Analytic Manifold Research Articles

Dimension rigidity in conformal structures

Let Λ be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily expressible sufficient condition, requiring that the limit set is not too much bigger than the radial limit set, for the following dichotomy: Λ is either a real-analytic manifold or a fractal in the sense of Mandelbrot (i.e. its Hausdorff dimension is strictly greater than its topological dimension).Our primary focus is on the infinite-dimensional case. An important component of the strategy of our proof comes from the rectifiability techniques of Mayer and Urbański ('03), who obtained a dimension rigidity result for conformal iterated function systems (including those with infinite alphabets). In order to handle the infinite dimensional case, both for Kleinian groups and for iterated function systems, we introduce the notion of pseudorectifiability, a variant of rectifiability, and develop a theory around this notion similar to the theory of rectifiable sets.Our approach also extends existing results in the finite-dimensional case, where it unifies the realms of Kleinian groups, conformal iterated function systems, and rational functions. For Kleinian groups, we improve on the rigidity result of Kapovich ('09) by substantially weakening its hypothesis of geometrical finiteness. Moreover, our proof, based on rectifiability, is entirely different than that of Kapovich, which depends on homological algebra.Another advantage of our approach is that it allows us to use the “demension” of Štan'ko ('69) as a substitute for topological dimension. For example, we prove that any dynamically defined version of Antoine's necklace must have Hausdorff dimension strictly greater than 1 (i.e. the demension of Antoine's necklace).

On low-dimensional manifolds with isometric $\widetilde{\mathrm{U}} (p, q)$-actions

Denote by $\widetilde{\mathrm{U}}(p,q)$ the universal covering group of $\mathrm{U}(p,q)$, the linear group of isometries of the pseudo-Hermitian space $\mathbb{C}^{p,q}$ of signature $p,q$. Let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $\widetilde{\mathrm{U}}(p,q)$-action and that satisfies $\dim M \leq n(n+2)$ where $n = p+q$. We prove that if the action of $\widetilde{\mathrm{SU}}(p,q)$ (the connected derived group of $\widetilde{\mathrm{U}}(p,q)$) has a dense orbit and the center of $\widetilde{\mathrm{U}}(p,q)$ acts non-trivially, then $M$ is an isometric quotient of manifolds involving simple Lie groups with bi-invariant metrics. Furthermore, the $\widetilde{\mathrm{U}}(p,q)$-action is lifted to $\widetilde{M}$ to natural actions on the groups involved. As a particular case, we prove that when $\widetilde{M}$ is not a pseudo-Riemannian product, then its geometry and $\widetilde{\mathrm{U}}(p,q)$-action are obtained from one of the symmetric pairs $(\mathfrak{su}(p,q+1), \mathfrak{u}(p,q))$ or $(\mathfrak{su}(p+1,q), \mathfrak{u}(p,q))$.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

Quaternionic-like manifolds and homogeneous twistor spaces.

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the 'quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the ρ-quaternionic manifolds. Moreover, the notion of 'heaven space' finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a ρ-quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.

Support theorems for the light ray transform on analytic Lorentzian manifolds

We study the weighted ray transform of integrating functions on a Lorentzian manifold over lightlike geodesics. We prove support theorems if the manifold and the weight are analytic.

Focal points and sup-norms of eigenfunctions

If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order o(\lambda) saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point x_0 \in M for the geodesic flow at which the associated Perron–Frobenius operator U_{x_0}\colon L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant L^2 function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.

THE MAYER VIETORIS SEQUENCE CALCULATING DE RHAM COHOMOLOGY

De Rham cohomology it is very obvious that it relies heavily on both topology as well as analysis. We can say it creates a natural bridge between the two. To understand and be able to explain what exactly de Rham cohomology is to the world of mathematics we need to know de Rham groups. This is the reasons to calculate the de Rham cohomology of a manifold. This is usually quite difficult to do directly. We work with manifold. Manifold is a generalization of curves and surfaces to arbitrary dimension. A topological space M is called a manifold of dimension k if : · M is a topological Hausdorff space . · M has a countable topological base. · For all m∈M there is an open neighborhood U⊂M such that U is homeomorphic to an open subset V of ℝk. There are many different kinds of manifolds like topological manifolds, ℂk - manifolds, analytic manifolds, and complex manifolds, we concerned in smooth manifolds. A smooth manifold can described as a topological space that is locally like the Euclidian space of a dimension known. An important definion is homeomorphism. Let X, Y be topological spaces, and let f: X⟶Y e a bijection. If both f and the inverse function f−1: X⟶Y are continuous, then f is called a homeomorphism We introduce one of the useful tools for this calculating, the Mayer – Vietoris sequence. Another tool is the homotopy axiom. In this material I try to explain the Mayer – Vietoris sequence and give same examples. A short exact sequence of cochain complexes gives rise to a long exact sequence in cohomology, called the Mayer - Vietories sequence. Cohomology of the circle (S^1), cohomology of the spheres (S^2). Homeomorphism between vector spaces and an open cover of a manifold. we define de Rham cohomology and compute a few examples.

On the Extendability of Locally Defined Isometries of a Pseudo-Riemannian Manifold

Let η be a stationary subalgebra of the Lie algebra ζ of all Killing vector fields on a pseudo-Riemannian analytic manifold, G be a simply connected Lie group generated by the algebra ζ, and H be its subgroup generated by the subalgebra η. Then the subgroup H is closed in G.

Homogeneous affine surfaces: Moduli spaces

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type A surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type B surfaces which are not Type A we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to 1.

Injectivity and Stability for a Generic Class of Generalized Radon Transforms

Let (M, g) be an analytic, compact, Riemannian manifold with boundary, of dimension n ge 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition (in: Quinto, Proceedings of conference “Seventy-five Years of Radon Transforms”, Hong Kong, 1994). Using analytic microlocal analysis, we prove a microlocal regularity theorem for generalized Radon transforms on analytic manifolds defined on an analytic family of hypersurfaces. We then show injectivity and stability for an open, dense subset of smooth generalized Radon transforms satisfying the Bolker condition, including the analytic ones.

On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schr\"odinger equations) on a compact, smooth Riemannian manifold, $(M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary doubling property, we prove upper bounds on the $\mathcal{H}^{n-1}$-measure of nodal domains of eigenfunctions of the generalized Dirichlet-to-Neumann map on analytic manifolds.

A Poincaré lemma for Whitney–de Rham complex

Let M be a real analytic manifold, Z a closed subanalytic subset of M . We show that the Whitney–de Rham complex over Z is quasi-isomorphic to the constant sheaf \mathbb C_Z .

Analytical and numerical manifolds in a symplectic 4-D map

We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschle map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.

Sequential gradient dynamics in real analytic Morse systems

ABSTRACTLet be a compact connected M-dimensional real analytic manifold with boundary and ϕ be a primal navigation function; i.e. a real analytic Morse function on Ω with a unique minimum and with minus gradient vector field G of ϕ on ∂Ω pointed inwards along each coordinate. Related to a robotics problem, we define a sequential hybrid process on Ω for G starting from any initial point q0 in the interior of Ω as follows: at each step, we restrict ourselves to an affine subspace where a collection of coordinates are fixed and allow the other coordinates change along an integral curve of the projection of G onto the subspace. We prove that provided each coordinate appears infinitely many times in the coordinate choices during the process, the process converges to a critical point of ϕ. That critical point is the unique minimum for a dense subset in primal navigation functions. We also present an upper bound for the total length of the trajectories close to a critical point.

Semiclassical Cauchy estimates and applications

In this note, we study solutions to semiclassical Schrodinger equations on a real analytic manifold with a real analytic potential and prove the semiclassical version of Cauchy estimates on derivatives. As an application, we use Donnelly and Fefferman's method to prove the upper and lower bounds for (n-1)-dimensional Hausdorff measure of the nodal sets of the solutions to semiclassical Schrodinger equations.

On the irreducible components of globally defined semianalytic sets

t In this work we present the concept of amenable C-semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable Csemianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets.

Heat flow, heat content and the isoparametric property

Let M be a Riemannian manifold and $$\Omega $$ a compact domain of M with smooth boundary. We study the solution of the heat equation on $$\Omega $$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? This problem is the analogous (for the heat equation) of the classical Serrin’s problem for harmonic domains. In this paper we give an affirmative answer to the above question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem.

The m-Derivations of Analytic Vector Fields Lie Algebras

We consider a (real or complex) analytic manifold M. Assuming that F is a ring of all analytic functions, full or truncated with respect to the local coordinates on M; we study the (m ≥ 2)-derivations of all involutive analytic distributions over F and their respective normalizers.

A remark on the intersections of subanalytic leaves

We discuss a new sufficient condition - weaker than the usual transversality condition - for the intersection of two subanalytic leaves to be smooth. It involves the tangent cone of the intersection and, as typically non-transversal, it is of interest in analytic geometry or dynamical systems. We also prove an identity principle for real analytic manifolds and subanalytic functions.

Local resolution of ideals subordinated to a foliation

Let M be a complex- or real-analytic manifold, $$\theta $$ be a singular distribution and $$\mathcal {I}$$ a coherent ideal sheaf defined on M. We prove the existence of a local resolution of singularities of $$\mathcal {I}$$ that preserves the class of singularities of $$\theta $$ , under the hypothesis that the considered class of singularities is invariant by $$\theta $$ -admissible blowings-up. In particular, if $$\theta $$ is monomial, we prove the existence of a local resolution of singularities of $$\mathcal {I}$$ that preserves the monomiality of the singular distribution $$\theta $$ .