[Paper Review] A short note on additive functions on Riemannian co-compact coverings

Minh Kha|arXiv (Cornell University)|Oct 31, 2015

advanced mathematical theories3 references3 citations

TL;DR

This paper presents a topological, invariant approach to constructing additive functions on Riemannian co-compact normal coverings using de Rham cohomology and harmonic forms. It establishes an isomorphism between the space of additive functions modulo periodic functions, the space of closed 1-forms on the base with exact lifts, and the real homomorphisms from the deck group to ℝ, providing a canonical construction independent of fundamental domains.

ABSTRACT

The main purpose of this note is to provide a topological approach to defining additive functions on Riemannian co-compact normal coverings.

Motivation & Objective

• To develop a topologically invariant method for defining additive functions on Riemannian co-compact normal coverings, avoiding dependence on arbitrary choices like fundamental domains.
• To establish a canonical isomorphism between the space of additive functions modulo periodic functions and the space of real homomorphisms from the deck group to ℝ.
• To connect additive functions with harmonic forms on the base manifold, enabling a canonical construction via Hodge theory.
• To generalize and formalize an invariant approach previously mentioned in brief form in earlier works.
• To provide a framework for constructing harmonic additive functions with prescribed transformation laws under the deck group action.

Proposed method

• Define the space $\Omega^1(M;G)$ as the kernel of the pullback map $\pi^*: H^1_{\text{dR}}(M) \to H^1_{\text{dR}}(X)$, consisting of closed 1-forms on $M$ whose lifts to $X$ are exact.
• Construct a smooth function $f_\omega(x) = \int_{x_0}^x \pi^*\omega$ for each $\omega \in \Omega^1(M;G)$, which satisfies the additive property $f_\omega(g\cdot x) - f_\omega(x) = \text{const}$.
• Introduce an equivalence relation $\sim$ on $\mathcal{A}(X)$ identifying functions that differ by a $G$-periodic function, leading to the quotient space $\mathcal{A}(X)/\sim$.
• Define a linear map $\Lambda: \Omega^1(M;G) \to \mathcal{A}(X)/\sim$ by $[\omega] \mapsto [f_\omega]$, and prove it is an isomorphism.
• Define a map $\Upsilon: \mathcal{A}(X)/\sim \to \operatorname{Hom}(G,\mathbb{R})$ by associating each additive function to its associated homomorphism $\ell_f$.
• Use Hodge theory to construct a harmonic representative in each cohomology class, ensuring existence and uniqueness of harmonic additive functions with prescribed transformation behavior.

Experimental results

Research questions

• RQ1 How can additive functions on a Riemannian co-compact normal covering be defined in a way that is independent of the choice of fundamental domain?
• RQ2 What is the precise relationship between the space of additive functions modulo periodic functions and the real homomorphisms from the deck group to ℝ?
• RQ3 Can harmonic additive functions be constructed canonically for any given homomorphism $\ell \in \operatorname{Hom}(G,\mathbb{R})$?
• RQ4 How do de Rham cohomology classes on the base manifold relate to the structure of additive functions on the covering space?
• RQ5 What is the role of the pullback of harmonic 1-forms on the base in constructing harmonic additive functions on the covering?

Key findings

• The space $\Omega^1(M;G)$ of closed 1-forms on $M$ with exact lifts to $X$ is isomorphic to $\operatorname{Hom}(G,\mathbb{R})$, providing a topological characterization of additive functions.
• The map $\Lambda: \Omega^1(M;G) \to \mathcal{A}(X)/\sim$ defined by $[\omega] \mapsto [f_\omega]$ is an isomorphism, establishing a canonical correspondence.
• The map $\Upsilon: \mathcal{A}(X)/\sim \to \operatorname{Hom}(G,\mathbb{R})$ is an isomorphism, confirming that additive functions modulo periodic functions are in one-to-one correspondence with group homomorphisms from $G$ to $\mathbb{R}$.
• For any $\ell \in \operatorname{Hom}(G,\mathbb{R})$, there exists a unique harmonic function $f$ on $X$ such that $f(g\cdot x) = f(x) + \ell(g)$, up to an additive constant.
• When $G = \mathbb{Z}^d$, there exists a smooth $\mathbb{R}^d$-valued additive function $h$ satisfying $h(g\cdot x) = h(x) + g$, which realizes the Albanese pseudo-metric as $d_G(x,y) = |h(x) - h(y)|$.
• The decomposition $\mathcal{A}(X) = \bigsqcup_{\alpha \in \operatorname{Hom}(G,\mathbb{R})} \{ f_\alpha + \varphi \mid \varphi \text{ is periodic} \}$ holds, where $f_\alpha$ is harmonic and vanishes at a fixed base point.