Current volume
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19521968 Revista de la Unión Matemática Argentina y de la Asociación Física Argentina
19441951 Revista de la Unión Matemática Argentina; órgano de la Asociación Física Argentina
19361944

Los dos fascículos que componen este volumen contienen trabajos presentados en el
II Encuentro de Geometría
Diferencial, realizado del 6 al 11 de junio 2005 en La Falda, Sierras de Córdoba,
Argentina.
Volumen 47, número 1 (2006)
Publicado en 2007.
II Encuentro de Geometría Diferencial.
Carlos E. Olmos
PDF

i 
Carnot manifolds.
Aroldo Kaplan
PDF
Transparencies of the talk given at egeo2005 (La Falda, Junio de
2005), slightly edited.

18 
4step Carnot spaces and the 2stein condition.
María J. Druetta
PDF
We consider the 2stein condition on kstep 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 2stein condition is not satisfied on a proper 4step
Carnot spaces S.

921 
An introduction to supersymmetry.
Vicente Cortés
PDF
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.

2328 
The special geometry of Euclidian supersymmetry: a survey.
Vicente Cortés
PDF
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.

2934 
Special metrics in G_{2} geometry.
Simon G. Chiossi and Anna Fino
PDF
We discuss metrics with holonomy G_{2} by presenting a
few crucial
examples and review a series of G_{2} 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 Ricciflat metric,
plus other features considered definitely worth investigating.
[1]. S. Chiossi, A. Fino, Conformally parallel G_{2}
structures on a class of solvmanifolds, Math. Z. 252 (4), 825848, 2006.

3549 
Dolbeault cohomology and deformations of nilmanifolds.
Sergio Console
PDF
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 KodairaSpencer theory, Dolbeault
cohomology can be still computed using invariant forms.

5160 
Taut submanifolds.
Claudio Gorodski
PDF
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 4manifolds are discussed at the end of the text.

6172 
Riemannian Gmanifolds as Euclidean submanifolds.
Ruy Tojeiro
PDF
We survey on some recent developments on the study of Riemannian
Gmanifolds as Euclidean submanifolds.

7383 
On complete spacelike submanifolds in the De Sitter space
with parallel mean curvature vector.
Rosa Maria S. Barreiro Chaves and Luiz Amancio M. Sousa Jr.
PDF
The text surveys some results concerning submanifolds with
parallel mean curvature vector immersed in the De Sitter space. We also propose
a semiRiemannian 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}.

8598 
Integrability of fstructures on generalized
flag manifolds.
Sofía Pinzón
PDF
Here we consider a generalized flag manifold F = U/K,
and a
differential structure F which satisfy
F^{3} + F = 0; these structures are called
fstructures. 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.

99113 
On the variety of planar normal sections.
Alicia N. García, Walter N. Dal Lago and Cristián U. Sánchez
PDF
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].

115123 
Connections compatible with tensors. A characterization
of leftinvariant LeviCivita connections in Lie groups.
Paolo Piccione and Daniel V. Tausk
PDF
Symmetric connections that are compatible with semiRiemannian
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 leftinvariant connection on a Lie group to be the
LeviCivita connection of some semiRiemannian metric on the group. As a
special case, we will consider constant connections in
R^{n}.

125134 
Spectral properties of elliptic operators on
bundles of Z^{k}_{2}manifolds.
Ricardo A. Podestá
PDF
We present some results on the spectral geometry of compact
Riemannian manifolds having holonomy group isomorphic to
Z^{k}_{2}, 1 ≤ k ≤ n − 1,
for the Laplacian on mixed forms and for twisted Dirac operators.

135149 
Volumen 47, número 2 (2006)
Publicado en 2007.
II Encuentro de Geometría Diferencial.
Carlos E. Olmos

i 
A short survey on biharmonic maps between
Riemannian manifolds.
S. Montaldo and C. Oniciuc
PDF

122 
On the geometry of a class of conformal harmonic maps of
surfaces into S^{n}_{1}.
Eduardo Hulett
PDF
This paper deals with certain advances in the understanding of
the geometry of superconformal harmonic maps of Riemann surfaces into De
Sitter space S^{n}_{1}. 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 GramSchmidt 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.

2338 
On the stability index of minimal and constant mean
curvature hypersurfaces in spheres.
Luis J. Alías
PDF
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 volumepreserving 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 selfcontained as possible.

3961 
Hypersurfaces with constant mean curvature.
Susana Fornari
PDF
The hypersurfaces with constant mean curvature (cmc) are studied
under different aspects:
 As critical Points of a Variational Problem.
 As solutions of a Dirichlet Problem.
 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.

6366 
Hipervariedades mínimas algebraicas en
S^{n}.
Oscar Mario Perdomo and Héber Mesa P.
PDF
En este artículo se discutirá la existencia de hipervariedades compactas
mínimas en la esfera ndimensional S^{n}
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 ∈ R^{n+1} 
x = 1, p(x) = 0} para algún polinomio
homogéneo irreducible
p : R^{n+1} → R.

6775 
Einstein metrics on flag manifolds.
Evandro C. F. dos Santos and Caio J. C. Negreiros
PDF
In this survey we describe new invariant Einstein metrics on flag
manifolds. Following closely San MartinNegreiros’s paper
[26] we state results
relating Kähler, (1,2)symplectic and Einstein structures on flags.
For the proofs see [11] and [10].

7784 
Stability of holomorphichorizontal maps and Einstein
metrics on flag manifolds.
Caio J. C. Negreiros
PDF
In this note we announce several results concerning the stability
of certain families of harmonic maps that we call holomorphichorizontal
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.

8594 
Classificatory problems in affine geometry approached by
differential equations methods.
Salvador Gigena
PDF
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.

95107 
Geodesics of the space of oriented lines of Euclidean space.
Marcos Salvai
PDF
For n = 3 or n = 7 let T^{n}
be the space of oriented lines in
R^{n}. In
a previous article we characterized up to equivalence the metrics on
T^{n} 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 pseudoRiemannian of split type) and described explicitly their geodesics.
In this short note we present the geometric meaning of the latter being null,
time or spacelike.
On the other hand, it is wellknown that T^{n}
is diffeomorphic to G(H^{n}),
the space of all oriented geodesics of the ndimensional hyperbolic space. For
n = 3 and n = 7, we compute now a pseudoRiemannian
invariant of T^{n}
(involving its periodic geodesics) that will be useful to show that
T^{n} and
G(H^{n}) 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.

109114 
Small oscillations on R^{2} and Lie theory.
Gabriela Ovando
PDF
Making use of Lie theory we propose a model for the simple harmonic
oscillator and for the linear inverse pendulum of R^{2}.
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 adinvariant metric. The corresponding quadratic
form induces the Hamiltonian and the associated Hamiltonian system is a
Lax equation.

115123 
Geometric approach to nonholonomic problems satisfying
Hamilton's principle.
Osvaldo M. Moreschi and Gustavo Castellano
PDF
The dynamical equations of motion are derived from Hamilton's
principle for systems which are subject to general nonholonomic 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.

125135 
Bifurcation theory and the harmonic content of oscillations.
Griselda R. Itovich, Federico I. Robbio and Jorge L. Moiola
PDF
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.

137150 
