In mathematics — specifically, in differential geometry — a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces More precisely, given two (pesudo-)Riemannian manifolds (M, g) and (N, h), a function φ : M → N is said to be a geodesic map if
- φ is a diffeomorphism of M onto N; and
- the image under φ of any geodesic arc in M is a geodesic arc in N; and
- the image under the inverse function φ−1 of any geodesic arc in N is a geodesic arc in M. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B
Examples
- If (M, g) and (N, h) are both the n-dimensional Euclidean space En with its usual flat metric, then any Euclidean isometry is a geodesic map of En onto itself. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M For the Mechanical engineering and Architecture usage see Isometric projection.
- Similarly, if (M, g) and (N, h) are both the n-dimensional unit sphere Sn with its usual round metric, then any isometry of the sphere is a geodesic map of Sn onto itself. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension.
- If (M, g) is the unit sphere Sn with its usual round metric and (N, h) is the sphere of radius 2 with its usual round metric, both thought of as subsets of the ambient coordinate space Rn+1, then the "expansion" map φ : Rn+1 → Rn+1 given by φ(x) = 2x induces a geodesic map of M onto N. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers
- There is no geodesic map from the Euclidean space En onto the unit sphere Sn, since they are not homeomorphic, let alone diffeomorphic. Topological equivalence redirects here see also Topological equivalence (dynamical systems.
- Let (D, g) be the unit disc D ⊂ R2 equipped with the Euclidean metric, and let (D, h) be the same disc equipped with a hyperbolic metric (as in the Poincaré disc model of hyperbolic geometry). In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is In In geometry the Poincaré disk model, also called the conformal disk model is a model of n -dimensional Hyperbolic geometry in which the points of the geometry are Then, although the two structures are diffeomorphic via the identity map i : D → D, i is not a geodesic map, since g-geodesics are always straight lines in R2, whereas h-geodesics can be curved. This article is about the Identity Map software design pattern
References
- Ambartzumian, R. V. (1982). Combinatorial integral geometry, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics. New York: John Wiley & Sons Inc. , pp. xvii+221. ISBN 0-471-27977-3. MR679133
- Kreyszig, Erwin (1991). Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Differential geometry. New York: Dover Publications Inc. , pp. xiv+352. ISBN 0-486-66721-9. MR1118149
External links
- Eric W. Weisstein, Geodesic mapping at MathWorld. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic