Topology of the Space of Nondegenerate Closed Curves
نویسنده
چکیده
A curve on a sphere, aane or projective space is called nondegenerate if its osculating frame is nondegenerate at every point. We calculate the number of connected components in the space of all closed nondegenerate curves immersed into S n ; R n or P n. In the cases of S n or R n it is equal to 4 for odd n > 3 and 6 for even n > 4 (for S 2 the answer is also 6). For projective space P n the number of connected components equals 10 for odd n > 3 and equals 3 for even n > 2. We calculate the number of homotopy classes of nondegenerate closed immersions of a circle into n-dimensional spheres, aane and projective spaces. An immersion in our sense is a C n-smooth immersion of a segment or a circle. Consider the C n-topology on the space of immersions. Denote by I the segment 0; 1]. A germ of any smooth immersion c : I ! R n determines at any moment t an oriented osculating ag V (t) = fV 1 (t); : : :; V n (t) = R n g, where V i (t) is generated by the i-tuple of vectors fc 0 (t); : : :; c (i) (t)g. Deenition. An immersion c : I ! R n (S n or P n) is called nondegenerate if its oriented osculat-ing ag at any moment is complete, i.e. its osculating frame fc 0 (t); : : :; c (n) (t)g forms a basis in T c(t) R n (T c(t) S n or T c(t) P n). Choose any orientation of R n (S n or P n if it is orientable). Deenition. A nondegenerate immersion c is called right-oriented or right if orientation of its oscu-lating frame at some (and thus at every) moment t coincides with the choosen one, and left-oriented or left otherwise. The main results of the article are as follows. Theorem 1. The number of the homotopy classes of nondegenerate right immersions S 1 ! R n equals 2 for even n > 4 and 3 for odd n > 3. Theorem 2. The number of the homotopy classes of nondegenerate right immersions S 1 ! S n equals 3 for even n > 2 and 2 for odd n > 3 (see g.1).
منابع مشابه
On the Number of Connected Components in the Space of Closed Nondegenerate Curves on S
The main definition. A parametrized curve γ : I → R is called nondegenerate if for any t ∈ I the vectors γ′(t), . . . , γ(t) are linearly independent. Analogously γ : I → S is called nondegenerate if for any t ∈ I the covariant derivatives γ′(t), . . . , γ(t) span the tangent hyperplane to S at the point γ(t) ( compare with the notion of n-freedom in [G]). Fixing an orientation in R or S we cal...
متن کاملDefinition of General Operator Space and The s-gap Metric for Measuring Robust Stability of Control Systems with Nonlinear Dynamics
In the recent decades, metrics have been introduced as mathematical tools to determine the robust stability of the closed loop control systems. However, the metrics drawback is their limited applications in the closed loop control systems with nonlinear dynamics. As a solution in the literature, applying the metric theories to the linearized models is suggested. In this paper, we show that usin...
متن کاملOn two problems concerning the Zariski topology of modules
Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...
متن کاملThe Wijsman structure of a quantale-valued metric space
We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...
متن کاملMore On λκ−closed sets in generalized topological spaces
In this paper, we introduce λκ−closed sets and study its properties in generalized topological spaces.
متن کامل