The {\em focal curve} of an immersed smooth curve $γ:s\mapsto γ(s)$, in Euclidean space $\R^{m+1}$, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of $γ$ (${\bf t},{\bf n}_1, ...,{\bf n}_m$), as $C_γ(s)=(γ+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}_m)(s)$, where the coefficients $c_1,...,c_{m-1}$ are smooth functions that we call the {\em focal curvatures} of $γ$. We discovered a remarkable formula relating the Euclidean curvatures $κ_i$, $i=1,...,m$, of $γ$ with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for $k=1,...,m$, necessary and sufficient conditions for the radius of the osculating $k$-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of $γ$ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of $γ$ with the Frenet frame and the Euclidean curvatures of its focal curve $C_γ$.
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.
An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $ρ^{\bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of…
We show that the focal radius of any submanifold $N$ of positive dimension in a manifold $M$ with sectional curvature greater than or equal to $1$ does not exceed $\frac{π}{2}.$ In the case of equality, we show that $N$ is totally geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a projective space with their standard metrics, provided $N$ is closed. Our results also hold for $k^{th}$--intermediate Ricci curvature, provided the submanifold has dimension $\geq k.$ Thus in a manifold with Ricci curvature $\geq n-1,$ all hypersurfaces have focal radius $\leq \frac{π}{2},$…
We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a $C^{1,α}$ compactness result for submanifolds, as well as a "soul-type" structure theorem for manifolds with nonnegative $k^{th}$--intermediate Ricci curvature that have a closed submanifold with dimension $\geq k$ and infinite focal radius. To prove these results, we use the comparison lemma for Jacobi fields from arXiv:1603…
It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of…
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.
It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions are given for all isoparametric hypersurfaces of space forms. In particular, for such hypersurfaces of the sphere, the exact collapsing time into a focal submanifold is given in terms of its dimension, the principal curvatures and their multiplicities.
In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group.
In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.
In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in irreducible symmetric spaces of compact type occur as principal orbits of Hermann actions. However, we investigate the flows conceptionally without use of this fact. Concretely the investigation is performed by investigating the mean curvature flows having the lifts of the submanifolds to an (infinite deiminsional separable) Hilbert space through a Riemannia…
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, these invariants expressed by focal curvatures of the curve. And then, we give the relationship between curvatures of evolute curve and shape curvatures. Morever, these invariants is given the geometric interpretation.
We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inflexions and vertices (maxima and minima of curvature) on the sections of a surface as the section passes through a tangential point. We also keep track on the evolution of the curvature of the curves at vertices and control its limit when the vertices collapse at singular points.…
Let $G=(V,E)$ be a finite, combinatorial graph. We define a notion of curvature on the vertices $V$ via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with curvature bounded from below by $K>0$ have diameter bounded from above. The Laplacian $L=D-A$ satisfies a Lichnerowicz estimate, there is a spectral gap $λ_2 \geq 2K$. We obtain matching two-sided bounds on the maximal commute time between any two vertices in terms of $|E| \cdot |V|^{-1} \cdot K^{-1}$. Moreover, we derive quantitative rates for the mixing ti…
In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin-Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally relate…
We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the $L$-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the $L$-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequenc…
We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.
In this paper, we study polyhedral 3-manifolds with nonnegative curvature and integral monodromy, two conditions motivated by Thurston's work in arXiv:math/9801088. We classify the 32 isometry types of codimension 3 singularities in such manifolds. We also show that the number of these singularities is bounded.
In this article, we introduce a notion of curvature, denoted by $ k_X(T)$, for a metric triple $T$ inside a (possibly discrete) metric space $X$. Such a notion enables us to consider curvature information of any metric space, including discrete metric spaces such as those generated by scientific data. To define the notion, we employ the information consisting of side lengths of the triple as well as the minimum total distance from vertices of the triple to points of the metric space. Those information provides us a unique number $k_X(T)$ such that the triple $T$ can be isometrically embedded i…
We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plan…
Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The Fáry-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4π$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs $Γ$, what limitations on the isotopy class of $Γ$ are implied by a bound on total curvature? What does ``total curvature" mean for a graph? We define a natural notion of net total curvature of a graph $Γ$ in $R^3$, and prove that if $Γ$ is homeomorphic to the $θ$…
