Local isometry

A local isometry between Riemannian manifolds is a local diffeomorphism $f$ with $f^{\ast }{g}_N=g_M$. Equivalently, its differentials are isometric linear isomorphisms between tangent spaces. It preserves the Levi-Civita connection, intrinsic sectional curvature, and affinely parametrized geodesics. A map whose differentials are merely isometric injections is an isometric immersion; the local diffeomorphism requirement distinguishes these notions.

The helicoid $H(s,v)=(\sinh s\cos v,\sinh s\sin v,v)$ and catenoid $C(s,v)=(\cosh s\cos v,\cosh s\sin v,s)$ have the same first fundamental form ${\cosh }^{2}s(d{s}^2+d{v}^2)$. Therefore matching equal parameter pairs gives a local isometry. The result is local: a full turn changes the height on the helicoid while the angular coordinate on the catenoid repeats. Equal intrinsic lengths do not require the same embedding into Euclidean space.

Two local isometries from a connected Riemannian manifold that have the same value and differential at one point agree everywhere. They preserve geodesics, so their equality follows on a normal neighborhood from $f({\exp }_{p}v)={\exp }_{f(p)}(d{f}_{p}v)$. The set of points where both value and differential agree is consequently open, and continuity makes it closed. Connectedness completes the argument.

A local isometry from a nonempty complete Riemannian manifold to a connected Riemannian manifold is a surjective covering map. Lift radial geodesics in a normal ball using completeness; the exponential map at each inverse-image centre maps a tangent ball diffeomorphically onto a sheet. Uniqueness of the geodesic initial-value problem makes these sheets disjoint and exhaustive. Surjectivity follows by lifting piecewise geodesic paths. Conversely a Riemannian covering of a complete target is complete, by lifting complete target geodesics and using the Hopf-Rinow theorem. If the target is disconnected, the first assertion gives a covering only of the union of components met by the map.

A local isometry intertwines the Levi--Civita covariant derivatives. It therefore sends every affinely parametrized geodesic to an affinely parametrized geodesic.

A nonunit Euclidean dilation sends straight-line geodesics to straight-line geodesics but scales their tangent inner products. Preservation of geodesics alone therefore does not imply local isometry.

A Riemannian isometry is a diffeomorphism that preserves the Riemannian metric, and hence lengths, areas, and intrinsic curvature.

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.

For the conethe first fundamental form isThe local change of polar coordinatesturns this into the Euclidean metric $d{R}^2+R^{2}d{\Theta }^2$.

For a circular cone with intrinsic metric $d{\rho }^2+{\lambda }^2{\rho }^{2}d{\theta }^2$, $0<\lambda \le 1$, local development uses plane angle $\varphi =\lambda \theta$. A plane straight segment that avoids the origin lifts to a geodesic. Choosing an angular difference at most $\pi$ gives plane difference at most $\pi \lambda$, so for $\lambda <1$ this constructs a joining geodesic between any two distinct points. Different permitted angle lifts can yield distinct joining geodesics; the missing apex cannot be used to join straight segments through the plane origin.