Analytic Riemannian Manifold Research Articles

Direct Images, Fields of Hilbert Spaces, and Geometric Quantization

Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family H s of Hilbert spaces, and the question arises if the spaces H s are canonically isomorphic. Axelrod et al. (J. Diff. Geo. 33:787–902, 1991) and Hitchin (Commun. Math. Phys. 131:347–380, 1990) suggest viewing H s as fibers of a Hilbert bundle H, introduce a connection on H, and use parallel transport to identify different fibers. Here we explore to what extent this can be done. First we introduce the notion of smooth and analytic fields of Hilbert spaces, and prove that if an analytic field over a simply connected base is flat, then it corresponds to a Hermitian Hilbert bundle with a flat connection and path independent parallel transport. Second we address a general direct image problem in complex geometry: pushing forward a Hermitian holomorphic vector bundle $${E \to Y}$$ along a non–proper map $${Y \to S}$$ . We give criteria for the direct image to be a smooth field of Hilbert spaces. Third we consider quantizing an analytic Riemannian manifold M by endowing TM with the family of adapted Kähler structures from Lempert and Szőke (Bull. Lond. Math. Soc. 44:367–374, 2012). This leads to a direct image problem. When M is homogeneous, we prove the direct image is an analytic field of Hilbert spaces. For certain such M—but not all—the direct image is even flat; which means that in those cases quantization is unique.

A new look at adapted complex structures

Given a closed real analytic Riemannian manifold, we construct and study a one parameter family of adapted complex structures on the manifold of its geodesics.

Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Consider a compact manifold M with boundary ∂ M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂ M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.

THE VOLUME OF A LOCAL NODAL DOMAIN

Let M either be a closed real analytic Riemannian manifold or a closed C∞-Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball, provided that this component enters the ball deeply enough.

Embedding into manifolds with torsion

We introduce a class of special geometries associated to the choice of a differential graded algebra contained in \({\Lambda^*\mathbb{R}^n}\). We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-Kähler and α-Einstein–Sasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case.

Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds

We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumptions, this can happen only on topological spheres. To cite this article: R. Grimaldi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A proof of Lens Rigidity in the category of Analytic Metrics

Consider a compact Riemannian manifold with boundary. If all maximally extended geodesics intersect the boundary at both ends, then to each geodesic γ(t) we can form the triple (γ(0), γ(T ), T ), consisting of the initial and final vectors of the segment as well as the length between them. The collection of all such triples comprises the lens data. In this talk, I will give a sketch of my proof that in the category of analytic Riemannian manifolds, the lens data uniquely determine the metric up to isometry. There are no convexity assumptions on the boundary, and conjugate points are allowed with very little restriction. ∗University of Washington

Semianalyticity of isoperimetric profiles

It is shown that, in dimensions <8, isoperimetric profiles of compact real analytic Riemannian manifolds are semianalytic.

Instabilities for supercritical Schrödinger equations in analytic manifolds

In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold ( M d , g ) , where the metric g is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.

Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold

For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$ we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ as a functional on the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting. As an application, we obtain a characterization of critical domains of the trace of the heat kernel under Dirichlet boundary conditions.

The Centroaffine Volume of Generalized Geodesic Balls Under Inversion at the Sphere

On an analytic Riemannian manifold (M,g), several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping. In the foregoing paper [1] we studied a more general structure (M,D,g), where D is a torsion-free and Ricci-symmetric connection. We calculated the Taylor expansion up to order (n+4) for the volume of what we called a generalized geodesic ball under the exponential mapping in case that all metric notions are Riemannian, while the exponential mapping is induced from the connection D. For the structure % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $(M,D,{\cal G})$ the coefficients of the Taylor expansion are much more complicated than in the Riemannian case. It is one of the main objectives of the present paper to study centroaffine hypersurfaces in Euclidean space, their geometric invariants which appear in the very complicated coefficient of order (n+4), and their behaviour under polarization (inversion at the unit sphere). Our results complement applications in the foregoing paper [1], where mainly the coefficients up to order (n+2) and geometric consequences have been studied.

Geometric Structures as Determined by the Volume of Generalized Geodesic Balls

Several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping of an analytic Riemannian manifold \( (M, {\cal G}) \). A more general structure \( (M, D{\cal G}) \), where D is a torsion-free and Ricci-symmetric connection, appears in several geometric situations. We study the Taylor expansion in this case, where all metric notions are Riemannian, while now the exponential mapping is induced from the connection D. We give many applications, in particular in different hypersurface theories.

Complexifications of nonnegatively curved manifolds

Define a good complexification of a closed smooth manifold M to be a smooth affine algebraic variety U over the real numbers such that M is diffeomorphic to U(R) and the inclusion U(R) → U(C) is a homotopy equivalence. Kulkarni showed that every manifold which has a good complexification has nonnegative Euler characteristic [16]. We strengthen his theorem to say that if the Euler characteristic is positive, then all the odd Betti numbers are zero. Also, if the Euler characteristic is zero, then all the Pontrjagin numbers are zero (see Theorem 1.1 and, for a stronger statement, Theorem 2.1). We also construct a new class of manifolds with good complexifications. As a result, all known closed manifolds which have Riemannian metrics of nonnegative sectional curvature, including those found by Cheeger [5] and Grove and Ziller [11], have good complexifications. We can in fact ask whether a closed manifold has a good complexification if and only if it has a Riemannian metric of nonnegative sectional curvature. The question is suggested by the work of Lempert and Szőke [17]. Lempert and Szőke, and independently Guillemin and Stenzel [12], constructed a canonical complex analytic structure on an open subset of the tangent bundle of M, given a real analytic Riemannian metric on M. We say that a real analytic Riemannian manifold has entire Grauert tube if this complex structure is defined on the whole tangent bundle TM. Lempert and Szőke found that every Riemannian manifold with entire Grauert tube has nonnegative sectional curvature. Also, a conjecture by Burns [4] predicts that for every closed Riemannian manifold M with entire Grauert tube, the complex manifold TM is an affine algebraic variety in a natural way; if this is correct, the complex manifold TM would be a good complexification of M in the above sense. The problem remains open at this writing despite the paper of Aguilar and Burns [2], because the proof of Proposition 2.1 there is not yet generally accepted. We can also ask how many Riemannian manifolds have entire Grauert tube. It is known to be a strong restriction: Szőke showed that of all surfaces of revolution diffeomorphic to the 2-sphere, only a one-parameter family of metrics of given volume have entire Grauert tube [22]. Aguilar showed that the quotient of a Riemannian manifold with entire Grauert tube by a group of isometries acting freely also has entire Grauert tube [1]. All known manifolds with entire Grauert

Newton's method on Riemannian manifolds: covariant alpha theory

In this paper, Smale's α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high‐order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given.

Symplectic geometry and the uniqueness of Grauert tubes

A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and Schwarz reflection.

Locally symmetric immersions

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.

Topological stability and Gromov hyperbolicity

We show that if the geodesic flow of a compact analytic Riemannian manifold $M$ of non-positive curvature is either $C^{k}$-topologically stable or satisfies the $\epsilon$-$C^{k}$-shadowing property for some $k &gt; 0$ then the universal covering of $M$ is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.

On the Holmgren-John uniqueness theorem for the wave equation with piecewise analytic coefficients

In this paper a uniqueness theorem is proved for the wave equation in the domain Q2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given on a part Γ0 × (0, 2T) of the space-time boundary of the cylinder Q2T, Γ0. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the wave equation. Bibliography:24 titles.

Gevrey Classes on Compact Real Analytic Riemannian Manifolds

Gevrey Classes on Compact Real Analytic Riemannian Manifolds

𝐺-deformations and some generalizations of H. Weyl’s tube theorem

We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under G G -deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only G G -deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.