Search

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{SU}(3)/\mathrm{T}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the \emph{tetrahedron} $\cal{T}:= x+y+z = 1$, in this paper, we describe all regions $\cal R \subset \cal T$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion's. We study the dynamics of such regions under the \emph{projected Ricci flow} in the plane $(x,y)$, concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $\mathrm{SU}(3)/\mathrm{T}^2$, further studying its evolution under the Ricci flow on the base., 15 pages, 12 figures. arXiv admin note: text overlap with arXiv:2305.06119

In this note, we show that the classical Wallach manifold $\mathrm{SU}(3)/\mathrm{T}^2$-admits metrics of positive intermediate Ricci curvature $(\Ricci_d >0)$ for $d = 1, 2, 3, 4, 5$ that lose these properties under the homogeneous Ricci flow for $d=1, 2, 3, 5$. We make the same analyses to the family of Riemannian flag manifolds $\mathrm{SU}(m+2p)/\mathrm{S}(\mathrm{U}(m)\times\mathrm{U}(p)\times \mathrm{U}(p))$, concluding similar results. These explicitly verify some claims expected to be true among experts (see \cite{BW}) for positive Ricci curvature and intermediate positive Ricci curvature. Our technique is only possible due to the global behavior understanding of the homogeneous Ricci flow for invariant metrics on these manifolds., This replacement fixes an imprecision on the former Theorem B (we kept the numbering), adding/changing the need in the remaining paper's content to make things accordingly. References were added, and small changes were made to the file

The Euler characteristic of a 2-dimensional compact manifold and the local behavior of smooth vector fields defined on it are related to each other by means of the Poincar\'e--Hopf Theorem. Despite of the importance of Filippov vector fields, concerning both their theoretical and applied aspects, until now, it was not known if this result it still true for Filippov vector fields. In this paper, we show it is. While in the smooth case the singularities consist of the points where the vector field vanishes, in the context of Filippov vector fields the notion of singularity also comprehend new kinds of points, namely, pseudo-equilibria and tangency points. Here, the classical index definition for singularities of smooth vector fields is extended to singularities of Filippov vector fields. Such an extension is based on an invariance property under a regularization process. With this new index definition, we provide a version of the Poincar\'e--Hopf Theorem for Filippov vector fields. Consequently, we also get a Hairy Ball Theorem in this context, i.e. ``any Filippov vector field defined on a sphere must have at least one singularity (in the Filippov sense)''., Comment: 19 pages, 3 figures

The Euler characteristic of a 2-dimensional compact manifold and the local behavior of smooth vector fields defined on it are related to each other by means of the Poincaré--Hopf Theorem. Despite of the importance of Filippov vector fields, concerning both their theoretical and applied aspects, until now, it was not known if this result it still true for Filippov vector fields. In this paper, we show it is. While in the smooth case the singularities consist of the points where the vector field vanishes, in the context of Filippov vector fields the notion of singularity also comprehend new kinds of points, namely, pseudo-equilibria and tangency points. Here, the classical index definition for singularities of smooth vector fields is extended to singularities of Filippov vector fields. Such an extension is based on an invariance property under a regularization process. With this new index definition, we provide a version of the Poincaré--Hopf Theorem for Filippov vector fields. Consequently, we also get a Hairy Ball Theorem in this context, i.e. ``any Filippov vector field defined on a sphere must have at least one singularity (in the Filippov sense)''., 19 pages, 3 figures

The Ricci flow was introduced by Hamilton and gained its importance through the years. Of special importance is the limiting behavior of the flow and its symmetry properties. Taking this into account, we present a novel normalization for the homogeneous Ricci flow with natural compactness properties. In addition, we present a characterization for Gromov-Hausdorff limits of homogeneous spaces. As a result, we present a detailed picture of the homogeneous Ricci flow for three-isotropy-summands flag manifolds: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. Moreover, we achieve a full classification of the possible Gromov-Hausdorff limits of the aforementioned lines of flow., Comment: 35 pages, 4 figures. Exposition improved. Minor mistakes corrected

In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a regular saddle that hits a discontinuity as a parameter is varied, is surprisingly complex. Bifurcation diagrams are presented here for non-resonant saddles in the plane, including an example in a forced pendulum., 39 pages, 35 figures