Narrow Results

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L²(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

In this paper, we study a coupled system of the Ricci–Bourguignon flow on a closed Riemannian manifold $M$ with the harmonic map flow. At the first, we will investigate the existence and uniqueness for solution of this flow on a closed Riemannian manifold and then we find evolution of some geometric structures of manifold along this flow.

This paper provides the optimal exponential decay rate of the lower bound for the first positive eigenvalue of the Laplacian operator on a compact Riemannian manifold with a negative lower bound on the Ricci curvature and with large diameter. For manifolds with boundary, suitable convexity conditions are assumed.

The Fisher–Rao metric on the parameter space for the set of lines in a two-dimensional convex image is approximated under the assumption that the errors in the measurements are small. The volume of the parameter space under the approximating metric is proportional to the area of the image under the Euclidean metric. In the case of a rectangular image, expressions for the approximating metric are obtained and an algorithm is given for sampling the parameter space. The sample points are used in an algorithm for detecting lines in a rectangular image. Experimental results are reported. In the case of a disc shaped image the parameter space for lines embeds isometrically, under the approximating metric, into three-dimensional Euclidean space.

A novel class of Einstein vacua is presented, which possess nonvanishing cosmological constant and accelerating horizon with the topology of S^D-3 fibration over S¹. After Euclideanization, the solution describes a conformally distorted S^D-1 fibration over S¹, which is smooth, compact and inhomogeneous and can be regarded as analog of Don Page's gravitational instanton.

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP² is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S³(1), solving a question posed by Gromov.

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the L^p Hardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C^∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of nonconstant solutions u ∈ H^1,p(M) to the quasilinear problem

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

We establish the existence of semiclassical states for a nonlinear Klein–Gordon–Maxwell–Proca system in static form, with Proca mass $1,$ on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of three-dimensional manifolds on which the system has semiclassical solutions for every exponent $p\in (2,\infty ).$ The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero.

In this paper, first we study a complete smooth metric measure space $(M^n,g,e^{-f}dv)$ with the ($\infty$)-Bakry–Émery Ricci curvature ${\mathrm{\text{Ric}}}_f\ge \frac{a}{2}g$ for some positive constant $a$. It is known that the spectrum of the drifted Laplacian ${\mathrm{\Delta }}_f$ for $M$ is discrete and the first nonzero eigenvalue of ${\mathrm{\Delta }}_f$ has lower bound $\frac{a}{2}$. We prove that if the lower bound $\frac{a}{2}$ is achieved with multiplicity $k\ge 1$, then $k\le n$, $M$ is isometric to ${\mathrm{\Sigma }}^{n-k}\times {ℝ}^k$ for some complete $(n-k)$-dimensional manifold $\mathrm{\Sigma }$ and by passing an isometry, $(M^n,g,e^{-f}dv)$ must split off a gradient shrinking Ricci soliton $({ℝ}^k,g_{\mathrm{\text{can}}},\frac{a}{4}|t{|}^2)$, $t\in {ℝ}^k$. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian ${ℒ}=\mathrm{\Delta }-\frac{1}{2}〈x,\nabla \cdot 〉$ for properly immersed self-shrinkers in the Euclidean space ${ℝ}^{n+p}$, $p\ge 1$ and show the discreteness of the spectrum of ${ℒ}$ and a logarithmic Sobolev inequality.

We refine and generalize several interpolation inequalities bounding the $L^p$ norm of a probability density with respect to the reference measure $\mu$ by its Sobolev norm and the Kantorovich distance to $\mu$ on a smooth weighted Riemannian manifold satisfying $\mathrm{\text{CD}}(0,\infty )$ condition.

We consider a Schrödinger differential expression L₀ = Δ_M + V₀ on a Riemannian manifold (M,g) with metric g, where Δ_M is the scalar Laplacian on M and V₀ is a real-valued locally square integrable function on M. We consider a perturbation L₀ + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L₀ + V to be essentially self-adjoint on . This is an extension of a result of T. Kappeler.

We introduce a study of Riemannian manifold M = ℝ² endowed with a metric of diagonal type of the form , where g is a positive function, of C^∞-class, depending on the variable x² only. We emphasize the role of metric in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

On a Riemannian almost product manifold (M, P, g), we consider a linear connection preserving the almost product structure P and the Riemannian metric g and having a totally skew-symmetric torsion. We determine the class of the manifolds (M, P, g) admitting such a connection and prove that this connection is unique in terms of the covariant derivative of P with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold (G, P, g) constructed by a Lie group G.

This paper aims to be a natural continuation of our very recent research, published in [Int. J. Geom. Meth. Mod. Phys.8(4) (2011) 783–796]. Using as geometric background a generalized Poincaré manifold, this work presents models of new explicit classes of manifolds within the class of gradient Ricci solitons. In Sec. 1 we shortly introduce a history of this topic to motivate its relevance both for theoretical and applied sciences, and in Sec. 2 our work contains its notations and assumptions. In Sec. 3 we give our framework and introduce the basic result, and in Sec. 4 we develop our theory. Finally, in Sec. 5, we conclude this paper and suggest possible further development. While the background in the first two sections is introductory, the theory in Secs. 3 and 4 is new as a whole, containing original results.

In our very recent published work [Int. J. Geom. Meth. Mod. Phys.8(4) (2011) 783–796], we considered the Riemannian manifold M = ℝ² endowed with the warped metric ḡ(x, y) = diag(g(y), 1), where g is a positive function, of C^∞-class, depending on the variable y only. Within this framework, we found a wide class of 2D gradient Ricci solitons and specialized our results to discuss some case studies. This research is a natural continuation, providing classification results for the subclass of steady gradient Ricci solitons.

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

In this study, we search slant submanifolds of almost product semi-Riemannian manifold. Accordingly, we investigate slant submanifolds of semi-Riemannian manifold by making the classifications as slant Riemannian, slant semi-Riemannian and slant lightlike submanifold. Moreover, we add the theorems which characterize existence of slant submanifold.