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Jacobi maps between Riemannian manifolds. (English) Zbl 0842.53026
The Jacobi operator on a Riemannian manifold \((M, g)\) is the selfadjoint operator \(R_X(u)= {R(u, X)X\over g(X, X)}\), where \(R\) denotes the curvature tensor of \((M, g)\) and \(X\in TM\), \(g(X, X)\neq 0\). A map \(f: (M, g)\to (\overline M, \overline g)\) is called a Jacobi map if it preserves the Jacobi operators: \(f_*\circ R_X= \overline R_{f_*(X)}\circ f_*\).
As their main result, the authors prove that – provided \((M, g)\) has non-vanishing sectional curvatures – any Jacobi map \(f: (M, g)\to (\overline M, \overline g)\) is conformal and for suitable function \(\sigma: M\to \mathbb{R}\) one has \(\overline R(f_* U, f_* X)(f_* Y)= e^{2\sigma} f_* (R(U, X)Y)\). Using this result, they conclude that \((M, g)\) has to be conformally flat.
53C20 Global Riemannian geometry, including pinching
53B20 Local Riemannian geometry
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