Tangent flow

A flow in the space $\Omega_k$ of orthonormal $k$-frames of an $n$-dimensional Riemannian manifold $M$, having the following property. Let $\omega(t)$ be an arbitrary trajectory of the flow; by definition of the space $\Omega_k$, $\omega(t)$ is some $k$-frame $\xi_1(t),\dots,\xi_k(t)$ at some point $x(t)\in M$ (that is, in the tangent space to $M$ at this point). It is required that $dx(t)/dt=\xi_1(t)$ (a variant: it is required that the moving frame of the parametrized curve $x(t)$ in $M$ has as its first $k$ vectors precisely $\xi_1(t),\dots,\xi_k(t)$). 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 (which is a particular case of a tangent flow, when $k=1$ and the covariant derivative $D\xi_1/dt=0$). See [1], [2].

Various types of flow in the tangent space to some manifold $M$ (or, if it is supposed that $M$ 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 (generally, a system of equations of the second order) on $M$ and the variational equation of a flow on $M$ were called tangent flows. But this terminology did not achieve wide application. More customary terminology has since been used.

References

[1]	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
[2]	V.I. Arnol'd, "Remarks on winding numbers" Sibirsk. Mat. Zh. , 2 : 6 (1961) pp. 807–813 (In Russian)