theorem. Comm. Math. Helv., pp. 389-401, 50 (1975).
[12] Karcher, H. - Schnittort und konvexe Mengen in vollstndigen Riemannschen Mannigfaltigkeiten. Math. Annalen, 117, pp. 105-121, (1968).
[13] Lima, E. L. - Commuting Vector Fields on S3. Ann. of Math. 81, pp. 70-81, (1965).
[14] Maeda, M. - On the injective radius of noncompact Riemannian Manifold. Proc. Japan Acad. 50, pp. 148-151, (1974).
[15] Nash, J. - The embedding problem for Riemannian manifolds. Ann. of Math. 63, pp. 20-63, (1956).
[16] Rodrigues L. - Geometria das subvariedades. Monografias do IMPA, (1976).
[17] Sacksteder, R. - On hypersurfaces with no negative sectional curvatures. Amer. J. Math. 82, pp. 609-630, (1960).
[18] Spivak M. - A conprehensive introduction to differential geometry, volume IV. Pulblish or Perish - Inc Boston Mass.
[19] Stokes, J. J. - ber die Gestalt der positiv gekrmmten offenen Flche. Compositio Mathematica 3, 55-88, (1936).
[20] Tribuzy, I. A. - A characterization of Rn. Archiv der Mathematik, maio (1978).
[21] Tribuzy, I. A. - Convexidade em Variedades Riemannianas, Tese IMPA, (1978).
[22] Tribuzy, I. A. - Convex immersions into positively-curved manifolds. Bol. Soc. Bras. Mat. Vol. 17 n0 1, pp. 21-39, (1986).
[23] Tribuzy, I. A. - Isosceles triangles in Riemannian geometry - a characterization of the n-sphere. Bol. Soc. Bras. Mat. New Series 38(4), pp. 573-583, (2007).
[24] Whitehead - Convex regions in geometry of paths. Quart. J. Math. Oxford, 3 (1932).
[25] Yang, C. T. - Odd-dimensional wiedersehen manifolds are spheres. J. Diff. Geometry, 15, pp. 91-96, (1980).
[26] Kazdan, A. - An isoperimetric inequality and wiedersehen manifolds - seminar on differential geometry. Annals of Math. Studies, 102, (1982).
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2017-05-02
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How to Cite
[1]
“On characterization of riemannian manifolds”, Proyecciones (Antofagasta, On line), vol. 27, no. 2, pp. 113–144, May 2017, doi: 10.4067/S0716-09172008000200001.