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Difference between revisions of "Geodesic mapping"

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''projective mapping''
 
''projective mapping''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441401.png" /> that transforms the geodesic lines of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441402.png" /> into the geodesic lines of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441403.png" />. A geodesic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441404.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441406.png" /> are spaces in which geodesics are defined, is a local homeomorphism (diffeomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441408.png" /> are smooth manifolds).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g0441409.png" /> to a connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g04414010.png" /> is a geodesic mapping under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g04414011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044140/g04414012.png" /> is a covector field. A space with an affine connection is projectively flat if and only if the projective curvature tensor vanishes.
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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 &amp; Wiley  (1979)  (In Russian)</TD></TR></table>
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<table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Hilbert's fourth problem" , Winston &amp; Wiley  (1979)  (In Russian)</TD></TR>
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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)
How to Cite This Entry:
Geodesic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_mapping&oldid=39679
This article was adapted from an original article by Yu.A. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article