Orientation-reversing Riemannian isometry ($h^{\ast }dV=-dV$)

An isometry between oriented Riemannian manifolds that reverses orientation. Its differential is orthogonal with determinant minus one in oriented orthonormal frames, so it reverses the oriented volume form while preserving the associated positive volume measure. Reflection in a geodesic of the hyperbolic plane is an example. An orientation-reversing isometry still identifies the scalar Laplace spectra, even though it does not preserve the chosen complex orientation of a Riemann surface.