A Hadamard theorem in transversely affine geometry with applications to affine orbifolds

TL;DR

This paper extends Hadamard-type theorems to the setting of transversely affine foliations by developing a robust transverse affine framework. It introduces transverse affine connections and their partner connections, establishing an equivalence between transverse affine structures, holonomy-invariant normal-bundle connections, and basic transverse frame-bundle connections. The authors define transverse-geodesics and transverse Jacobi fields to relate leaf-space geometry to transverse geometry, and prove a Hadamard-like splitting result for the universal cover under pseudoconvexity, disprisonment, and absence of transverse conjugate points, yielding a product decomposition $\tilde{M} \cong \tilde{L} \times B$ with $B$ contractible. As a key application, the results extend Beem–Parker-type theorems to affine orbifolds, showing developability and contractible universal covers under analogous hypotheses. Collectively, the work unifies and extends transverse-geometric methods to an affine context, with potential impact on the study of affine orbifolds and foliations with compact leaves.

Abstract

We introduce and investigate a novel notion of transversely affine foliation, comparing and contrasting it to the previous ones in the literature. We then use it to give an extension of the classic Hadamard's theorem from Riemannian geometry to this setting. Our main result is a transversely affine version of a well-known "Hadamard-like" theorem by J. Hebda for Riemannian foliations. Alternatively, our result can be viewed as a foliation-theoretic analogue of the Hadamard's theorem for affine manifolds proven by Beem and Parker. Namely, we show that under the transverse analogs of pseudoconvexity and disprisonment for the family of geodesics in the transverse affine geometry, together with an absence of transverse conjugate points, the universal cover of a manifold endowed with a transversely affine foliation whose leaves are compact and with finite holonomy is diffeomorphic to the product of a contractible manifold with the universal cover of a leaf. This also leads to a Beem-Parker-type Hadamard-like theorem for affine orbifolds.

Figures (1)

• Figure 1: Geodesics $\gamma$ and $\beta$ have coinciding projection up until they reach leaves $L_-$ and $L_+$, respectively.

Theorems & Definitions (71)

• Theorem 1.1: Flaherty
• Definition 1.2
• Theorem 1.3: Beem--Parker
• Theorem 1.4: Hebda
• Theorem 1
• Corollary 2: Affine Hadamard theorem for orbifolds
• Lemma 2.1
• proof
• Remark 2.2
• Remark 3.1