The X-ray transform for a generic family of curves

On one family of 13-dimensional closed Riemannian positively curved manifolds

An infinite family of pairwise nonhomeomorphic 13-dimensional positively curved manifolds is constructed

Curve Shortening Flow and Smooth Projective Planes

In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective plane. In addition, as a consequence of our main result, we show that any two smooth embedded curves on $RP^2$ which intersect transversally at exactly one point converge to two different geodesics under the flow.

A Note on Curve Counting Scheme in an Algebraic Family and The Admissible Decomposition Classes

Mcduff had proposed in 1997 a way to modify the definition of Taubes' version of Gromov invariant when multiple coverings of -1 curves appear. In this paper we generalize Mcduff's proposal to the family case, as is needed in the discussion of family Seiberg-Witten theory. For simplicity, the discussion has been formulated in the algebraic category.

Surfaces Family With Common Null Asymptotic

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this frame and analysed the necessary and sufficient condition for that curve to satisfy the asymptotic requirement. We illustrate the method by giving some examples.

Parametric Representation of a Hypersurface Family With a Common Spatial Geodesic

In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface as a linear combination of this frame and analyse the necessary and sufficient condition for that curve to be geodesic. We illustrate this method by presenting some examples.

Similar Curves With Variable Transformations

In this paper, we define a new family of curves and call it a {\it family of similar curves with variable transformation} or briefly {\it SA-curves}. Also we introduce some characterizations of this family and we give some theorems. This definition introduces a new classification of a space curve. Also, we use this definition to deduce the position vectors of plane curves, general helices and slant helices, as examples of a similar curves with variable transformation.

On the $FRS$-generic family of space cusps

We consider in this paper the $FRS$-deformations of a family of space curves with codimension $\leq 3$. Some geometric aspects of a space curve such as flattenings, vertices and twistings points has been studied.

Family Switching Formula and the $-n$ Exceptional Rational Curves

In this paper, we derive the family switching formula of -n two-sphere fiber bundle embedded in a smooth four-manifold fiber bundle. In the smooth category, it is a partial generalization of Fintushel-Stern's argument for four-manifolds. We also derive an algebraic analogue of the family switching formula, allowing the P^1 fibration to carry singular fibers. At the end, we apply the idea of switching to compare the canonical algebraic Kuranishi models of different classes on the so-called "universal family".

通过椭圆曲线研究一族恰当双调和映射的零度

The Nullity of a Family of Proper Biharmonic Maps via Elliptic Curves

研究从平坦二维环面到二维球面的一族恰当双调和映射的零度猜想，通过揭示谱几何与算术几何联系，构造椭圆曲线同构转化问题，确定莫德尔 - 韦伊群，证实猜想并得出该族映射零度为5的结论。

The X-ray transform on a general family of curves on Finsler surfaces

We consider a general family of curves $Γ$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $φ\in C^{\infty}(M)$ and $ω$ a smooth 1-form on $M$. We show that $$\int_{γ(t)}\{φ(γ(t))+ω_{γ(t)}(\dotγ(t))\}\,dt=0$$ holds for every $γ\inΓ$ whose endpoints belong to $\partial M$, $γ(a)\in\partial M$, $γ(b)\in\partial M$ if and only if $φ=0$ and $ω$ is exact. Similar results were proved when $M$ is closed and some additional conditions on Gaussian curvature are imposed. We also study the cohomological equations of Anosov generelized thermostats on a closed Finsler surface.…

Null Similar Curves with Variable Transformations in Minkowski 3-space

In this study, we define a family of null curves in Minkowski 3-space and called null similar curves. We obtain some properties of these special curves. We show that two null curves are null similar curves if and only if these curves form a null Bertrand pair. Moreover, we obtain that the family of null geodesics and null helices form the families of null similar curves with variable transformation.

Intrinsic representation curves

The purpose of this article is to find a family of curves parametrized by arc length and that depend on an angular function and an intrinsic fraction function, which is defined as the quotient between torsion and curvature. We find for this family of curves explicit formulas of curvature, torsion and geodetic curvature, in terms of the angular function and the intrinsic fraction function. Applications are found for the case of the general helices and slant helices.

Constrained elastic curves and surfaces with spherical curvature lines

In this paper we consider surfaces with one or two families of spherical curvature lines. We show that every surface with a family of spherical curvature lines can locally be generated by a pair of initial data: a suitable curve of Lie sphere transformations and a spherical Legendre curve. We then provide conditions on the initial data for which such a surface is Lie applicable, an integrable class of surfaces that includes cmc and pseudospherical surfaces. In particular we show that a Lie applicable surface with exactly one family of spherical curvature lines must be generated by the lift of …

Chern-Weil theory for line bundles with the family Arakelov metric

We prove a result of Chern-Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J.I. Burgos Gil, J. Kramer and U. Kühn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full level structure, equipped with the Petersson metric. Our main tool, as in the work by Burgos Gil, Kramer and Kühn, is the notion of a b-divisor.