Infinite-dimensional Banach Manifolds Research Articles

Wilcox Formula for Vector Fields on Banach Manifolds

MSC: 93C10, 45G15 DOI: 10.21538/0134-4889-2021-27-3-271-285 We obtain an analogue of Wilcox–Snyder formula for flows of diffeomorphisms of $C^m$-smooth vector fields on infinite-dimensional Banach manifolds. For classical linear system this formula can be efficiently used, for example, to obtain Magnus expansion of solutions. The generalized Wilcox formula is obtained by using

An Alternative Surface Measures Construction in Finite-Dimensional Spaces and its Consistency with the Classical Approach

Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds. Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae. Methods. Basic results of mathematical analyses, measure theory and differential geometry are used. Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area. Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.

Infinite-dimensional statistical manifolds based on a balanced chart

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_{\lambda},\lambda\in [2,\infty))$, retain many of the features of finite-dimensional information geometry; in particular, the $\alpha$-divergences are of class $C^{\lceil\lambda\rceil-1}$, enabling the definition of the Fisher metric and $\alpha$-derivatives of particular classes of vector fields. Manifolds of probability measures, $(M_{\lambda},\lambda\in [2,\infty))$, based on centred versions of the charts are shown to be $C^{\lceil\lambda \rceil-1}$-embedded submanifolds of the $\tilde{M}_{\lambda}$. The Fisher metric is a pseudo-Riemannian metric on $\tilde{M}_{\lambda}$. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of $\alpha$-covariant derivatives. $\tilde{M}_{\lambda}$ and $M_{\lambda}$ provide natural settings for the study and comparison of approximations to posterior distributions in problems of Bayesian estimation.

Fréchet geometry via projective limits

Fréchet spaces of sections arise naturally as configurations of a physical field. Some recent work in Fréchet geometry is briefly reviewed and some suggestions for future work are offered. An earlier result on the structure of second tangent bundles in the finite-dimensional case was extended to infinite-dimensional Banach manifolds and Fréchet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Fréchet structure needed for applications.

On the Equivariant Implicit Function Theorem with Low Regularity and Applications to Geometric Variational Problems

AbstractWe prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite-dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-) Riemannian geodesics and constant mean curvature hypersurfaces.

BANACH DIRAC BUNDLES

Banach Dirac bundles are introduced within the framework of infinite-dimensional Banach manifolds and their basic properties are derived. Banach Lie algebroids are geometrical objects recently introduced by the first author. It is the main goal of the present paper to reveal the most important links between Banach Dirac bundles and Banach Lie algebroids, respectively.

On the structure of conformally compact Einstein metrics

Let M be an (n + 1)-dimensional manifold with non-empty boundary, satisfying π 1(M, ∂ M) = 0. The main result of this paper is that the space of conformally compact Einstein metrics on M is a smooth, infinite dimensional Banach manifold, provided it is non-empty. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric and stress–energy tensor at conformal infinity, again in dimension 4. This result also holds for Lorentzian–Einstein metrics with a positive cosmological constant.

Genericity of Nondegeneracy for Light Rays in Stationary Spacetimes

Given a Lorentzian manifold (M, g), an event p and an observer U in M, then p and U are light conjugate if there exists a lightlike geodesic γ : [0, 1] → M joining p and U whose endpoints are conjugate along γ. Using functional analytical techniques, we prove that if one fixes p and U in a differentiable manifold M, then the set of stationary Lorentzian metrics in M for which p and U are not light conjugate is generic in a strong sense. The result is obtained by reduction to a Finsler geodesic problem via a second order Fermat principle for light rays, and using a transversality argument in an infinite dimensional Banach manifold setup.

On boundary value problems for Einstein metrics

On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein moduli space is unobstructed. The Dirichlet and Neumann boundary maps to data on the boundary are smooth, but not Fredholm. Instead, one has natural mixed boundary-value problems which give Fredholm boundary maps. These results also hold for manifolds with compact boundary which have a finite number of locally asymptotically flat ends, as well as for the Einstein equations coupled to many other fields.

The Hunter–Saxton Equation: A Geometric Approach

We provide a rigorous foundation for the geometric interpretation of the Hunter–Saxton equation as the equation describing the geodesic flow of the $\dot{H}^1$ right-invariant metric on the quotient space $Rot(\mathbb{S})\backslash\mathcal{D}^k(\mathbb{S})$ of the infinite-dimensional Banach manifold $\mathcal{D}^k(\mathbb{S})$ of orientation-preserving $H^k$-diffeomorphisms of the unit circle $\mathbb{S}$ modulo the subgroup of rotations $Rot(\mathbb{S})$. Once the underlying Riemannian structure has been established, the method of characteristics is used to derive explicit formulas for the geodesics corresponding to the $\dot{H}^1$ right-invariant metric, yielding, in particular, new explicit expressions for the spatially periodic solutions of the initial-value problem for the Hunter–Saxton equation.

Approximation by smooth functions with no critical points on separable Banach spaces

We characterize the class of separable Banach spaces X such that for every continuous function f : X → R and for every continuous function ε : X → ( 0 , + ∞ ) there exists a C 1 smooth function g : X → R for which | f ( x ) − g ( x ) | ⩽ ε ( x ) and g ′ ( x ) ≠ 0 for all x ∈ X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X ∗ . We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class C p , for p = 1 , 2 , … , + ∞ . In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces ℓ p ( N ) and L p ( R n ) . Some important consequences of the above results are (1) the existence of a non-linear Hahn–Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.

On orientability and degree of Fredholm maps

The degree of a map between two manifolds has played important roles in various mathematical areas. Certain orientability is always required in order to make sense of the concept of degree. In the case of finite-dimensional nonorientable manifolds, this goes back to Hopf, Olum, and Steenrod, after Brouwer’s pioneering work on orientable manifolds (cf. [9] and references therein). Elworthy and Tromba [4] took the first study in the case of infinite-dimensional Banach manifolds, where they introduced the degree on orientable Fredholm manifolds. This orientability restriction on manifolds is, however, often too severe and unnatural. It was Fitzpatrick, Pejsachowicz, and Rabier [6] who pointed out explicitly that the only requirement was the orientability of maps involved rather than that of manifolds. (The finitedimensional version was in Olum’s work.) Their approach is based on the concept of parity of paths, which makes it particularly useful in problems dealing with crossing singular strata. Indeed this is often the only practical way to check the orientability of a map. More recently, Benevieri and Furi [1] took another approach to orienting Fredholm maps that is conceptually more clear and seems more natural, since it comes directly from pointwise orientations of all Fredholm operators. The approach taken in this paper has a more geometric flavor and also provides an instance where geometry and analysis interact nicely. The use of a determinant line bundle that arises from geometry links conveniently the notions of Benevieri– Furi and Fitzpatrick–Pejsachowicz–Rabier. In fact, many properties in [1], [2], and [6] become much easier to understand through our new approach. Conversely, the geometric approach allows us to apply functional analysis tools to some problems in gauge theory involving a real structure, where the relevant manifolds are often nonorientable or with no natural orientation, hence making it necessary to orient relevant maps instead. More details will appear in [10].