Difference between revisions of "Geodesic mapping"
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''projective mapping'' | ''projective mapping'' | ||
− | A mapping | + | A mapping $f$ that transforms the geodesic lines of a space $U$ into the geodesic lines of a space $V$. A geodesic mapping $f : U \rightarrow V$, where $U$ and $V$ are spaces in which geodesics are defined, is a [[local homeomorphism]] (diffeomorphism if $U$ and $V$ are smooth manifolds). |
A space that locally permits a geodesic mapping into a Euclidean space is called projectively flat. Geodesic mappings of one Riemannian space into another exist in exceptional cases. Among the Riemannian spaces only those of constant curvature are projectively flat [[#References|[1]]]. The description of all Riemannian projectively-flat metric spaces constitutes Hilbert's fourth problem [[#References|[2]]]. | A space that locally permits a geodesic mapping into a Euclidean space is called projectively flat. Geodesic mappings of one Riemannian space into another exist in exceptional cases. Among the Riemannian spaces only those of constant curvature are projectively flat [[#References|[1]]]. The description of all Riemannian projectively-flat metric spaces constitutes Hilbert's fourth problem [[#References|[2]]]. | ||
− | In the theory of spaces with an affine connection one does not speak of geodesic mappings but rather of geodesic transformations of a connection, which means a transition to another connection on the same manifold with preservation of the geodesics. The transition from a connection | + | In the theory of spaces with an [[affine connection]] one does not speak of geodesic mappings but rather of geodesic transformations of a connection, which means a transition to another connection on the same manifold with preservation of the geodesics. The transition from a connection $\Gamma^i_{jk}$ to a connection $\bar\Gamma^i_{jk}$ is a geodesic mapping under the condition $\bar\Gamma^i_{jk} = \Gamma^i_{jk} + A^i_k\psi_j + A^i_j\psi_k$, where $\psi$ is a covector field. A space with an affine connection is projectively flat if and only if the projective curvature tensor vanishes. |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Pogorelov, "Hilbert's fourth problem" , Winston & Wiley (1979) (In Russian)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)</TD></TR> | ||
+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Pogorelov, "Hilbert's fourth problem" , Winston & Wiley (1979) (In Russian)</TD></TR> | ||
+ | </table> | ||
+ | |||
+ | {{TEX|done}} |
Latest revision as of 21:52, 6 November 2016
projective mapping
A mapping $f$ that transforms the geodesic lines of a space $U$ into the geodesic lines of a space $V$. A geodesic mapping $f : U \rightarrow V$, where $U$ and $V$ are spaces in which geodesics are defined, is a local homeomorphism (diffeomorphism if $U$ and $V$ are smooth manifolds).
A space that locally permits a geodesic mapping into a Euclidean space is called projectively flat. Geodesic mappings of one Riemannian space into another exist in exceptional cases. Among the Riemannian spaces only those of constant curvature are projectively flat [1]. The description of all Riemannian projectively-flat metric spaces constitutes Hilbert's fourth problem [2].
In the theory of spaces with an affine connection one does not speak of geodesic mappings but rather of geodesic transformations of a connection, which means a transition to another connection on the same manifold with preservation of the geodesics. The transition from a connection $\Gamma^i_{jk}$ to a connection $\bar\Gamma^i_{jk}$ is a geodesic mapping under the condition $\bar\Gamma^i_{jk} = \Gamma^i_{jk} + A^i_k\psi_j + A^i_j\psi_k$, where $\psi$ is a covector field. A space with an affine connection is projectively flat if and only if the projective curvature tensor vanishes.
References
[1] | J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German) |
[2] | A.V. Pogorelov, "Hilbert's fourth problem" , Winston & Wiley (1979) (In Russian) |
Geodesic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_mapping&oldid=19263