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Connection one-form of an oriented orthonormal frame

Definition

Let $(M, g)$ be an oriented Riemannian surface, let $U \subseteq M$ be open, and let $(E_{1}, E_{2})$ be a specified smooth positively oriented $g$ -orthonormal frame on $U$ . The connection one-form in this frame and sign convention is the smooth one-form $ω \in Ω^{1} (U)$ defined by $ω (X) = g (\nabla_{X} E_{1}, E_{2})$ for each smooth vector field $X$ on $U$ . Its frame equations are $\nabla_{X} E_{1} = ω (X) E_{2}, \nabla_{X} E_{2} = - ω (X) E_{1} .$

Lee's and Datar's frame convention is the negative one: $ω_{s t d} (X) = g (E_{1}, \nabla_{X} E_{2}) = - ω (X)$ . The explicit sign choice here is used by the later structure-equation and rotation-law items.

Facts & Assumptions

Given: An oriented Riemannian surface, an open subset $U$ , its Levi–Civita connection, and a specified smooth positively oriented orthonormal frame $(E_{1}, E_{2})$ on $U$ .

[F1]

An oriented Riemannian surface carries the supplied metric on its oriented two-manifold (Oriented Riemannian surface and positive quarter-turn).

[F2]

The Levi–Civita connection is metric compatible, so $X g (Y, Z) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z)$ for local fields (Levi civita connection).

[F3]

An affine connection is function-linear in its differentiating direction $X$ (Affine connection on a smooth manifold).

[F4]

A smooth differential $k$ -form is a smooth section of $⋀^{k} T^{*} M$ ; for $k = 1$ this is a smooth one-form (A smooth differential $k$ -form).

Proof

technique · direct

1.1F1F3F4given

Define $ω (X) = g (\nabla_{X} E_{1}, E_{2})$ . By [F3], $ω (f X) = f ω (X)$ , so its value at a point depends linearly only on the tangent vector there. In a local coordinate frame, the coefficients $g (\nabla_{\partial_{i}} E_{1}, E_{2})$ are smooth because the metric, connection, and frame are smooth. Thus $ω$ is a smooth section of $T^{*} U$ , hence a smooth one-form by [F4].

2.1F2step 1.1

Since $g (E_{1}, E_{1}) = 1$ , metric compatibility [F2] gives $0 = X g (E_{1}, E_{1}) = 2 g (\nabla_{X} E_{1}, E_{1})$ . The coefficient of $E_{1}$ in $\nabla_{X} E_{1}$ is therefore zero, while its coefficient of $E_{2}$ is the defining value $ω (X)$ . Hence $\nabla_{X} E_{1} = ω (X) E_{2}$ .

3.1F2step 2.1∎

The same calculation gives $g (\nabla_{X} E_{2}, E_{2}) = 0$ . Differentiating $g (E_{1}, E_{2}) = 0$ and using [F2] yields $0 = g (\nabla_{X} E_{1}, E_{2}) + g (E_{1}, \nabla_{X} E_{2}) = ω (X) + g (E_{1}, \nabla_{X} E_{2})$ . Thus $\nabla_{X} E_{2} = - ω (X) E_{1}$ , and the equality also gives $ω_{s t d} = - ω$ in the Lee/Datar convention.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, §“The Gauss–Bonnet Formula,” printed p. 165, equations (9.4), defines $ω_{s t d} (X) = g (E_{1}, \nabla_{X} E_{2}) = - g (\nabla_{X} E_{1}, E_{2})$ and obtains $\nabla_{X} E_{1} = - ω_{s t d} (X) E_{2}$ and $\nabla_{X} E_{2} = ω_{s t d} (X) E_{1}$ . Datar, Lectures on Riemannian Geometry, Lecture 2, §2.1, printed p. 11, uses the same form and frame equations. In both sources the sign is opposite to the explicitly defined $ω$ above.

Depends on

Used by

Flat torus and zero Euler characteristic Example
Gauss-Bonnet for a spherical cap Example
Projective-plane curvature via a hemisphere Example
Total curvature of a round sphere Example
Tangent-angle formula for geodesic curvature Proposition
Gaussian curvature structure equation Theorem
Rotation law for the surface connection form Theorem

Dependency tree · two levels

11 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)