Transparencies of the talk given at egeo2005 (La Falda, Junio de 2005), slightly edited.

We consider the 2-stein condition on k-step Carnot spaces S. These spaces are a subclass in the class of solvable Lie groups of Iwasawa type of algebraic rank one and contain the homogeneous Einstein spaces within this class. They are obtained as a semidirect product of a graded nilpotent Lie group N and the abelian group R.

We show that the 2-stein condition is not satisfied on a proper 4-step Carnot spaces S.

This is a short introduction to supersymmetry based on the first of two lectures given at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005.

This is a survey about recent joint work with Christoph Mayer, Thomas Mohaupt and Frank Saueressig on the special geometry of Euclidian supersymmetry. It is based on the second of two lectures given at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005.

We discuss metrics with holonomy G₂ by presenting a few crucial examples and review a series of G₂ manifolds constructed via solvable Lie groups, obtained in [1]. These carry two related distinguished metrics, one negative Einstein and the other in the conformal class of a Ricci-flat metric, plus other features considered definitely worth investigating.

[1]. S. Chiossi, A. Fino, Conformally parallel G₂ structures on a class of solvmanifolds, Math. Z. 252 (4), 825-848, 2006.

In these notes I review some classes of invariant complex structures on nilmanifolds for which the Dolbeault cohomology can be computed by means of invariant forms, in the spirit of Nomizu's theorem for de Rham cohomology.

Moreover, deformations of complex structures are discussed. Small deformations remain in some cases invariant, so that, by Kodaira-Spencer theory, Dolbeault cohomology can be still computed using invariant forms.

This is a short, elementary survey article about taut submanifolds. In order to simplify the exposition, we restrict to the case of compact smooth submanifolds of Euclidean or spherical spaces. Some new, partial results concerning taut 4-manifolds are discussed at the end of the text.

We survey on some recent developments on the study of Riemannian G-manifolds as Euclidean submanifolds.

The text surveys some results concerning submanifolds with parallel mean curvature vector immersed in the De Sitter space. We also propose a semi-Riemannian version of an important inequality obtained by Simons in the Riemannian case and apply it in order to obtain some results characterizing umbilical submanifolds and a product of submanifolds in the (n+p)-dimensional De Sitter space S_p^n+p.

Here we consider a generalized flag manifold F = U/K, and a differential structure F which satisfy F³ + F = 0; these structures are called f-structures. Such structure F determines in the tangent bundle of F some ad(K)-invariant distributions. Since flag manifolds are homogeneous reductive spaces, they certainly have combinatorial properties that allow us to make some easy calculations about integrability conditions for F itself and the distributions that it determines on F. An special case corresponds to the case U = U(n), the unitary group, this is the geometrical classical flag manifold and in fact tools coming from graph theory are very useful.

In the present paper we present a survey of results concerning the variety X[M] of planar normal sections associated to a natural embedding of a real flag manifold M^m. The results included are those that, we feel, better describe the nature of this algebraic variety of RP^{m − 1}. In particular we present results concerning its Euler characteristic showing that it depends only on dim M and not on the nature of M itself. Furthermore, when M is the manifold of complete flags of a compact simple Lie group, we present what is, in some sense, its dimension and a large class of submanifolds of RP^{m − 1} contained in X[M].

Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi-Civita connection of some semi-Riemannian metric on the group. As a special case, we will consider constant connections in Rⁿ.

We present some results on the spectral geometry of compact Riemannian manifolds having holonomy group isomorphic to Z^k₂, 1 ≤ k ≤ n − 1, for the Laplacian on mixed forms and for twisted Dirac operators.

This paper deals with certain advances in the understanding of the geometry of superconformal harmonic maps of Riemann surfaces into De Sitter space Sⁿ₁. The character of these notes is mainly expository and we made no attempt to provide complete proofs of the main results, which can be found in reference [12]. Our main analytic tool to study superconformal harmonic maps is a Gram-Schmidt algorithm to produce adapted frames for such maps. This allows us to compute the normal curvatures and obtain identities which are used to study their geometry. Some global properties such as fullness and rigidity are considered and a highest order Gauss transform or polar map is constructed and its main properties are discussed.

The study of minimal and, more generally, constant mean curvature hypersurfaces in Riemannian space forms is a classical topic in differential geometry. As is well known, minimal hypersurfaces are critical points of the variational problem of minimizing area. Similarly, hypersurfaces with constant mean curvature are also solutions to that variational problem, when restricted to volume-preserving variations. In this paper we review about the stability index of both minimal and constant mean curvature hypersurfaces in Euclidean spheres, including some recent progress by the author, jointly with some of his collaborators. One of our main objectives on writing this paper has been to make it comprehensible for a wide audience, trying to be as self-contained as possible.

The hypersurfaces with constant mean curvature (cmc) are studied under different aspects:

1. As critical Points of a Variational Problem.
2. As solutions of a Dirichlet Problem.
3. Under the point of view of harmonicity of the Gauss map.

We explain, in a short wave, the principal technical and some results obtained in each aspects.

En este artículo se discutirá la existencia de hipervariedades compactas mínimas en la esfera n-dimensional Sⁿ que se pueden escribir como la hipersuperficie de nivel de un polinomio homogéneo de grado k. Es decir estamos interesados en hipervariedades mínimas de la forma M = {x ∈ Rⁿ⁺¹ | |x| = 1, p(x) = 0} para algún polinomio homogéneo irreducible p : Rⁿ⁺¹ → R.

In this survey we describe new invariant Einstein metrics on flag manifolds. Following closely San Martin-Negreiros’s paper [26] we state results relating Kähler, (1,2)-symplectic and Einstein structures on flags. For the proofs see [11] and [10].

In this note we announce several results concerning the stability of certain families of harmonic maps that we call holomorphic-horizontal frames, with respect to families of invariant Hermitian structures on flag manifolds. Special emphasis is given to the Einstein case. See [23] for additional detail and the proofs of the results mentioned in this survey.

We present in this survey article an account of some examples on the use of nonlinear differential equations, both partial and ordinary, that have been applied to the treatment of classificatory problems in affine geometry of hypersurfaces. Locally strongly convex, complete affine hyperspheres is the first topic explained, then hypersurfaces of decomposable type, and, finally, those with parallel second fundamental (cubic) form.

For n = 3 or n = 7 let Tⁿ be the space of oriented lines in Rⁿ. In a previous article we characterized up to equivalence the metrics on Tⁿ which are invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions (they exist only in such dimensions and are pseudo-Riemannian of split type) and described explicitly their geodesics. In this short note we present the geometric meaning of the latter being null, time- or space-like.

On the other hand, it is well-known that Tⁿ is diffeomorphic to G(Hⁿ), the space of all oriented geodesics of the n-dimensional hyperbolic space. For n = 3 and n = 7, we compute now a pseudo-Riemannian invariant of Tⁿ (involving its periodic geodesics) that will be useful to show that Tⁿ and G(Hⁿ) are not isometrically equivalent, provided that the latter is endowed with any of the metrics which are invariant by the canonical action of the identity component of the isometry group of H.

Making use of Lie theory we propose a model for the simple harmonic oscillator and for the linear inverse pendulum of R². In both cases the phase space are orbits of the coadjoint representation of the Heisenberg Lie group. These orbits and the Heisenberg Lie algebra are included in a solvable Lie algebra admitting an ad-invariant metric. The corresponding quadratic form induces the Hamiltonian and the associated Hamiltonian system is a Lax equation.

The dynamical equations of motion are derived from Hamilton's principle for systems which are subject to general non-holonomic constraints. This derivation generalizes results obtained in previous works which either only deal with the linear case or make use of the D'Alembert's or Chetaev's conditions.

The harmonic content of periodic solutions in ODEs is obtained using standard techniques of harmonic balance and the fast Fourier transform (FFT). For the first method, the harmonic content is attained in the vicinity of the Hopf bifurcation condition where a smooth branch of oscillations is born under the variation of a distinguished parameter. The second technique is applied directly to numerical simulation, which is assumed to be the correct solution. Although the first method is local, it provides an excellent tool to characterize the periodic behavior in the unfoldings of other more complex singularities, such as the double Hopf bifurcation (DHB). An example with a DHB is analyzed with this methodology and the FFT algorithm.