Volume 16, Issue 3

A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

Let M be a compact Riemannian manifold without boundary and V : M → ℝ a smooth function. Denote by P t and d μ = e V dx the semigroup and symmetric measure of the second order differential operator L = ∆ + ∇ V · ∇. For some suitable convex function Ф :𝓙 → ℝ defined on an interval 𝓙, we consider the Ф -entropy of P t f (with respect to μ ) for any f ∈ C ∞ (M , 𝓙). We show that an integral form curvature-dimension condition is equivalent to an estimate on the rate of change of the Ф -entropy. We also generalize this result to bounded smooth domains of a complete Riemannian manifold.

The optimal density function assigns to each symplectic toric manifold M a positive number d ≤ 1 obtained by considering the ratio between the maximum volume of M which can be filled by symplectically embedded disjoint balls and the total symplectic volume of M . In the toric version, M is toric and the balls need to be embedded respecting the toric action on M . We give a brief survey of toric symplectic manifolds and the recent constructions of moduli space structure on them, and then we recall how to define a natural density function on this moduli space. We review previous work which explains how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is 4.

We introduce, in the Riemannian setting, the notion of conformal Ricci soliton , which includes as particular cases Einstein manifolds , conformal Einstein manifolds and (generic and gradient) Ricci solitons . We provide necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of (0, 3)-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor D recently introduced by Cao and Chen. We derive commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.

Using Oprea’s optimization methods on submanifolds, we give another proof of the inequalities relating the normalized δ -Casoraticurvature δ^$\hat \delta $ c ( n −1) for submanifolds in real space forms. Also, inequalities relating the normalized δ -Casorati curvature δC ( n − 1) for submanifolds in real space forms are obtained. Besides, we characterize a kind of Casorati ideal hypersurface of Euclidean 4-space. We also show that this kind of Casorati ideal hypersurface is rigid.

An r -hyperconvex body is a set in the d -dimensional Euclidean space 𝔼 d that is the intersection of a family of closed balls of radius r . We prove the analogue of the classical Blaschke–Santaló inequality for r -hyperconvex bodies, and we also establish a stability version of it. The other main result of the paper is an r -hyperconvex version of the reverse isoperimetric inequality in the plane.

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that for d ≥ 4 there are superexponentially many combinatorially distinct neighborly d -polytopes on n vertices that admit realizations inscribed in the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations in ℝ d for d ≥ 3.

In this paper we obtain a formula for the boundary map of the cellular ℤ-homology of the real maximal isotropic Grassmannians of type B , C and D which are obtained as minimal flag manifolds of split real forms of semi-simple Lie algebras of the respective type. We use the model of shifted Young diagrams presented in [9] to describe the Schubert varieties and the cellular decompositions of such flag manifolds. The results are applied to determine the orientability of the real maximal isotropic Grassmannians.

Being a Lie group, the group SE(3) of orientation preserving motions of the real Euclidean 3-space becomes a symmetric space (in the sense of O. Loos) when endowed with the multiplication µ ( g , h ) = gh −1 g . In this note we classify all connected symmetric subspaces of SE(3) up to conjugation. Moreover, we indicate some of its important applications in robot kinematics.

Let Γ be a metric graph of genus g and let r be an integer with 2 ≤ r ≤ g − 2 such that Γ has a linear system g2rr$g_{2r}^r$. Then Γ has a linear system g21$g_{2}^1$. For algebraic curves this is part of Clifford’s well-known Theorem.