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Jacobi operator for leaf geodesics. (English) Zbl 0848.57030

This is a continuation of a paper published earlier by the author [Ergodic Theory Dyn. Syst. 8, 637-650 (1988; Zbl 0663.53028)]. Let \({\mathcal F}\) be a foliation of a Riemannian manifold \((M, g)\) and \(\Omega\) the space of piecewise smooth curves \(c: [0, 1]\to M\) tangent to the leaves of \({\mathcal F}\). Let \(T_c \Omega\) \((c\in \Omega)\) be the space of continuous piecewise smooth vector fields along \(c\). The author defines a differential operator \[ J: JZ=- Z''+ R(\dot c, Z)\dot c+ (\nabla_Z B)(\dot c, \dot c)+ 2B (Z^{\prime \top}, \dot c) \] for any vector field \(Z\) along \(c\). The Jacobi operator \(J\) depends on the curvature \(R\) of \(M\) as well as on the second fundamental form \(B\) of \({\mathcal F}\). Some properties of \(J\) are studied. Example: “Let \(X= Y+Z\) satisfy \(JX =0\), \(Y^\bot= 0\) and \(Z^\top =0\). Then \(X\in T_c \Omega\) if and only if \(Z(0) =-A^{\bot \dot c(0)} Z(0)\).” \(A^\bot\) denotes the Weingarten operator of the orthogonal distribution \(T^\top{\mathcal F}\). Two remarkable cases are examined: Riemannian foliations and totally geodesic foliations.

MSC:

57R30 Foliations in differential topology; geometric theory
53C40 Global submanifolds

Citations:

Zbl 0663.53028
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