I am starting to study the Frechet distance, and the expression "orientation-preserving homeomorphisms" is very used, but I have not found a formal definition...For instance, for two given to parametrised surfaces $A,B:[0,1]^{2}\longrightarrow X$, $(X,d)$ being a metric space, the Fréchet distance is defined as
$\inf_{\phi,\psi\in\mathrm{Aut}([0,1]^{2})} \max_{x\in[0,1]^{2}} d(A(\phi(x),B(\psi(x)))$,
where $\mathrm{Aut}([0,1]^{2}) $ denotes the set of orientation-preserving homeomorphisms of $[0,1]^{2}$. What means, in this context (or replacing $[0,1]^{2}$ by $[0,1]^{d}$), the expression "orientation-preserving homeomorphisms"?
Many thanks in advance for your comments.
The concept of "orientation" is too subtle to give a comprehensive explanation in this answer. You certainly know it from linear algebra (an isomorphism $f : \mathbb{R}^n \to \mathbb{R}^n$ is orientation preserving if $det(f) > 0$). In the context of your question we deal with homeomorphims $h$ between manifolds (with boundary). Locally a manifold looks like Euclidean space $\mathbb{R}^n$ (or, at the boundary, like a half-space $[0,\infty) \times \mathbb{R}^{n-1}$) and it is possible to give a meaning to "$h$ is locally orientation preserving".
In your special case an orientation preserving homeomorphism $h : [0,1] \to [0,1]$ is one such that $h(0) = 0 , h(1) = 1$; if $h(0) = 1, h(1) = 0$ it is orientation reversing. On $[0,1]^d$ it is not that easy to explain, but if $h$ is continously differentiable on the interior of $[0,1]^d$ (i.e. on $(0,1)^d$), then it means that the Jacobian matrix of $h$ has a positive determinant for all $x_0 \in (0,1)^d$. Intuitively, an orientation preserving homeomorphism is obtained by "deforming the identity", an orientation reversing homeomorphism by "deforming a reflection of $[0,1]^d$ at same hyperplane $H_k = \{ x = (x_1,..,,x_n) \in \mathbb{R}^n \mid x_k = 1/2 \}$".