View source for Tangent flow

A flow in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921401.png" /> of orthonormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921402.png" />-frames of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921403.png" />-dimensional [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921404.png" />, having the following property. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921405.png" /> be an arbitrary trajectory of the flow; by definition of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921407.png" /> is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921408.png" />-frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t0921409.png" /> at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214010.png" /> (that is, in the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214011.png" /> at this point). It is required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214012.png" /> (a variant: it is required that the moving frame of the parametrized curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214014.png" /> has as its first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214015.png" /> vectors precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214016.png" />). To obtain interesting results on tangent flows it is necessary to impose various extra conditions. The results obtained generalize certain of the properties of a [[Geodesic flow|geodesic flow]] (which is a particular case of a tangent flow, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214017.png" /> and the covariant derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214018.png" />). See [[#References|[1]]], [[#References|[2]]].

Various types of flow in the tangent space to some manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214019.png" /> (or, if it is supposed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214020.png" /> is endowed with a Riemannian or a Finsler metric, in the space of unit tangent vectors) were sometimes called tangent flows. For example, a [[Spray|spray]] (generally, a system of equations of the second order) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214021.png" /> and the variational equation of a flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092140/t09214022.png" /> were called tangent flows. But this terminology did not achieve wide application. More customary terminology has since been used.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,   "Some remarks on flows of line elements and frames"  ''Soviet Math. Dokl.'' , '''2'''  (1961)  pp. 562–564  ''Dokl. Akad. Nauk SSSR'' , '''138''' :  2  (1961)  pp. 255–257</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Arnol'd,   "Remarks on winding numbers"  ''Sibirsk. Mat. Zh.'' , '''2''' :  6  (1961)  pp. 807–813  (In Russian)</TD></TR></table>