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Rotation law for the surface connection form

Statement

Let $(M, g)$ be an oriented Riemannian surface and $U \subseteq M$ open with specified smooth positively oriented $g$ -orthonormal frames $(E_{1}, E_{2})$ and $(E_{1}^{'}, E_{2}^{'})$ , with connection forms $ω$ and $ω^{'}$ in the convention $ω (X) = g (\nabla_{X} E_{1}, E_{2})$ . On each open patch $V \subseteq U$ where a smooth angle lift $φ : V \to R$ is supplied with $E_{1}^{'} = \cos φ E_{1} + \sin φ E_{2}, E_{2}^{'} = - \sin φ E_{1} + \cos φ E_{2},$ the forms satisfy $ω^{'} ∣_{V} = ω ∣_{V} + d φ$ , where $d φ (X) = X (φ)$ . Any two such lifts of the same frame rotation differ by a locally constant element of $2 π Z$ , so their differentials agree on overlaps; no global angle lift is asserted.

Facts & Assumptions

Given: An oriented Riemannian surface with two specified positive orthonormal frames on an open set $U$ , and an open patch $V \subseteq U$ carrying a smooth real function $φ$ that expresses the rotation from the first frame to the second.

[F1]

The connection forms of the two frames are defined by $ω (Y) = g (\nabla_{Y} E_{1}, E_{2})$ and $ω^{'} (Y) = g (\nabla_{Y} E_{1}^{'}, E_{2}^{'})$ , and the frame equations $\nabla_{Y} E_{1} = ω (Y) E_{2}$ and $\nabla_{Y} E_{2} = - ω (Y) E_{1}$ hold (Connection one-form of an oriented orthonormal frame).

[F2]

For a smooth function $f$ on a manifold, $d f (X) = X f$ for every smooth vector field $X$ (The exterior derivative of a function is its differential).

Proof

technique · Differentiate the rotated frame vector with the product rule and read off the defining inner product

1.1F1givenalgebra

Fix a smooth vector field $X$ on $V$ . Expanding $E_{1}^{'} = \cos φ E_{1} + \sin φ E_{2}$ and differentiating with the product rule, the frame equations of [F1] give $\nabla_{X} E_{1}^{'} = - \sin φ (X φ) E_{1} + \cos φ ω (X) E_{2} + \cos φ (X φ) E_{2} - \sin φ ω (X) E_{1} = (ω (X) + X φ) E_{2}^{'}$ .

2.1F1step 1.1algebra

Since $(E_{1}^{'}, E_{2}^{'})$ is a $g$ -orthonormal frame, $g (E_{2}^{'}, E_{2}^{'}) = 1$ ; substituting the expansion of step 1.1 into the definition $ω^{'} (X) = g (\nabla_{X} E_{1}^{'}, E_{2}^{'})$ from [F1] yields $ω^{'} (X) = ω (X) + X φ$ for every smooth $X$ on $V$ .

3.1F2step 2.1

By [F2] applied to the smooth function $φ$ on $V$ , the one-form $d φ$ satisfies $d φ (X) = X φ$ ; hence step 2.1 states the identity of one-forms $ω^{'} ∣_{V} = ω ∣_{V} + d φ$ on $V$ .

4.1givenstep 2.1step 3.1algebra

Let $\tilde{φ}$ be a second smooth angle lift of the same rotation on a connected open set $W \subseteq V$ . Expanding both expressions for $E_{1}^{'}$ in the basis $(E_{1}, E_{2})$ gives $\cos \tilde{φ} = \cos φ$ and $\sin \tilde{φ} = \sin φ$ ; hence $\tilde{φ} - φ$ takes values in $2 π Z$ , and by continuity it is constant on $W$ , so $d \tilde{φ} = d φ$ on $W$ . If $W$ is empty the identity is vacuous, and if $φ$ is constant then $d φ = 0$ , so $ω^{'} ∣_{V} = ω ∣_{V}$ there.

5.1step 3.1step 4.1∎

Steps 3.1 and 4.1 prove the stated identity on every supplied patch $V$ and its independence of the choice of lift on overlaps; no global angle lift is constructed or required.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, §“The Gauss–Bonnet Formula,” printed p. 165, equations (9.4), records the frame equations in the opposite sign convention $ω_{s t d} = - ω$ ; Datar, Lectures on Riemannian Geometry, Lecture 2, §2.1, printed p. 11, records the same frame equations. The rotation identity is derived above from those frame equations in the present sign convention; the sources are not claimed to state the identity in this sign.

Depends on

Used by

A surface connection form depends on its frame False statement
The Gauss-Bonnet expression is independent of the metric Lemma
Total turning with connection and corner terms Proposition

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)