I understand that some non Euclidean Geometries, such as Hyperbolic and Spherical Geometry, that are can be approximated with ordinary Euclidean Geometry over very tiny increments of space. For instance in Euclidean space, while the distance between two lines may change the rate of change in the distance between two lines remains constant outside any points of intersection, and over infinitesimal areas the change in the rate of change in the distance between two geodesics tends to be negligible even in Spherical and Hyperbolic Geometry. Also as the size of an equilateral triangle approaches $0$ the sum of the angles of the triangle approach $180°$. The fifth postulate as I understand it is the only one of Euclid's postulates that doesn't hold in Spherical or Hyperbolic Geometry.
The spacetime curvature described by General Relativity cannot be approximated with ordinary Euclidean Geometry even over very tiny hyper volumes of spacetime, as it instead needs to be approximated with Minkowski Geometry. While in Minkowski Geometry the parallel postulate holds like in Euclidean Geometry, and the rate of change in the distance between two world lines remains constant, and if there is a right triangle, with one of the legs being space like, and the other being time like, then the plus sign in the formula for the hypotenuse is replaced by a minus sign. Also in Minkowski Geometry if the a space like direction and a timeline direction are perpendicular then the angle between them is infinite rather than being $90°$ as one would expect from Euclidean Geometry.
What I'm wondering is if there's a name for geometries that can be approximated by Euclidean Geometry over very small areas of space, and similarly a name for geometries that can be approximated by Minkowski Geometry over very small areas of spacetime.