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It is known from differential geometry that the necessary and sufficient conditions for a planar curve (of length w) to be closed are the equalities f coS (footk(t)dt) dt = /sin (footk(t)dt) dr = O, where t is the natural parameter, k is the curvature, and the integral is taken over an arbitrary segment of length w. These conditions are obtained by explicit integration of the Frenet equations. The equalities yield the necessary condition f k dt E 2rrZ.In 1951, Fenchel [1] posed the problem of constructing criteria for the closedness of a space curve by given curvature and torsion as functions of the natural parameter. (This problem was also discussed by S. S. Cherny and N. V. Efimov.) In this case one can also integrate the Frenet equations; however, the resulting criterion is that a series of integrals (a multiplicative integral) be equal to zero [2]; this series does not seem to be represented in a "finite" form (i.e., with finitely many integrals and terms). Aminov [3] established the criterion for the closedness of a space broken line (in terms of corresponding angles). In [4] and in subsequent papers, Aminov studied the connection of curvature and curve torsion given by a trigonometric polynomial of finite order.As stated in a number of papers, it is impossible to solve Fenchel's problem effectively. In a certain sense, we confirm this opinion.We mention that for a curve to be closed it is necessary (and sufficient, if the curve is bounded) that its tangent indicatrix be closed. For this reason, solution of the problem on the closedness of a curve on the unit sphere in E 3 plays an essential role for Fenchel's problem.It is understood that a spherical curve is defined up to its motion by its geodesic curvature (using the latter, one can effectively calculate the space curvature and torsion). Thus, we arrive at the following problem (similar to the corresponding planar problem): find a criterion for the closedness of a curve on a unit sphere in E 3 by its geodesic curvature as a function of the natural parameter.The obvious necessary condition is the periodicity. We prove that in the class of simple natural integral expressions the necessary conditions are either quite empty or are far from sufficient ones.Suppose that w > 0 is fixed. We denote the class of closed C2-curves of length w on the unit sphere in E 3 by C2(w), the natural parameter by t, and the geodesic curvature by k(t

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