Summary of the three lectures. These notes are available electronically at http://www.ma.huji.ac.il/~drorbn/Talks/Srni-9901/notes.html.

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].

Let $M$ be a $C^{\infty }$-manifold with a Riemannian conformal structure $C$. Given a regular curve $\gamma$ on $M$, the authors define a linear operator on the space of (differentiable) vector fields along $\gamma$, only depending on $C$, called the Fermi-Walker connection along $\gamma$. Then, the authors introduce the concept of Fermi-Walker parallel vector field along $\gamma$, proving that such vector fields set up a linear space isomorphic to the tangent space at a point of $\gamma$. This allows to consider the Fermi-Walker horizontal lift of...

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.