Mobius transformations map circles to circles.


Wiki says 'Möbius transformations can be more generally defined in spaces of dimension n>2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere'.



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Explicit generalizations of Mobius transformations?
Yes, and, indeed, these examples special cases of the situation that $G$ is a simple real Lie group, $K$ is a maximal compact subgroup, and we look at the action of $G$ on the left on the Riemannian symmetric space $G/K$.


For the so-called classical groups $G$, the spaces $G/K$ are model-able in terms of matrices, vectors, quadratic forms, and other geometric algebra. For example, the $n$-dimensional real hyperbolic space is isomorphic to $X=O(n,1)/O(n)$, with orthogonal groups of the indicated signatures. Just as the upper half-plane is isomorphic to the unit disc, with a slightly different group acting by linear fractional transformations, there is the real $n$-ball model of $X$, as well as an upper half-space model, using, respectively, different models of $O(n,1)$.


Several examples are carried out in considerable detail in some of the notes linked-to from my page <http://www.math.umn.edu/~garrett/m/lie/> In particular, the essays "Classical homogeneous spaces" and "Classical groups, domains, cones" treat these and other examples very explicitly.


EDIT: Perhaps the easiest example is to give the action of $G=\{g\in GL\_{n+1}(\mathbb R): g^TQg=Q\}$, where $Q=\pmatrix{-1\_n & 0 \cr 0 & 1}$, on the unit ball in $\mathbb R^n$, by $\pmatrix{a & b \cr c & d}(x)=(ax+b)(cx+d)^{-1}$, where $a$ is $n$-by-$n$, $b$ is $n$-by-$1$, etc., and $x$ is $n$-by-$1$. This is a model of real hyperbolic $n$-space. The upper half-space model is similar.


