Hilbert Manifold Research Articles

On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.

Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity

Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of quantum theory are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then analyze the origin of physical dimensions, provide a geometric interpretation of Planck's constant, and find a simple interpretation of the two-slit experiment and the process of measurement.

Cocycle superrigidity and harmonic maps with infinite-dimensional targets

We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property $(T)$. In particular, our results apply to SP(1,n) and $F_4^{-20}$ and lattices in those groups. The main technical step is to prove a very general result concerning existence of harmonic maps into infinite dimensional spaces, namely a class of simply connected, homogeneous, aspherical Hilbert manifolds. This builds on previous work of Corlette-Zimmer and Korevaar-Schoen. Our result is new because we consider more general targets than previous authors and make no assumption concerning the action. In particular, the target space is not assumed to have non-positive curvature and in important cases has significant positive curvature. We also do not make a reductivity assumption concerning absence of fixed points at infinity. The proof of cocycle superrigidity given here, unlike previous ergodic theoretic ones, is effective. The straightening of the cocycle is explicitly a limit of a heat flow. This explicit construction should yield further applications.

Unitary Orbits in a Full Matrix Algebra

The Hilbert manifold ∑ ∞ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits Ω⊂∑ ∞ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set Ω, and we study the convexity of this set for the geodesic structures that arise when we give ∑ ∞ different Riemannian metrics.

The geometry of Kato Grassmannians

Abstract We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.

On smooth Chas–Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In [A.J. Baker, C. Ozel, Complex cobordism of Hilbert manifolds with some applications to flag varieties, Contemp. Math. 258 (2000) 1–19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [C. Ozel, On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in preparation], by using Quinn's Transversality Theorem [F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. 15 (1970) 213–222], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [M. Chas, D. Sullivan, String topology, math.GT/9911159, 1999], Chas and Sullivan described an intersection product on the homology of loop space LM. In [R.L. Cohen, J.D.S. Jones, A homotopy theoretic realization of string topology, math.GT/0107187, 2001], R. Cohen and J. Jones described a realization of the Chas–Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. In this paper, we will extend this product on cobordism and bordism theories.

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1–19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213–222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.

Phase Space for the Einstein Equations

A Hilbert manifold structure is described for the ADM phase space of asymptotically flat initial data $(g,\pi)$ with local $H^2\times H^1$ Sobolev regularity. Solutions of the constraint equations form a Hilbert submanifold. A regularized RT Hamiltonian is defined and smooth on the full phase space and generates the Einstein evolution for any lapse-shift asymptotic to a (time) translation at infinity. Critical points for the total (ADM) mass, considered as a function on the Hilbert manifold of constraint solutions, arise precisely at initial data generating stationary vacuum spacetimes.

Linear algebra and differential geometry on abstract Hilbert space

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n‐dimensional vector spaces. However, while n‐dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple observation is used to construct a fruitful formalism of local coordinates on Hilbert manifolds. Images of charts on manifolds in the formalism are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations then describe families of functional equations on various spaces of functions. The formalism itself and its applications in linear algebra, differential equations, and differential geometry are carefully analyzed.

A bordism approach to string topology

Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extending previous results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the prop of these reduced diagrams. Some of these operations are extended to spaces of maps from a sphere to a compact manifold.

A Morse complex for infinite dimensional manifolds—part I

In this paper and in the forthcoming Part II, we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle—or more generally an essential subbundle—of the tangent bundle of M, suitably related with the gradient flow of f. This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W, possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W, classification, up to topological conjugacy, of the gradient flow on the closure of W, in the case dim W = 2 .

Properly Discontinuous Isometric Actions on the Unit Sphere of Infinite Dimensional Hilbert Spaces

We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with constant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously, and in the case of finitely generated Abelian groups the necessary and sufficient conditions are given.

The exponential map of a weak Riemannian Hilbert manifold

We prove the Focal Index Lemma and the Rauch--Berger Comparison Theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop space $\Omega (M^n)$ with the $L^2$ (weak) Riemannian structure.

Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space X into $\mathbb{R}^{m}$ can be uniformly approximated by C∞-smooth mappings with no critical points. This kind of result can be regarded as a sort of strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold that is a level set of a smooth function with no critical points. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are C∞-smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modeled on X.

Singular Points and Singular Values of Mappings of Hilbert Manifolds

Singular Points and Singular Values of Mappings of Hilbert Manifolds

When the Morse index is infinite

Let f be a smooth Morse function on an infinite-dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and coindex. For any critical point x, choose an integer a(x arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x,y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x)−a(y.

Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions

We consider a Riemannian manifold $(\mathcal M,g)$ and a codimension one distribution $\Delta\subset T\mathcal M$ on $\mathcal M$ which is the orthogonal of a unit vector field $Y$ on $\mathcal M$. We do not make any nonintegrability assumption on $\Delta$. The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional $E$ on the space of horizontal curves, i.e. everywhere tangent to the distribution $\Delta$. We consider the case of horizontal curves joining a smooth submanifold $\mathcal P$ of $\mathcal M$ and a fixed point $q\in\mathcal M$. Under the assumption that $\mathcal P$ is transversal to $\Delta$, it is known (see [P. Piccione and D. V. Tausk, < i> Variational aspects of the geodesic problem is sub-Riemannian geometry< /i> , J. Geom. Phys. < b> 39< /b> (2001), 183–206]) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of $E$ are the so called {\it normal extremals} (see [W. Liu and H. J. Sussmann, < i> Shortest paths for sub-Riemannian metrics on rank–$2$ distribution< /i> , Mem. Amer. Math. Soc. < b> 564< /b> (1995)]). We compute the second variation of $E$ at its critical points, we define the notions of $\mathcal P$-Jacobi field, of $\mathcal P$-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of $E$ under the assumption of completeness for $(\mathcal M,g)$.

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold ( M, g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p, q∈ M that are non-conjugate in a suitable sense, under the assumption that ( M, g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.

Monotonicity, resolvents and Yosida approximations of operators on Hilbert manifolds

Monotonicity, resolvents and Yosida approximations of operators on manifolds of class C 3 are considered. Strong convergence of difference approximations to differential equations is shown by using resolvents. Our motivation is to give an elementary step toward constructing nonlinear semigroup theory on infinite dimensional manifolds.

Ordinary differential operators in Hilbert spaces and Fredholm pairs

Let $A(t)$ be a path of bounded operators on a real Hilbert space, hyperbolic at $\pm \infty$ . We study the Fredholm theory of the operator $F_A=d/dt-A(t)$ . We relate the Fredholm property of $F_A$ to the stable and unstable linear spaces of the associated system $X^{\prime}=A(t)X$ . Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimensional framework still hold. Our motivation is to develop the linear theory which is necessary for the set-up of Morse homology on Hilbert manifolds.