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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
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##### References:
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