[Paper Review] Extension of de Rham decomposition theorem via non-Euclidean development

Yacine Chitour, Mauricio Godoy Molina+1 more|arXiv (Cornell University)|Mar 3, 2012

Geometric Analysis and Curvature Flows14 references3 citations

TL;DR

This paper extends the de Rham decomposition theorem to Riemannian manifolds rolling against hyperbolic space by introducing a non-Euclidean development framework. It establishes a necessary and sufficient condition for the reducibility of the hyperbolic holonomy group, showing that the manifold decomposes as a warped product with specific warping functions—either $ e^{-s} $ or $ \sinh(s), \cosh(s) $—and proves that such decomposition implies complete controllability of the rolling system.

ABSTRACT

In the present paper, we give a necessary and sufficient condition for a Riemannian manifold $(M,g)$ to have a reducible action of a hyperbolic analogue of the holonomy group. This condition amounts to a decomposition of $(M,g)$ as a warped product of a special form, in analogy to the classical de Rham decomposition theorem for Riemannian manifolds. As a consequence of these results and Berger's classification of holonomy groups, we obtain a simple necessary and sufficient condition for the complete controllability of the system of $(M,g)$ rolling against the hyperbolic space.

Motivation & Objective

• To generalize the de Rham decomposition theorem to the setting of rolling against hyperbolic space, replacing the classical Euclidean holonomy framework.
• To characterize when the hyperbolic analogue of the holonomy group acts reducibly on a Riemannian manifold.
• To establish a necessary and sufficient condition for complete controllability of the rolling system between a Riemannian manifold and hyperbolic space.
• To extend Berger’s holonomy classification to the context of non-Euclidean development and rolling dynamics.

Proposed method

• Introduces a non-Euclidean development construction based on Cartan’s original idea, adapted to hyperbolic target spaces.
• Defines a connection $ \nabla^{-1} $ on the bundle $ TM \oplus \mathbb{R} $, whose holonomy group $ \mathcal{H}^{-1} $ encodes the rolling dynamics.
• Analyzes the invariance of light-like subbundles $ V_1 $ under parallel transport with respect to $ \nabla^{-1} $, indicating reducibility of $ \mathcal{H}^{-1} $.
• Uses warped product structures: $ (I \times M_1, ds^2 \oplus_{f} g_1) $ with $ f(s) = e^{-s} $, or $ (I \times M_2 \times M_1, ds^2 \oplus_{\sinh(s)} g_2 \oplus_{\cosh(s)} g_1) $, to model the decomposition.
• Applies the theory of affine connections and holonomy to the rolling system, leveraging the $ G_c(n) $-principal bundle structure for space forms with curvature $ c = -1 $.
• Employs the exponential map on $ \mathbb{H}^k $ to construct local models and extend local reducibility to global reducibility via continuity.

Experimental results

Research questions

• RQ1 Under what conditions does the hyperbolic holonomy group of a Riemannian manifold act reducibly?
• RQ2 How can the de Rham decomposition theorem be generalized beyond the Euclidean setting to include hyperbolic target spaces?
• RQ3 What warped product structures in the metric correspond to reducible holonomy in the non-Euclidean development framework?
• RQ4 How does the reducibility of the holonomy group $ \mathcal{H}^{-1} $ relate to the complete controllability of the rolling system on hyperbolic space?
• RQ5 Can the rolling system between a Riemannian manifold and $ \mathbb{H}^n $ be completely controllable if and only if the manifold’s holonomy is full?

Key findings

• A Riemannian manifold $ (M,g) $ has a reducible action of the hyperbolic holonomy group if and only if it is locally isometric to a warped product $ (I \times M_1, ds^2 \oplus_{e^{-s}} g_1) $ with $ f(s) = e^{-s} $.
• If $ (M,g) $ is isometric to $ (I \times M_2 \times M_1, ds^2 \oplus_{\sinh(s)} g_2 \oplus_{\cosh(s)} g_1) $, then the holonomy group $ \mathcal{H}^{-1} $ is reducible.
• When $ (M,g) $ is isometric to a warped product $ (O \times M_1, \mathbf{g}_{k;-1} \oplus_{\cosh(d(\cdot))} g_1) $ over a normal neighborhood in $ \mathbb{H}^k $, the holonomy $ \mathcal{H}^{-1} $ is reducible by continuity from the local structure.
• The rolling system between $ (M,g) $ and $ \mathbb{H}^n $ is completely controllable if and only if $ (M,g) $ has full affine holonomy, which holds precisely when the holonomy is irreducible.
• The construction of the connection $ \nabla^{-1} $ on $ TM \oplus \mathbb{R} $ allows the holonomy group $ \mathcal{H}^{-1} $ to detect reducibility via invariant light-like subbundles.
• The proof establishes that $ \nabla^{-1}_{X}(L,1) = g(X,L)(L,1) $, showing that $ V_1 = \mathbb{R}(L,1) $ is preserved under parallel transport, hence $ \mathcal{H}^{-1} $ is reducible.