We consider a mechanical system of three ants on the floor, which move according to two independt rules: Rule A - forces the velocity of any given ant to always point at a neighboring ant, and Rule B ...- forces the velocity of every ant to be parallel to the line defined by the two other ants. We observe that Rule A equips the 6-dimensional configuration space of the ants with a structure of a homogeneous (3,6) distribution, and that Rule B foliates this 6-dimensional configuration space onto 5-dimensional leaves, each of which is equiped with a homogeneous (2,3,5) distribution. The symmetry properties and Bryant-Cartan local invariants of these distributions are determined. In the case of Rule B we study and determine the singular trajectories (abnormal extremals) of the corresponding distributions. We show that these satisfy an interesting system of two ODEs of Fuchsian type.
We show that compact embedded starshaped r-convex hypersurfaces of certain warped products satisfying Hr = aH + b with a 0, b > 0, where H and Hr are respectively the mean curvature and r-th mean ...curvature is a slice. In the case of space forms, we show that without the assumption of starshapedness, such Weingarten hypersurfaces are geodesic spheres. Finally, we prove that, in the case of space forms, if Hr − aH − b is close to 0 then the hypersurface is close to geodesic sphere for the Hausdorff distance. We also prove an anisotropic version of this stability result in the Euclidean space.
é condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure ...limits of manifolds with lower Ricci curvature bounds.>
Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations of positive-definite ...matrices (here called covariance matrices). While the potential applications of such models have attracted significant attention, a major obstacle still stands in the way of these applications: there seems to exist no practical method of computing the normalising factors associated with Riemannian Gaussian distributions on spaces of high-dimensional covariance matrices. The present paper shows that this missing method comes from an unexpected new connection, with random matrix theory.
Canonical metrics on holomorphic Courant algebroids Garcia‐Fernandez, Mario; Rubio, Roberto; Shahbazi, Carlos ...
Proceedings of the London Mathematical Society,
September 2022, Letnik:
125, Številka:
3
Journal Article
Recenzirano
Odprti dostop
The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi–Yau manifold X$X$ admits a metric with holonomy contained in SU(n)$\mathrm{SU}(n)$, and that these metrics are ...parameterized by the positive cone in H1,1(X,R)$H^{1,1}(X,\mathbb {R})$. In this work, we give evidence of an extension of Yau's theorem to non‐Kähler manifolds, where X$X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid Q$Q$ of Bott–Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull–Strominger system, whereas the role of H1,1(X,R)$H^{1,1}(X,\mathbb {R})$ is played by an affine space of ‘Aeppli classes’ naturally associated to Q$Q$ via Bott–Chern secondary characteristic classes.
The Riemannian product M1(c1)×M2(c2), where Mi(ci) denotes the 2-dimensional space form of constant sectional curvature ci ∈ R, has two different Spin c structures carrying each a parallel spinor. ...The restriction of these two parallel spinor fields to a 3-dimensional hypersurface M characterizes the isometric immersion of M into M1(c1) × M2(c2). As an application, we prove that totally umbilical hypersurfaces of M1(c1) × M1(c1) and totally umbilical hypersurfaces of M1(c1) × M2(c2) (c1 = c2) having a local structure product, are of constant mean curvature.
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