The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.
An isometry group is a mathematical structure that consists of all isometries (distance-preserving transformations) of a metric space. In more formal terms, given a metric space \((X, d)\), the isometry group of that space is the group of all bijective mappings \(f: X \to X\) such that for any points \(x, y \in X\): \[ d(f(x), f(y)) = d(x, y).
The group of all transformations that preserve some bilinear form, notable examples:
- orthogonal group preserves the inner product
- unitary group preserves a Hermitian form
- Lorentz group preserves the Minkowski inner product
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