×

zbMATH — the first resource for mathematics

Courbures et basculements des sous-variétés Riemanniennes. (Curvatures and balances of Riemannian submanifolds). (French) Zbl 0595.53023
This is a continuation of the author’s paper [Geom. Dedicata 12, 1-15 (1982; Zbl 0495.53003)], in which submanifolds V of dimension p and codimension q in a Riemannian manifold \(\tilde V\) of dimension \(n=p+q\) are studied. In the previous paper an endomorphism \(\beta\) of balance (endomorphisme de bascule) is defined as an endomorphism of the tangent space \(T_ xV\) at each point x of V which measures the angular variation of the tangent space to V considered as a subspace of \(T_ x\tilde V.\)
In this paper various properties of V are investigated which are related to this endomorphism. The geometric meaning of det \(\beta\) is also explained and it is called the areal curvature (courbure aréolaire) of V. The pitching (tangage) \(t_ u\) along a tangent vector u of V is also defined. The present paper consists of five chapters among which Chapter 0 is a recall of the previous paper. Some of the contents of the succeeding chapters are given below.
I. The geometric meaning of the bilinear form associated with \(\beta^ 2\) is given in relation to the metric of a Grassmann manifold. For isobalanced \(\beta\), namely, if it is homothetic, V is investigated precisely. When V is compact an inequality is obtained between the integral of det \(\beta\) and the Betti numbers of V.
II. Here rolling (roulis) \(r_ u\) is defined for a tangent vector u at a point x of V given by the angular variation of the space spanned by u and the normal space to V at x when x moves in the direction of u. The relation \(\| \beta (u)\|^ 2=(r_ u)^ 2+(t_ u)^ 2\) holds. A line of curvature is defined (even in the case \(q>1)\) as a curve on V such that the tangent vector to the curve satisfies \(r_ u=0\). The theorem of Joachimsthal holds with respect to this line of curvature. The notion of rolling is related to the curvatures of \(\tilde V\) and the normal connection.
III. The case \(\tilde V=R^ n\) is studied. A Killing field \(X_ u\) of \(R^ n\) is defined from the second fundamental form of V and a tangent vector u to V, and then the notions of central subspace, central point, and instantaneous translation are defined and studied. IV. Tubes around V and \(\tilde V\) and especially longitudinal submanifolds are studied.
Reviewer: Y.Muto

MSC:
53B25 Local submanifolds
53C40 Global submanifolds
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] R.L. BISHOP , R.J. CRITTENDEN , Geometry of Manifolds , Academic Press ( 1964 ). MR 29 #6401 | Zbl 0132.16003 · Zbl 0132.16003
[2] B.Y. CHEN , Geometry of submanifolds , Marcel Dekker, Inc. New-York ( 1973 ). Zbl 0262.53036 · Zbl 0262.53036
[3] B.Y. CHEN , Geometry of Submanifolds and its Applications , Science Univ. of Tokyo ( 1981 ). Zbl 0474.53050 · Zbl 0474.53050
[4] B.Y. CHEN , L. VANHECKE , Differential geometry of geodesic spheres , Journal für die reine und angewandte Mathematik, Band 325, p. 28-67. MR 82m:53038 | Zbl 0503.53013 · Zbl 0503.53013 · doi:10.1515/crll.1981.325.28 · crelle:GDZPPN002198487 · eudml:152356
[5] S.S. CHERN , La géométrie des sous-variétés d’un espace euclidien à plusieurs dimensions , Enseignement Math. 40, ( 1951 - 1954 ), p. 26-46. MR 16,856g | Zbl 0064.17504 · Zbl 0064.17504
[6] S.S. CHERN , N. KUIPER , Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space , Ann. of Math. 56 ( 1952 ), p. 422-430. MR 14,408e | Zbl 0049.23402 · Zbl 0049.23402 · doi:10.2307/1969650
[7] S.S. CHERN , R.K. LASHOF , On the total curvature of immersed manifolds , Amer. J. Math. 79 ( 1957 ). MR 18,927a | Zbl 0078.13901 · Zbl 0078.13901 · doi:10.2307/2372684
[8] J. DIEUDONNE , Eléments d’Analyse , tome 4, Gauthier-Villars ( 1971 ). Zbl 0217.00101 · Zbl 0217.00101
[9] C. GODBILLON , Géométrie différentielle et Mécanique analytique , Hermann ( 1969 ). MR 39 #3416 | Zbl 0174.24602 · Zbl 0174.24602
[10] A. GRAY , Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula , Topology, Vol. 21, n^\circ 2, p. 201-228 ( 1982 ). MR 83c:53064 | Zbl 0487.53035 · Zbl 0487.53035 · doi:10.1016/0040-9383(82)90005-2
[11] A. GRAY , L. VANHECKE , The volumes of tubes in a riemannian manifold , Rend. Sem. Mat. Univers. Politecn. Torino, vol. 39, 3, p. 1-50 ( 1981 ). MR 84i:53053 | Zbl 0511.53059 · Zbl 0511.53059
[12] W. GREUB , S. HALPERIN , R. VANSTONE , Connections, Curvature, and Cohomology , Academic Press ( 1973 ). Zbl 0335.57001 · Zbl 0335.57001
[13] S. KOBAYASHI , K. NOMIZU , Foundations of differential geometry , Interscience Publishers ( 1969 ). MR 38 #6501 | Zbl 0175.48504 · Zbl 0175.48504
[14] R. LANGEVIN , Courbures, feuilletages et surfaces , Thèse, Orsay ( 1980 ). MR 82j:53117 | Zbl 0466.53036 · Zbl 0466.53036
[15] H. MAILLOT , Endomorphisme de bascule et courbure aréolaire d’une sous-variété d’une variété riemannienne , Geometriae Dedicata 12 ( 1982 ), p. 1-15. MR 83e:53059 | Zbl 0495.53003 · Zbl 0495.53003 · doi:10.1007/BF00147326
[16] H. MAILLOT , Sur la courbure des sous-variétés d’un espace euclidien , Compte rendu Acad. Sci. Paris, t. 297 ( 1983 ), Série I, p. 651. MR 85b:53058 | Zbl 0533.53002 · Zbl 0533.53002
[17] H. MAILLOT , Sur les droites privilégiées en un point d’une sous-variété V d’un espace euclidien et les directions principales des tubes autour de V . Compte rendu Acad. Sci. Paris, t. 298, Série I, n : 3, 1984 , p. 51. MR 85i:53061 | Zbl 0562.53002 · Zbl 0562.53002
[18] H. MAILLOT , Sur certains champs de sous-espaces vectoriels et sur les sous-variétés riemanniennes dont la connexion normale est plate , Compte rendu Acad. Sci. Paris, t. 297, p. 497 ( 1983 ). MR 85b:53042 | Zbl 0534.53045 · Zbl 0534.53045
[19] J. MILNOR , Morse Theory , Princeton Univ. Press ( 1963 ). MR 29 #634 | Zbl 0108.10401 · Zbl 0108.10401
[20] J.M. MORVAN , Quelques propriétés géométriques et topologiques des sous-variétés riemanniennes , Thèse, Limoges ( 1979 ).
[21] J. SIMONS , Minimal varieties in Riemannian manifolds , Ann. of Math. 88 ( 1968 ), p. 62-105. MR 38 #1617 | Zbl 0181.49702 · Zbl 0181.49702 · doi:10.2307/1970556
[22] M. SPIVAK , A comprehensive Introduction to Differential Geometry , Publish or Perish, Inc. ( 1975 ). Zbl 0306.53003 · Zbl 0306.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.