The bisymplectomorphism group of a bounded symmetric domain

An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma.Comment: 19 pages. Version 2: minor correction

A note on the determinant 308 in Proskuryakov's linear algebra book

We put in evidence and correct a mistake in the formula for the determinant 308 in Proskuryakov’s linear algebra book. We apply this formula to reprove the well-known fact that the Fubini-Study metric on the complex projective space is Einstein

Surfaces in R4 with constant principal angles with respect to a plane

We study surfaces in R4 whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of $\R^2$. We classify all surfaces with one principal angle equal to $0$ and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in R4 with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector fiel

HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY

We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e., where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e., the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and at invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent

Constant-angle surfaces in liquid crystals

We discuss some properties of surfaces in R3 whose unit normal has constant angle with an assigned direction field. The constant angle condition can be rewritten as an Hamilton-Jacobi equation correlating the surface and the direction field. We focus on examples motivated by the physics of interfaces in liquid crystals and of layered fluids, and discuss the properties of the constant-angle surfaces when the direction field is singular along a line (disclination) or at a point (hedgehog defect

Fluctuation-dissipation relations and energy landscape in an out-of-equilibrium strong glass-forming liquid

We study the out-of-equilibrium dynamics following a temperature-jump in a model for a strong liquid, BKS-silica, and compare it with the well known case of fragile liquids. We calculate the fluctuation-dissipation relation, from which it is possible to estimate an effective temperature $T_{eff}$ associated to the slow out-of-equilibrium structural degrees of freedom. We find the striking and unexplained result that, differently from the fragile liquid cases, $T_{eff}$ is smaller than the bath temperature

Saddles in the energy landscape probed by supercooled liquids

We numerically investigate the supercooled dynamics of two simple model liquids exploiting the partition of the multi-dimension configuration space in basins of attraction of the stationary points (inherent saddles) of the potential energy surface. We find that the inherent saddles order and potential energy are well defined functions of the temperature T. Moreover, decreasing T, the saddle order vanishes at the same temperature (T_MCT) where the inverse diffusivity appears to diverge as a power law. This allows a topological interpretation of T_MCT: it marks the transition from a dynamics between basins of saddles (T>T_MCT) to a dynamics between basins of minima (T<T_MCT).Comment: 4 pages, 3 figures, to be published on PR

Electrophysiological and arrhythmogenic effects of 5-hydroxytryptamine on human atrial cells are reduced in atrial fibrillation

5-Hydroxytryptamine (5-HT) is proarrhythmic in atrial cells from patients in sinus rhythm (SR) via activation of 5-HT&lt;sub&gt;4&lt;/sub&gt; receptors, but its effects in atrial cells from patients with atrial fibrillation (AF) are unknown. The whole-cell perforated patch-clamp technique was used to record L-type Ca&lt;sup&gt;2+&lt;/sup&gt; current (&lt;i&gt;I&lt;/i&gt;&lt;sub&gt;CaL&lt;/sub&gt;), action potential duration (APD) and arrhythmic activity at 37 °C in enzymatically isolated atrial cells obtained from patients undergoing cardiac surgery, in SR or with chronic AF. In the AF group, 5-HT (10 μM) produced an increase in &lt;i&gt;I&lt;/i&gt;&lt;sub&gt;CaL&lt;/sub&gt; of 115 ± 21% above control (&lt;i&gt;n&lt;/i&gt; = 10 cells, 6 patients) that was significantly smaller than that in the SR group (232 ± 33%; &lt;i&gt;p&lt;/i&gt; 0.05; &lt;i&gt;n&lt;/i&gt; = 27 cells, 12 patients). Subsequent co-application of isoproterenol (1 μM) caused a further increase in &lt;i&gt;I&lt;/i&gt;&lt;sub&gt;CaL&lt;/sub&gt; in the AF group (by 256 ± 94%) that was greater than that in the SR group (22 ± 6%; p &#60; 0.05). The APD at 50% repolarisation (APD&lt;sub&gt;50&lt;/sub&gt;) was prolonged by 14 ± 3 ms by 5-HT in the AF group (&lt;i&gt;n&lt;/i&gt; = 37 cells, 14 patients). This was less than that in the SR group (27 ± 4 ms; &lt;i&gt;p&lt;/i&gt; &#60; 0.05; &lt;i&gt;n&lt;/i&gt; = 58 cells, 24 patients). Arrhythmic activity in response to 5-HT was observed in 22% of cells in the SR group, but none was observed in the AF group (p &#60; 0.05). Atrial fibrillation was associated with reduced effects of 5-HT, but not of isoproterenol, on &lt;i&gt;I&lt;/i&gt;&lt;sub&gt;CaL&lt;/sub&gt; in human atrial cells. This reduced effect on &lt;i&gt;I&lt;/i&gt;&lt;sub&gt;CaL&lt;/sub&gt; was associated with a reduced APD&lt;sub&gt;50&lt;/sub&gt; and arrhythmic activity with 5-HT. Thus, the potentially arrhythmogenic influence of 5-HT may be suppressed in AF-remodelled human atrium