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.

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.

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.