Narrow Results

It is not known whether or not the length of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n)vo{l}^{\frac{1}{n}}$, or $\tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper, we will prove that for each $𝜖>0$ one can find such estimates for the length of a geodesic loop with angle between $\pi -𝜖$ and $\pi$ with an explicit constant that depends both on $n$ and $𝜖$.

That is, let $𝜖>0$, and let $a=\lceil \frac{1}{sin(\frac{𝜖}{2})}\rceil +1$. We will prove that there exists a “wide” (i.e. with an angle that is wider than $\pi -𝜖$) geodesic loop on $M^n$ of length at most $2n!a^{n}d$. We will also show that there exists a “wide” geodesic loop of length at most $2(n+1){!}^2{a}^{{(n+1)}^3}FillRad\le 2\cdot n(n+1){!}^2{a}^{{(n+1)}^3}vo{l}^{\frac{1}{n}}$. Here $FillRad$ is the Filling Radius of $M^n$.

Some families of H-valued vector fields with calculable Lie brackets are given. These provide examples of vector fields on path spaces with a divergence and we show that versions of Bismut type formulae for forms on a compact Riemannian manifold arise as projections of the infinite dimensional theory.

It is believed that the family of Riemannian manifolds with negative curvatures is much richer than that with positive curvatures. In fact there are many results on the obstruction of furnishing a manifold with a Riemannian metric whose curvature is positive. In particular any manifold admitting a Riemannian metric whose Ricci curvature is bounded below by a positive constant must be compact. Here we investigate such obstructions in terms of certain functional inequalities which can be considered as generalized Poincaré or log-Sobolev inequalities. A result of Saloff-Coste is extended.

It has been shown in Refs. 2–6 that two natural definitions of surface measures, on the space of continuous paths in a compact Riemannian manifold embedded into ℝⁿ, introduced in the paper by Smolyanov¹ are equivalent; this means that there exists a natural object — the surface measure, which we call the Smolyanov surface measure. Moreover, it has been shown^2–6 that this surface measure is equivalent to the Wiener measure and the corresponding density has been found. But the known proof of the equivalence of the two definitions of the surface measure is rather nonexplicit; in fact the densities of the measures corresponding to the two different definitions were found independently and only a posteriori it was discovered that those densities coincided.

Our aim is to give a direct proof of this fact. We introduce a more restrictive definition of the surface measure as the weak limit of a standard Brownian motion in ℝⁿ conditioned to be in the tubular ε-neighborhood of the manifold at times 0=t₀<t₁<⋯<t_n-1<t_n= 1 as both ε and the diameter of the partition tend to zero. Letting ε and then the diameter of the partition tend to zero and vice versa, we arrive at the two definitions above. We prove the existence of the Smolyanov surface measure using our definition, show that this measure is equivalent to the law of a Brownian motion on the manifold, and compute the corresponding density in terms of the curvature of the manifold. As a special case of this, we again obtain the results of Refs. 2–6.

We prove the existence and uniform bounds for the electrostatic Klein–Gordon–Maxwell systems in the inhomogeneous context of a compact Riemannian manifold when the mass potential, balanced by the phase, is small in a quantified sense. Phase compensation for the electrostatic Klein–Gordon–Maxwell systems and the positive mass theorem are used in a crucial way.

Non-Euclidean, or incompatible elasticity, is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e. there are no bending contributions.

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)_n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)_n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)_n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)₄ spacetime manifolds.

In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named (ZRF)_n. The main result of the paper is that the closedness property of the associated covector is achieved also for rank(Z_kl) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for(ZRF)_n manifolds with rank(Z_kl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric (PZS)_n and weakly Z-symmetric manifolds (WZS)_n in the case of non-singular Z tensor. In the last sections we study special conformally flat (ZRF)_n and give a brief account of Z recurrent forms on Kaehler manifolds.

In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds (PS)_n and pseudo-concircular symmetric manifolds is defined. This is named pseudo-Q-symmetric and denoted with (PQS)_n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat (PQS)_n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of (PQS)_n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a (PQS)_n scalar field space-time is considered, and interesting properties are pointed out.

We study the bound state problem for $N$ attractive point Dirac $\delta$-interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds ${ℍ}^2$ and ${ℍ}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on ${ℝ}^2$ and ${ℍ}^2$.

In this paper, we study the biharmonic submanifolds of Riemannian manifolds endowed with metallic and complex metallic structures. In case of both the structures, we obtain the necessary and sufficient conditions for a submanifold to be biharmonic. Particularly, we find the estimates for mean curvature of Lagrangian and complex surfaces.

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.

A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on the Stiefel manifolds SO(2k+ l)/SO(l).

Dynamical systems on Riemannian manifolds are examined which are either generated by a semi-flow or by a C¹-smooth map. The exponential growth rate of the k-volumina provides a lower bound of the topological entropy of the system if its phase space is compact and has box-counting dimension k. If the system possesses an equivariant sub-bundle instead of the tangent map its restriction onto the fibers of this sub-bundle can be considered. Inequalities for systems with not necessarily compact phase space are also given. Examples address linear maps in ℝ^m, the time-reversed Lorenz system, and the geodesic flow on a (not necessarily compact) Riemannian manifold without conjugate points.

We consider a periodic magnetic Schrödinger operator on a noncompact Riemannian manifold M such that H¹(M,ℝ) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We review a general scheme of a proof of existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit, which was suggested in our previous paper, and some applications of this scheme. Then we apply these methods to establish similar results in the case when the wells have regular hypersurface pieces.