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?



A sufficient condition for isometrically embedding of manifolds in the Euclidean space they have already sat
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.


