throughout the following question, whenever I'm wrong please correct me!
Recently I came across the notions of symmetry and isometry. Though, there is something obscure (definitely in my head) concerning the distinction of those two things. For instance, let's say that $\Sigma \subset \mathbb{R}^{n},$ is an arbitrary geometric object (I'm looking for the general point view of the notion of symmetries and isometries, hence am going to assume that at the moment no further structure has been assumed on this object). Then the group of symmetries of $\Sigma$, is $$Symm(\Sigma)= \{ \sigma \in Isom(\mathbb{R}^n) \thinspace | \thinspace \sigma(\Sigma) = \Sigma \},$$ whilst isometries of $\mathbb{R}^n$ are defined as usually, being distance-preserving maps $\sigma : \mathbb{R}^n \rightarrow \mathbb{R}^n,$ which turn out to be continuous, one-to-one and onto (hence homeomorphisms of the underlying topological structure induced by the metric). Now, let's say that a geometric figure $\Sigma \subset \mathbb{R}^{n},$ is given on its own again and someone asks, "What's the group of isomotries and symmetries of $\Sigma$?". Then there are two possibilities:
- $\Sigma$, inherits the metric by $\mathbb{R}^n,$ as a subspace.
- $\Sigma$, becomes a metric space with some other metric and we examine it by forgetting any ambient space.
Now, for the first one, I think the symmetries and isometries coincide, right (if no, a counterexample suffices)? It's just another name for the same map $\sigma: \Sigma \rightarrow \Sigma$ which preserves distances. But what happens for the second case?
For instance, for the last question I have in my mind the distinctive case $\Sigma= \mathbb{S}^{n-1},$ the $(n-1)$-dimensional sphere which naturally inherits a metric by $\mathbb{R}^n$, but it can be equipped with another metric too, hence the isometries change in those two cases since different type of measurment is being applied each time. What happens if moreover we assume some differentiable structure and someone asks for the isomotries/symmetries of Riemmanian manifolds instead of subsets of $\mathbb{R}^n$? What about the symmetries and isometries in that case?
Thank you, I hope haven't done something wrong, because is my first post! If yes, do let me know.