[Paper Review] Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them

Ekaterina Pervova|arXiv (Cornell University)|Jan 2, 2016

Advanced Operator Algebra Research7 references5 citations

TL;DR

This paper develops a diffeological analog of Riemannian metrics for vector pseudo-bundles by introducing pseudo-metrics that are minimally degenerate at each fiber, and establishes conditions under which such metrics exist and are preserved under diffeological gluing. It shows that a pseudo-metric on a glued pseudo-bundle arises from compatible pseudo-metrics on the components, and induces a natural dual pseudo-metric on the dual pseudo-bundle under local triviality.

ABSTRACT

Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in arXiv:1509.03023. We give more details regarding the behavior of this operation with respect to gluing, also providing some details omitted from arXiv:1509.03023, and pay more attention to the relations with the spaces of smooth maps. We also show that a usual smooth vector bundle over a manifold that admits a finite atlas can be seen as a result of a diffeological gluing, and thus deduce that its usual dual bundle is the same as its diffeological dual. We then consider the notion of a pseudo-metric, the fact that it does not always exist (which seems to be related to non-local-triviality condition), construction of an induced pseudo-metric on a pseudo-bundle obtained by gluing, and finally, the relation between the spaces of all pseudo-metrics on the factors of a gluing, and on its result. We conclude by commenting on the induced pseudo-metric on the pseudo-bundle dual to the given one.

Motivation & Objective

• To extend the concept of Riemannian metrics to diffeological vector pseudo-bundles, where local triviality is not required.
• To address the failure of standard scalar products on non-standard finite-dimensional diffeological vector spaces by introducing minimally degenerate symmetric bilinear forms.
• To develop a theory of pseudo-metrics on diffeologically glued vector pseudo-bundles and relate them to the pseudo-metrics on the original components.
• To define and study the induced pseudo-metric on the dual pseudo-bundle of a given pseudo-bundle with a pseudo-metric, under local triviality.

Proposed method

• Introduces the notion of a diffeological vector pseudo-bundle as a smooth vector bundle-like object without requiring local triviality.
• Defines a pseudo-metric as a smooth section of the symmetric tensor product of the dual pseudo-bundle, with minimal degeneracy at each fiber.
• Applies the diffeological gluing construction from [9] to build new pseudo-bundles from compatible data on overlapping domains.
• Establishes a map $\mathcal{P}$ from pairs of compatible pseudo-metrics on the glued components to the pseudo-metric on the result, showing injectivity.
• Uses the natural pairing map $\Phi: V \to V^*$ induced by a pseudo-metric $g$, defined by $\Phi(v) = g(\pi(v))(v, \cdot)$, to construct the dual pseudo-metric.
• Proves smoothness of $\Phi$ and uses local triviality to show that the induced dual pseudo-metric $g^*$ is smooth via local diffeomorphisms and evaluation maps.

Experimental results

Research questions

• RQ1 Under what conditions does a diffeological vector pseudo-bundle admit a pseudo-metric, and when is it induced by gluing?
• RQ2 How do compatible pseudo-metrics on the components of a diffeological gluing relate to the pseudo-metric on the resulting pseudo-bundle?
• RQ3 Can a pseudo-metric on a pseudo-bundle induce a well-defined, smooth pseudo-metric on its dual pseudo-bundle?
• RQ4 Is the map $\mathcal{P}$ from compatible pairs of pseudo-metrics to the glued pseudo-metric always surjective, or only injective?
• RQ5 Does the existence of a dual pseudo-metric require the assumption of local triviality, and is this assumption necessary?

Key findings

• A finite-dimensional diffeological vector pseudo-bundle that admits a pseudo-metric allows for a well-defined, smooth induced pseudo-metric on its dual pseudo-bundle, provided the original bundle is locally trivial.
• The map $\mathcal{P}$ from compatible pseudo-metrics on the glued components to the pseudo-metric on the result is always injective, as the restriction to each fiber recovers the original metric.
• The induced dual pseudo-metric $g^*$ is defined pointwise by $g^*(x)(\Phi(v), \Phi(w)) := g(x)(v,w)$, and is well-defined due to the kernel of $\Phi$ being the isotropic subspace of $g(x)$.
• The smoothness of $\Phi$ follows from the smoothness of $g$, and the induced $g^*$ is smooth over a neighborhood where local triviality holds.
• The construction of $g^*$ relies on the existence of a subspace $V_0 \subset \pi^{-1}(x)$ such that $\Phi|_{V_0}$ is a diffeomorphism onto the dual fiber, enabling local trivialization of the dual bundle.
• The necessity of local triviality for the smoothness of $g^*$ remains an open question, as the paper cannot confirm whether it is strictly required.