Do the following Riemannian metrics on $GL(n,\mathbb{R})$ give us isometric structures?Do they generate the same volume forms? Is $O(n)$ a totally geodesic submanifold with respect to these metrics?

1. The metric with orthonormal frame $A\otimes A$ at each point $A\in GL(n,\mathbb{R})$


2. The metric with orthonormal frame $A\otimes A^{tr}$  for $A\in GL(n,\mathbb{R})$

Note that the tangent space $T_A GL(n,\mathbb{R})$  at each point $A$ is identified with $M_{n}(\mathbb{R})$ hence with $\mathbb{R}^{n^2}$.(Note that $GL(n,\mathbb{R})$ is counted as an open subset of Euclidean space hence has trivial tangent bundle so the above identification has an obvious desribtion). So at each point $A\in GL(n,\mathbb{R})$, each column of the tensor product matrix $A\otimes A$ (or $A\otimes A^{tr}$) can be considered as a tangent vector in $T_A GL(n,\mathbb{R})$. So $A\otimes A$ is obviously a fram, that is a base for the tangent space.This frame define a unique well defined Riemannian metric on $GL(n,\mathbb{R})$.



**Edit:** According to the comments on this question we clarify the identification $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$:
The identification is based on the lexicographic order on the index $i,j$ in $(a_{ij})$. For example $$ \begin{pmatrix} a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix}$$

is identified with $$(a_{11}, a_{12},a_{21}, a_{22})$$


**Remark:** For a homotopic version of this question please see [this MO post](https://mathoverflow.net/questions/349671/homotopicity-of-two-certain-sections-of-frame-bundle-of-gln-mathbbr).