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Let $(M,g)$ and $(N,h)$ be two complete Riemannian manifolds and $f:M\rightarrow N$ be a differmorphism. If $\nabla$ is the Levi-civita connection of $M$ then is it true that the push-forward of $\nabla$ $f_*\nabla$ is the Levi-civita connection of $N$? If this statement true then is it possible that $F\circ \gamma$ is a geodesic in $N$ with connection $F_*\nabla$ where $\gamma$ is a geodesic in $M$?

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  • $\begingroup$ if $f$ is an isometry then certainly yes, but for a general $f$ the answer is no (there may be $f$'s which are not isometries and still preserve the connection, but a general $f$ will not) $\endgroup$ – user8268 May 15 '17 at 9:31
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No, you can generate many different connections $f_* \nabla$ by choosing different $f$, while the Levi-Civita connection is unique. The $f$ that this does hold for are known as affine transformations, which include isometries but in some cases other maps too - consider $\mathbb R^n$, where dilation and shear mappings preserve the connection but not the metric.

In the case that $f$ is affine, then yes, it sends geodesics to geodesics.

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