Given:
- two Euclidean spaces $\mathcal{P},\mathcal{P'}$ (with their usual smooth structure) and a injective smooth mapping $f:\mathcal{P}\to\mathcal{P'}$
- a Riemannian manifold $(\mathcal{M}\subset \mathcal{P},g_\mathcal{M})$ (the tangent spaces of $\mathcal{M}$ lies in $\mathcal{P}$)
- a space $\mathcal{N} = f(\mathcal{M}) \subset \mathcal{P'}$. $f$ is an isomorphism $\mathcal{M}$ and $\mathcal{N}$ and $f$ is $C^\infty$ on $\mathcal{M}$
Is it direct to say that:
- $\mathcal{N}$ is a smooth manifold
- For $x\in\mathcal{M}$, $\mathcal{T}_{f(x)}\mathcal{N}= f(\mathcal{T}_x \mathcal{M})$ (tangent spaces)
- For $y\in \mathcal{N}$, $u_1,u_2\in \mathcal{T}_y \mathcal{N}$, the metric tensor $g_\mathcal{N}$ defined by:
$$g_\mathcal{N}(u_1,u_2) \triangleq g_\mathcal{M}(f^{-1}(u_1),f^{-1}(u_2))$$
is such that $(\mathcal{N},g_\mathcal{N})$ is a Riemanian Manifold with:
- Same notion of distance $d_\mathcal{M}(x_1,x_2) = d_\mathcal{N}(f(x_1),f(x_2))$
- Same notion of geodesic $\gamma_\mathcal{M}(x_1\to x_2) = f^{-1}(\gamma_\mathcal{N}(f(x_1)\to f(x_2))$
Or do I need additional properties to prove these points ? My goal is to use the mapping $f$ to endow $\mathcal{N}$ with a Riemannian structure. A solution could be to use local maps to show properties 1., 2. and 3. but it is maybe unnecessary in this "simple" case ?
[Edited following comments of Anthony Carapetis]
If $f$ is a linear bijection $\mathcal M \to \mathcal N$ then it extends to a linear isomorphism from $V =\operatorname{span} \mathcal M$ to $V' = \operatorname{span} \mathcal N$. Since these subspaces are the only relevant parts of $\mathcal P, \mathcal P'$ to the manifold structure (tangent vectors will also lie in these subspaces), we can transport all structure from $\mathcal M$ to $\mathcal N$ via the ambient space identification $V \simeq V'$ as you describe.