Canonical connection and a geometric proof of the Frobenius theorem

TL;DR

The paper introduces a canonical connection $\nabla$ on a Riemannian manifold $(M^n,g)$ relative to a distribution $E^r$, characterized as the unique $g$-compatible affine connection whose torsion $\tau$ satisfies $\tau(X,Y)=\tau(\xi,\eta)=0$, $g(\tau(\xi,X),Y)=\tfrac12 (L_{\xi} g)(X,Y)$, and $g(\tau(\xi,X),\eta)=-\tfrac12 (L_X g)(\xi,\eta)$, reducing to the Levi-Civita connection when $E=TM$. A central feature is that if $E$ is involutive, then $E$ is totally geodesic with respect to $\nabla$, which allows geometric construction of integral submanifolds via the exponential map and, under geodesic completeness, global descriptions. The work contrasts this connection with the Schouten–Van Kampen, Vranceanu, and Bott connections, highlighting its metric compatibility and the property that Bott’s normal connection is generally not metric-compatible. Using curvature and Jacobi-field arguments, the authors obtain a geometric proof of Frobenius’ theorem: an involutive distribution is integrable, with local Frobenius coordinates given by $E|_U=\mathrm{span}\{\partial/\partial x^1,\dots,\partial/\partial x^r\}$, and possible global descriptions under geodesic completeness.

Abstract

In this paper, we introduce a new canonical connection on Riemannian manifold with a distribution. Moreover, as an application of the connection, we give a geometric proof of the Frobenius theorem.

Theorems & Definitions (16)

• Theorem 1.1: Frobenius theorem
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Remark 2.1
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm-connection']}
• proof : Proof of Theorem \ref{['thm-total-geo']}
• Remark 2.2