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Assume that $M$ is a submanifold of $\mathbb{R}^n$ and is equipped with a Riemannian metric such that the parallel transports associated with corresponding LC conection preserve the inner products of tangent spaces which they inherit from the standard metric of $\mathbb{R}^n$.

Does this imply that $(M,g)$ is embedded in $\mathbb{R}^n$, isometrically?

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Consider $M = \mathbb R^1$ with the standard inner product, embedded into $\mathbb R^1$ by the multiplication by $2$ map $x \mapsto 2x$. Then distance is not preserved, but parallel transport remains the same.

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  • $\begingroup$ Thank you. So can we say that M is isometrically embedded up to a constant rescalling? $\endgroup$
    – Ali Taghavi
    Commented Aug 1, 2018 at 22:51
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    $\begingroup$ In dimension $n$ you can rescale by a matrix $A \in {\rm GL}_n$. If you assume that the connection induced by the metric and the connection induced by the embedding are equal, then by parallel transport it is enough for the metrics to agree at the tangent space to a single point-- so the ambiguity is exactly $GL_n/ O_n$. However, with your hypothesis, I wouldn't expect anything (though I don't have a counter example in mind). $\endgroup$
    – Phil Tosteson
    Commented Aug 1, 2018 at 23:28

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