Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2 π. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map , or equivalently the winding number of the unit tangent (which … Zobacz więcej In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, f : M → N is an immersion if Zobacz więcej A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H : M … Zobacz więcej A k-tuple point (double, triple, etc.) of an immersion f : M → N is an unordered set {x1, ..., xk} of distinct points xi ∈ M with the same image … Zobacz więcej A far-reaching generalization of immersion theory is the homotopy principle: one may consider the immersion condition (the rank of the derivative is always k) as a partial differential relation (PDR), as it can be stated in terms of the partial derivatives of the function. … Zobacz więcej Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2m < n + 1 every map f : M → N of an m-dimensional … Zobacz więcej • A mathematical rose with k petals is an immersion of the circle in the plane with a single k-tuple point; k can be any odd number, but if even must be a multiple of 4, so the figure … Zobacz więcej • Immersed submanifold • Isometric immersion • Submersion Zobacz więcej Witryna11 kwi 2016 · By arbitrariness of U and continuity of \(k_\gamma \) and k, it follows that \(k_\gamma (t_0)\leqslant k(t_0)\). \(\square \) The variant of Theorem 1 for closed curves (see Corollary 1) generalizes a result due to McAtee [], who proved that there exists a \(C^2\) knot of constant curvature in each isotopy class building upon the …

Averaged Mean Curvature Flow

WitrynaJ. Scott Carter, Extending immersed circles in the sphere to immersed disks in the ball, Comment. Math. Helv. 67 (1992), no. 3, 337–348. MR 1171298, DOI … Witryna28 kwi 2024 · As far as I know, immersions become more relevant in the context of manifolds (of which curves are a special case). In general, if you have a map $f : M \to N$ between manifolds which is an immersion, it means the derivative $df_x : T_x M \to T_ {f (x)}N$ is injective at each point $x \in M$. florida state university mis

On the curve diffusion flow of closed plane curves SpringerLink

WitrynaAbstract. This is an expository paper describing the recent progress in the study of the curve shortening equation. {X_ { {t\,}}} = \,kN. ( (0.1)) Here X is an immersed curve … WitrynaShortening embedded curves By MArrHEw A. GRAYSON* 0. Introduction The curve shortening problem is to analyze the long-term behavior of smooth curves, immersed in a Riemannian surface, which evolve by their curvature vectors. Although evolution by curvature is a natural way to shorten curves, it leads to a number of complex problems. Witrynato immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure over kru;vs{puvq, can be viewed as a set of immersed curves. With this observation as a starting point, given a knot K in S 3, we realize the immersed curve invariant yHFpS r ˚pKqq[4] florida state university masters online