PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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Total turning with connection and corner terms

Statement

Let $(M, g, J)$ be an oriented Riemannian surface, let $U \subseteq M$ be open with a specified smooth positively oriented $g$ -orthonormal frame $(E_{1}, E_{2})$ , and let $ω (X) = g (\nabla_{X} E_{1}, E_{2})$ be its connection form in the convention of this page. Let $γ : [a, b] \to U$ be a closed piecewise $C^{2}$ regular unit-speed curve with matching unit tangents at the identified endpoint and with finitely many ordinary corners at parameters $a < t_{1} < \dots < t_{m} < b$ . Suppose each smooth arc $γ ∣_{[t_{j - 1}, t_{j}]}$ carries a $C^{1}$ angle lift $θ_{j}$ with $\dot{γ} = \cos θ_{j} E_{1} + \sin θ_{j} E_{2}$ . Then $\sum_{j = 1}^{m + 1} (θ_{j} (t_{j}) - θ_{j} (t_{j - 1})) = \int_{γ} k_{g} d s - \int_{γ} ω,$ where $t_{0} = a$ , $t_{m + 1} = b$ , the $α_{j}$ are the signed exterior angles of the corners, defined here for this possibly self-intersecting curve as the unique $α_{j} \in (- π, π)$ satisfying $T_{+} = \cos α_{j} T_{-} + \sin α_{j} J T_{-}$ ; ordinary means $T_{+} \neq - T_{-}$ . This extends the same signed-angle convention from regular-region boundaries, without requiring a region bounded by $γ$ . Also $\int_{γ} ω : = \sum_{j} \int_{t_{j - 1}}^{t_{j}} ω (\dot{γ} (t)) d t$ . Consequently the total geodesic turning $\int_{γ} k_{g} d s + \sum_{j} α_{j}$ equals $\sum_{j} (θ_{j} (t_{j}) - θ_{j} (t_{j - 1})) + \sum_{j} α_{j} + \int_{γ} ω$ . This latter expression is independent of the chosen positive frame. The same identity holds when $γ$ is covered by finitely many positive frames and the connection and angle increments are computed separately on each framed arc: a frame transition changes the angle increment by the negative of the transition-angle increment and the connection integral by its positive increment, so their sum is unchanged.

Facts & Assumptions

Given: An oriented Riemannian surface, an open set $U$ with a specified smooth positive orthonormal frame, and a closed piecewise $C^{2}$ regular unit-speed curve in $U$ with finitely many ordinary corners, matching endpoint tangents, and a $C^{1}$ angle lift on each smooth arc.

[F1]

Tangent-angle formula: on a connected subinterval where $\dot{γ} = \cos θ E_{1} + \sin θ E_{2}$ with $θ$ of class $C^{1}$ , $k_{g} = θ^{'} + ω (\dot{γ})$ , with one-sided derivatives at included endpoints (Tangent-angle formula for geodesic curvature).

[F2]

For a positively oriented regular-region boundary, at an ordinary corner the signed exterior angle is the unique $α \in (- π, π)$ with $T_{+} = \cos α T_{-} + \sin α J T_{-}$ , where $T_{-}, T_{+}$ are the incoming and outgoing unit tangents (Signed exterior angle at an ordinary corner).

[F3]

A second smooth positive orthonormal frame $(E_{1}^{'}, E_{2}^{'})$ on a patch $V$ with supplied smooth angle lift $φ$ satisfying $E_{1}^{'} = \cos φ E_{1} + \sin φ E_{2}$ and $E_{2}^{'} = - \sin φ E_{1} + \cos φ E_{2}$ has connection form $ω^{'} ∣_{V} = ω ∣_{V} + d φ$ (Rotation law for the surface connection form).

[F4]

On a unit-speed curve, $d s = d t$ and $k_{g} = g (A_{γ}, J \dot{γ})$ is the signed geodesic curvature (Signed geodesic curvature).

Proof

technique · integrate the angle-derivative formula on each smooth arc, add the corner jumps, then verify that a change of frame moves both sides by the same amount

1.1F1F4given

On each smooth arc the curve is unit speed, so [F4] gives $d s = d t$ and $k_{g} d s = k_{g} d t$ ; by [F1] applied to the arc's angle lift, $θ_{j}^{'} = k_{g} - ω (\dot{γ})$ there, with one-sided derivatives at the interior endpoints.

2.1step 1.1algebra

Summing the fundamental theorem of calculus over the $m + 1$ arcs gives $\sum_{j} (θ_{j} (t_{j}) - θ_{j} (t_{j - 1})) = \int_{γ} k_{g} d s - \int_{γ} ω$ .

3.1F2step 2.1algebra

At a corner, $(T_{-}, J T_{-})$ is a positive orthonormal basis, so the unit vector $T_{+}$ has coordinates $(\cos α_{j}, \sin α_{j})$ for a unique angle in $(- π, π)$ because $T_{+} \neq - T_{-}$ . This is the local definition in the Statement and agrees with [F2] when the curve is a regular-region boundary. Add the sum of these corner terms and $\int_{γ} ω$ to both sides of step 2.1. The result is the stated expression for total geodesic turning $\int_{γ} k_{g} d s + \sum_{j} α_{j}$ . If some smooth arc has constant tangent direction relative to the chosen frame, its angle increment is $0$ and the identity remains valid.

4.1F2F3step 3.1algebra∎

Frame independence. Suppose on an overlap $V$ a second positive frame is given with angle lift $φ$ as in [F3]. Since $\dot{γ} = \cos θ_{j} E_{1} + \sin θ_{j} E_{2} = \cos (θ_{j} - φ) E_{1}^{'} + \sin (θ_{j} - φ) E_{2}^{'}$ on the overlap, the angle lift for the primed frame is $θ_{j} - φ$ , and [F3] gives $ω^{'} = ω + d φ$ there. On each framed subarc the angle increment changes by $- Δ φ$ and the connection-form integral changes by $+ Δ φ$ , so their sum is unchanged. The corner angle $α_{j}$ is defined from the tangent vectors themselves and is also unchanged. Applying this on a finite cover by framed subarcs proves the asserted frame independence of the total geodesic turning expression.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, §“The Gauss–Bonnet Formula,” equations (9.3)–(9.4) and the proof of Theorem 9.3, printed pp. 164–166, integrates the tangent-angle derivative along a boundary curve and adds the corner contributions; Datar, Lectures on Riemannian Geometry, Lecture 2, §2.1, Lemma 2.1.2 and §2.2, follows the same route. Both sources state the result in the opposite connection-form sign $ω_{s t d} = - ω$ ; the identity above is its transcription into this page's convention, with the transition-angle bookkeeping for covers by frames proved locally.

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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)