[Paper Review] Geometric interpretations of the symmetric product in affine differential geometry

María Barbero-Liñán, Andrew D. Lewis|arXiv (Cornell University)|Apr 6, 2011

Control and Dynamics of Mobile Robots12 references3 citations

TL;DR

This paper provides a coordinate-free, intrinsic geometric interpretation of the symmetric product in affine differential geometry using parallel transport, analogous to the flow-based interpretation of the Lie bracket. It establishes that distributions closed under the symmetric product are precisely those invariant under the geodesic flow, offering a novel intrinsic proof of this fundamental equivalence using the Baker–Campbell–Hausdorff formula and horizontal lifts.

ABSTRACT

The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product is used to provide an intrinsic proof of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow.

Motivation & Objective

• To provide a geometric, coordinate-free interpretation of the symmetric product in affine differential geometry.
• To establish an intrinsic proof of the equivalence between distributions closed under the symmetric product and those invariant under the geodesic flow.
• To extend the flow-based intuition of the Lie bracket to the symmetric product using parallel transport and the Baker–Campbell–Hausdorff formula.

Proposed method

• Introduce an intrinsic definition of the symmetric product using the horizontal lift of vector fields and the geodesic spray.
• Use the Baker–Campbell–Hausdorff formula to analyze compositions of flows and derive infinitesimal characterizations of the symmetric product.
• Define a path $\Upsilon^{Z}_{3}(t)$ in the tangent bundle whose second derivative at $t=0$ yields the symmetric product via horizontal lifts.
• Employ horizontal and vertical lifts of vector fields to analyze tangency conditions in the tangent bundle $TM$.
• Use the geodesic invariance condition to show that the horizontal lift of $\langle X:Y\rangle$ lies in the distribution $\mathcal{D}$.
• Apply the polarization identity and Corollary 4.2 to reduce the problem to verifying vanishing of the projection onto the base manifold.

Experimental results

Research questions

• RQ1 How can the symmetric product be geometrically interpreted in a coordinate-free manner using parallel transport, analogous to the flow interpretation of the Lie bracket?
• RQ2 What intrinsic condition ensures that a distribution is invariant under the geodesic flow of an affine connection?
• RQ3 How does the symmetric product relate to the horizontal and vertical lifts of vector fields in the tangent bundle?
• RQ4 Can the equivalence between symmetric product closure and geodesic invariance be proven without coordinates or frame bundles?
• RQ5 What role does the Baker–Campbell–Hausdorff formula play in deriving infinitesimal formulas for the symmetric product?

Key findings

• The symmetric product $\langle X:Y\rangle = \nabla_X Y + \nabla_Y X$ is intrinsically characterized via the second derivative of a path constructed from parallel transport and geodesic flows.
• A distribution $\mathcal{D}$ is geodesically invariant if and only if it is closed under the symmetric product of its sections.
• The intrinsic proof of geodesic invariance avoids coordinate charts and frame bundles, relying instead on horizontal lifts and the Baker–Campbell–Hausdorff formula.
• The horizontal lift of the symmetric product $\langle X:Y\rangle$ lies in the tangent space to $\mathcal{D}$ whenever $X,Y$ are sections of $\mathcal{D}$ and $\mathcal{D}$ is geodesically invariant.
• The condition $\nabla_X X \in \Gamma^\infty(\mathcal{D})$ for all $X \in \Gamma^\infty(\mathcal{D})$ is equivalent to geodesic invariance, providing a minimal criterion.
• The use of the path $\Upsilon^{Z}_{3}(t)$ and its second derivative yields a novel infinitesimal formula for the symmetric product in terms of parallel transport and geodesic flows.