Given a Hörmander system $X=\{X_1,\cdots ,X_m\}$ on a domain $\Omega \subseteq {𝐑}^n$ we show that any subelliptic harmonic morphism $\phi$ from $\Omega$ into a $\nu$-dimensional riemannian manifold $N$ is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also $\phi$ is a submersion provided that $\nu \le m$ and $X$ has rank $m$. If $\Omega ={𝐇}_n$ (the Heisenberg group) and $X=\left\{\frac{1}{2}\left(L_{\alpha }+L_{\overline{\alpha }}\right),\frac{1}{2i}\left(L_{\alpha }-L_{\overline{\alpha }}\right)\right\}$, where $L_{\overline{\alpha }}=\partial /\partial {\overline{z}}^{\alpha }-i{z}^{\alpha }\partial /\partial t$ is the Lewy operator, then a smooth map $\phi :\Omega \to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi :(C\left({𝐇}_n\right),F_{{\theta }_0})\to N$ is a harmonic morphism, where $S^1\to C\left({𝐇}_n\right)\stackrel{\pi }{\to }\to {𝐇}_n$ is the canonical circle bundle and $F_{{\theta }_0}$ is the Fefferman...

In the mid fifties, Charles Ehresmann defined Geometry as "the theory of more or less rich structures, in which algebraic and topological structures are generally intertwined". In 1973 he defined it as the theory of differentiable categories, their actions and their prolongations. Here we explain how he progressively formed this conception, from homogeneous spaces to locally homogeneous spaces, to fibre bundles and foliations, to a general notion of local structures, and to a new foundation of differential...

In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties: $1_{}$. if M is a linear manifold, then (M) contains convex functions, $2_{}$. (·) is invariant under diffeomorphisms, $3_{}$. each f ∈ (M) is differentiable on a dense $G_{\delta }$-set, is investigated.

We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.

We prove short time existence for the Ricci flow on open manifolds of non-negative complex sectional curvature without requiring upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger–Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with non-negative complex sectional curvature which subconverge to a Ricci flow on the open manifold. Furthermore,...

The gradient flow of bending energy for plane curve is studied. The flow is considered under two kinds of constraints; one is under the area and total-length constraints; the other is under the area and local-length constraints. The fundamental results (the local existence and uniqueness) were obtained independently by Kurihara and the second author for the former one; by Okabe for the later one. For the former one the global existence was shown for any smooth initial curves, but the asymptotic...

In this paper we consider the modified wave equation associated with a class of radial Laplacians $L$ generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra $\mathbf{c}$-function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of...

In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well...

In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as...

We prove the short-time existence of the hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of ${ℝ}^{n+1}$ ($n\ge 2$) is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces are shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic...