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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.
##### MSC:
 53C20 Global Riemannian geometry, including pinching 53B20 Local Riemannian geometry
##### Keywords:
conformal map; Jacobi operator; Jacobi map; conformally flat
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