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Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.

Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open Riemannian submanifold of $\mathbb{R}^{n^2}$.

We have two metrics on $GL_n^+$ (in the sense of metric spaces); intrinsic $d^{int}$ (when we only allow paths which stay inside $GL_n^+$) and extrinsic $d^{ext}$ (the standard Euclidean metric).

**Clarification:** $d^{int}$ is the Riemann distance function on $GL_n^+$ induced by the standard Riemannian metric on $\mathbb{R}^{n^2}$ (i.e I consider $GL_n^+$ as a Riemannian submanifold of $\mathbb{R}^{n^2})$

Of course $d^{ext} \le d^{int}$. For some matrices $d^{ext}(A,B) < d^{int}(A,B)$;

If $\gamma(t)=A+t(B-A)$, then $\det(\gamma(t))$ is a polynomial which can be negative on a subinterval of $[0,1]$. (i.e the minimizing geodesic in the Euclidean space, is contained in $GL^{-}$ for some time)

(see this answer to this question for details on this specific case, and here for a cleaner proof that "nearly minimizing paths are nearly geodesic")

Note that $d^{ext},d^{int}$ both generate the same topology on $GL_n^+$. Moreover, both are incomplete.

So, the following natural question arises:

**Are $d^{ext},d^{int}$ strongly equivalent? i.e; Is it true that**

$d^{int} \le C \cdot d^{ext} \text{ for some } C \in \mathbb{R}$?