The Arnold’s theorem (generalizing a consideration by Jacobi) states that on a generic Riemannian surface, which is sufficiently close to a sphere, the k-th caustic of a generic point has at least four semi-cubical vertices. We prove this fact by the methods of the Morse theory; in particular we replace the previous analytical condition of the “sufficient closeness to the sphere” by a geometric one, which probably is considerably less restrictive. Let M be a compact smooth Riemannian 2-dimensional manifold, p ∈ M, and Φ : (TpM, 0) → (M, p) the geodesic map, sending any central ray of TpM into the geodesic line, passing from p to the corresponding direction so that in the restriction to this ray the map Φ is isometric. The k-th caustic Ck(p) of the point p is the union of k-th conjugate points on all these geodesic lines, see. [6], [3]. Let the Riemannian metric on M be elliptic and generic, then for any k Ck(p) is a compact curve in M, all whose singularities are transversal self-intersections and semicubical cusps only. Denote by C(p) ⊂ TpM the union of critical points of the geodesic map Φ. This set splits into the union of curves Ck(p), homeomorphic to circles and consisting of the k-th intersection points of our rays with C(p). Then Ck(p) ≡ Φ(Ck(p)); this map has the fold singularity over the non-singular points of the caustic, and the Whitney cusp singularities over the cusps, see [4], §3.1. In addition, in TpM the norm function is defined by our metric, and the cusps of Ck(p) are exactly the images of extrema of the restriction of this function to Ck(p). Any locally non-singular branch of the caustic has the standard co-orientation (i.e. orientation of normal directions): to the side, on which the geodesic map locally has more pre-images. The standard unit sphere S is an example of a non-generic surface: any its set Ck(p) is not a curve but, depending on k, either p itself or its opposite point. Nevertheless all the critical sets Ck(p) are still non-singular: they are concentric circles of radii πk. C.-G. Jacobi has noticed that on a generic ellipsoid the first caustic of a generic point always has at least four cusps. V.I. Arnold [1], [2], using some ideas of S.L. Tabachnikov [7], has proved the following theorem. Theorem 1. For any k and any generic surface, sufficiently C∞-close to the standard sphere, the k-th caustic has at least 4 cusps (where the condition of closeness to the sphere strengthens when k grows). The proof in [1] is analytic. Below we give a topological proof. The conditions of “closeness to the sphere” used in it also are topological and probably much less Date: 8th June 2011. Supported in part by the grant NSh-8462.2010.1 of the President of Russia for the support of leading scientific schools.