We prove that a four-dimensional, connected, simply connected and complete Riemannian manifold which is curvature homogeneous up to order two is a homogeneous Riemannian space.

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. Examples of one particular family have ends which collapse asymptotically...

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n 2 are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n 2)-dimensional submanifolds in ${ℂ}^{n+n^2}$ for all n > 1. In...

The aim of this paper is to construct a canonical nonlinear connection $\Gamma =(M_{\left(\alpha \right)\beta }^{\left(i\right)},N_{\left(\alpha \right)j}^{\left(i\right)})$ on the 1-jet space $J^1(T,M)$ from the Euler-Lagrange equations of the quadratic multi-time Lagrangian function $$L=h^{\alpha \beta }\left(t\right)g_{ij}(t,x)x_{\alpha }^i{x}_{\beta }^j+U_{\left(i\right)}^{\left(\alpha \right)}(t,x)x_{\alpha }^i+F(t,x).$$

The concept of the Ricci soliton was introduced by R. S. Hamilton. The Ricci soliton is defined by a vector field and it is a natural generalization of the Einstein metric. We have shown earlier that the vector field of the Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an application of the theory of infinitesimal harmonic transformations.

We construct a family of almost quaternionic Hermitian structures from an almost contact metric 3-structure and also do three kinds of quaternionic Kähler structures from a Sasakian 3-structure. In particular we have a generalization of the second main result of Boyer-Galicki-Mann [5].

It is shown that Fueter regular functions appear in connection with the Eells condition for harmonicity. New conditions for mappings from 4-dimensional conformally flat manifolds to be harmonic are obtained.

We show that $n$-dimensional $(n⩾2)$ complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure $n$-space (i.e. the Euclidean metric $n$-space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the map we construct...

We show that ${\pi }_1(Symp(M,\omega ))$ can be nontrivial for $M$ that does not admit any symplectic circle action.