[Paper Review] An intrinsic formulation for rolling pseudo-Riemannian manifolds

Irina Markina, F. Silva Leite|arXiv (Cornell University)|Oct 11, 2012

Geometric Analysis and Curvature Flows29 references3 citations

TL;DR

This paper presents an intrinsic formulation for rolling pseudo-Riemannian manifolds without slipping or twisting, using only metric data rather than extrinsic embeddings. It establishes a smooth distribution on the configuration space encoding kinematic constraints and proves that the causal character of rolling curves is preserved under certain conditions, extending Riemannian rolling theory to the pseudo-Riemannian setting with new geometric features.

ABSTRACT

In the present work we define the rolling of one pseudo-Riemannian manifold over another without slipping and twisting. We compare the definition of the rolling without slipping and twisting of two manifolds isometrically embedded into a pseudo-Euclidean space with the rolling defined only by the intrinsic data, namely by the metric tensors on manifolds. The smooth distribution on the configuration space, encoding the no-slipping and no-twisting kinematic conditions is constructed. Some results concerning the causal character of the rolling curves are also included. Several examples are presented along the paper to illustrate concepts and help to understand the theoretical results.

Motivation & Objective

• To develop a rigorous intrinsic formulation of rolling for pseudo-Riemannian manifolds, independent of extrinsic embeddings.
• To compare the intrinsic rolling definition—based solely on metric tensors—with the classical extrinsic approach using isometric embeddings in pseudo-Euclidean space.
• To construct a smooth distribution on the configuration space that encodes the no-slipping and no-twisting kinematic constraints.
• To analyze the causal character of rolling curves and identify conditions under which it is preserved during the motion.
• To generalize prior Riemannian rolling frameworks to the pseudo-Riemannian case, revealing new geometric phenomena due to indefinite metrics.

Proposed method

• Define rolling via an intrinsic approach using only the metric tensors of the manifolds, avoiding dependence on ambient pseudo-Euclidean space.
• Construct the configuration space as a smooth fiber bundle with structure group isomorphic to the pseudo-orthogonal group O(μ,ν−μ).
• Derive the kinematic constraints (no-slip and no-twist) as a smooth distribution on the configuration space using Christoffel symbols and metric compatibility.
• Use matrix representations of right and left invariant vector fields on the structure group to describe the dynamics of the rolling map.
• Analyze the causal character of rolling curves by comparing the induced metric on the curve with the causal structure of the base manifolds.
• Establish commutation relations for the right-invariant vector fields on the structure group using the matrix basis $E_{ij}$ and the signature matrix $J$.

Experimental results

Research questions

• RQ1 How can rolling without slipping and twisting be defined intrinsically on pseudo-Riemannian manifolds using only their metric tensors?
• RQ2 What is the relationship between the intrinsic rolling formulation and the classical extrinsic formulation based on isometric embeddings in pseudo-Euclidean space?
• RQ3 Under what conditions is the causal character of a rolling curve preserved during the motion?
• RQ4 How is the configuration space of the rolling system structured as a fiber bundle, and what is the role of the pseudo-orthogonal group in this structure?
• RQ5 What are the algebraic and geometric differences in rolling dynamics when the metric is indefinite (pseudo-Riemannian) compared to positive-definite (Riemannian)?

Key findings

• The intrinsic rolling formulation is defined solely by the metric tensors of the manifolds, eliminating dependence on ambient space embeddings.
• A smooth distribution on the configuration space is constructed that encodes the no-slipping and no-twisting constraints via the metric and its Christoffel symbols.
• The causal character of a rolling curve is preserved if the induced metric on the curve maintains the same causal type (spacelike, timelike, or lightlike) as the base manifold.
• The configuration space of rolling is naturally structured as a smooth fiber bundle with typical fiber isomorphic to the pseudo-orthogonal group O(μ,ν−μ).
• The right-invariant vector fields on the structure group are expressed using matrix basis $E_{ij}$ and the signature matrix $J$, enabling explicit computation of the rolling dynamics.
• Commutation relations for the vector fields $W_{ij}$ are derived, showing that the Lie algebra structure depends on the signature of the metric through the signs $\varepsilon_i$.