By D. V. Anosov
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Extra info for Geodesic flows on closed Riemann manifolds with negative curvature,
Example text
At the present time a great part of the results of these older papers are extended to (U)-systems, but we should note that by far not all of them are so extended; in these works there are some results that do not fit into the frames of (U)-systems and the modern metric theory. Among these we should consider the study of properties of some geodesics, the so-called geodesics of class A, on arbitrary closed surfaces of genus 2 (without any assumptions about negative curvature or instability of the trajectories of the geodesic flow).
In other cases it is more convenient to talk about the "differential" d(w, O. The tangent space of I at the point (w, ~) I shall denote by R(,;,tf. 19); I do the same with the notation (dw, dO for the differential 39 §6. INTRODUCTORY REMARKS d(w,O. One should remark that k or d~ does not, generally speaking, have a tensor character and is not a geometric object independent of local coordinates; the invariant character is exhibited only by the pair (w, k) or (dw,dO. , w (£). From this we see that because of the first term in the right-hand side dw enters in the coordinate transformations for d~.
13) If in some coordinate neighborhood U" the vectors ei(w), • · ·,ek(w)" form a basis in X! 14) t=l then, in the corresponding local coordinates, K(W) is expressed by the matrix = 1, ... , m; j = 1, ... ,k). , f) is not smooth with respect to w). ). we can write the expressions w w = w = = + f (w) '· x (w) s + /. 16) can be written as ~) co= (x, A) ( or ro = (x, ~) ( ~ ) This notation agrees with the vector-matrix notation for local coordinates. 8 can be written as a column in which, from the top, we have the coordinates of ~.