What are the most special spaces between which rigid transformations preserve the structures of the spaces

Question

An affine transformation is a linear transformation followed by a translation. They are morphism between affine spaces.

A rigid transformation consists of a rotation and a translation. I was wondering what are the spaces between which the rigid transformations can become morphisms?

what are the spaces between which the rotation transformations can become morphisms?

Same question for similarity transformation and projective transformation.

Thanks and regards!

Answer 1

If by "rotation" you mean "orientation-preserving isometry of $\mathbb{R}^n$ fixing the origin," then

• the spaces between which rigid transformations are morphisms are the "affine oriented (finite-dimensional) inner product spaces" over $\mathbb{R}$, by which I mean torsors over a finite-dimensional oriented inner product space over $\mathbb{R}$,
• the spaces between which rotations are morphisms are the oriented finite-dimensional inner product spaces over $\mathbb{R}$, and
• the spaces between which projective transformations are morphisms are the projective spaces over $\mathbb{R}$.

I cannot off the top of my head think of a good name for the spaces between which similarities are morphisms. I think you are looking for "conformal affine spaces," e.g. torsors over $\mathbb{R}^n$ equipped with a notion of oriented angle (but not the full inner product).