curvature in nLab

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Idea

The formal notion of curvature is a formalization and generalization of the intuitive notion of the (“extrinsic”) curvature of a surface embedded in a Cartesian space ℝ n\mathbb{R}^n.

This extrinsic curvature of a surface is called Gaussian curvature?. It may also be understood intrinsically as a property of just the surface without reference to the ambient Cartesian space that it is embedded in: the canonical metric on ℝ n\mathbb{R}^n induces a Riemannian metric on the surface and the surface’s curvature is encoded in the Levi-Civita connection on the tangent bundle of the surface.

This notion of curvature of a Levi-Civita connection in turn generalizes straightforwardly to a notion of curvature of any connection on a bundle and thus gives the name to the general concept.

For instance in the first-order formulation of gravity the curvature of spacetime is literally the curvature of the Levi-Civita connection on spacetime in this sense. But also the Yang-Mills field is a connection on a principal bundle and its curvature encodes the field strength of the Yang-Mills field, which is a concept rather remote from the intuition of a curved surface (thogh not unrelated).

Even more generally, the notion of a connection on a bundle and of a Lie algebra-valued 1-form generalizes to connections on principal 2-bundles and principal ∞-bundles and curvature of ∞-Lie algebroid valued differential forms.

In Eilenberg-Steenrod-type differential cohomology describing abelian such higher connections these curvatures appear in the form of generalized Chern character curvature characteristic forms.

Extrinsic curvature of embedded manifolds

Curvature κ(γ)\kappa(\gamma) of a smooth curve γ\gamma at a point pp of a smooth curve is the (signed) inverse radius of the (oriented) circle having tangency of order 1 with the curve at the point pp. Every smooth curve in a 3-dimensional space is determined up to isometry by its (parametrised) curvature and torsion. Intuitively, the curvature measures how much curves are bent, when measured in some metrics. Frenet-Serret formulas express the derivative of Frenet moving frame with respect to the parameter of a naturally parametrized curve in nn-dimensional Euclidean space as an antisymmetric matrix times the Frenet moving frame. The nonzero coefficients of the matrix are, up to the sign, the curvature of the curve, torsion and higher analogues. For higher dimensional surfaces one can look at normal curvature which is the curvature of the curve which is intersection of a plane determined by the normal vector to the surface and a given tangent vector. For any 2-dimensional tangent plane, the normal curvature has two extreme values. Their product is called the Gaussian curvature.

Sectional curvature of a higher dimensional smooth surface at its point pp in an Euclidean space along a tangent 2-dimensional plane is the Gaussian curvature of the curve which is the intersection of the surface with the plane. As a plane is determined by two vectors, the sectional curvature is determined by a surface and a pair of vectors, and all possible sectional curvatures can be read form that 2-form; therefore we talk about the operator of the curvature.

Curvature can be described also intrinsically, without recourse to the ambient space and its metric. Therefore it makes sense in Riemannian geometry based on the metric tensor just on a manifold. In 1917, Herman Weyl postulated a more fundamental quantity than a Riemannian metric, the connection on a fibre bundle, giving hence rise to a modern, generalized idea of the curvature. While Riemannian metric gives rise to a Levi-Civita connection on the tangent bundle of the Riemannian manifold, not every connection on a vector or principal bundles is induced by metrics. In that sense the connection is a more basic notion in geometry.

Curvature of a connection

The curvature of a connection on a bundle measures how the connection is locally non-trivial.

In as far as the notion of connection on a bundle is generalized by the notion of a cocycle in differential cohomology, curvature is essentially the Chern character.

In as far as cocycles in differential cohomology represent gauge fields in physics, the curvature is the field strength of these gauge fields.

After the conception of gauge theory, the term curvature was firmly established in its generalization from this special case to the case of connections on all kinds of bundles and higher bundles.

Curvature of a Lie-algebra valued form

• A connection on a trivial line bundle on a space XX is just a 1-form

  A∈Ω 1(X). A \in \Omega^1(X) \,.

  The curvature in this case is the 2-form F=dAF = d A.

• A connection on a trivial GG-principal bundle for GG a Lie group with Lie algebra 𝔤\mathfrak{g} is a 𝔤\mathfrak{g}-valued 1-form (see groupoid of Lie-algebra valued forms)

  A∈Ω 1(X)⊗𝔤. A \in \Omega^1(X) \otimes \mathfrak{g} \,.

  Its curvature is the Lie-algebra valued 2-form

  F A=dA+[A∧A], F_A = \mathbf{d} A + [A \wedge A] \,,

  where [−,−][-,-] is the Lie bracket in 𝔤\mathfrak{g}.

• According to the discussion at ∞-Chern-Weil theory, a connection on a trivial principal ∞-bundle is given by a collection of ∞-Lie algebroid valued differential forms. The notion of curvature in this general context is discussed at curvature of ∞-Lie algebroid valued differential forms.

Curvature characteristic forms

• curvature characteristic form

Geometric interpretation of curvature 2-forms

For the geometric interpretation of the curvature 2-form of a 𝔤\mathfrak{g}-valued 1-form

Ω •(X)←W(𝔤):A \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A

it is useful to recall that both the deRham complex Ω •(X)\Omega^\bullet(X) as well as the Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) are naturally interpreted as function algebras on infinitesimal objects, as discussed at ∞-Lie algebroid.

The deRham complex may be thought of as the algebra of functions on the infinitesimal path ∞-groupoid Π inf(X)\Pi^{inf}(X). This has as objects the points of XX, as morphisms infinitesimal paths

x→y x \to y

in XX

as 2-morphisms infinitesimal little surfaces

x → y 1 ↓ ↘ ↓ y 2 → z \array{ x &\to& y_1 \\ \downarrow & \searrow & \downarrow \\ y_2 &\to& z }

in XX, and so on.

On the other hand, CE(𝔤)CE(\mathfrak{g}) is the algebra of functions on the infinitesimal version BG (1)\mathbf{B}G_{(1)} of what is called the delooping groupoid BG\mathbf{B}G of the Lie group of which 𝔤\mathfrak{g} is a Lie algebra. This has a single object *{*}, and a morphism is an infinitesimal group element

*→e+λ at a* {*} \stackrel{e + \lambda^a t_a}{\to} {*}

for ee the neutral element of the group (the identity), for t at_a an element of the Lie algebra as before and for λ a\lambda^a some coefficient.

A 2-morphism is an infinitesimal surface bounded by such infinitesimal 1-morphisms such that going either way around the surface

* →e+λ 1 at a * ↓ λ 3 at a ↘ ↓ λ 2 at a * →λ 4 at a * \array{ {*} &\stackrel{e + \lambda^a_1 t_a}{\to}& {*} \\ \downarrow^{\rlap{\lambda_3^a t_a}} & \searrow & \downarrow^{\rlap{\lambda_2^a t_a}} \\ {*} &\stackrel{\lambda_4^a t_a}{\to}& {*} }

produces the same result when then morphisms are composed using the product in the Lie group: the top right way around the square here yields

(e+λ 1 at a))(e+λ 2 bt b))=e+λ 1 at a+λ 2 bt b+λ 1 aλ 2 bt at b (e + \lambda_1^a t_a) )(e + \lambda_2^b t_b) ) = e + \lambda_1^a t_a + \lambda_2^b t_b + \lambda_1^a \lambda_2^b t_a t_b

and the other way round yields

(e+λ 3 at a))(e+λ 4 bt b))=e+λ 3 at a+λ 4 bt b+λ 3 aλ 4 bt at b. (e + \lambda_3^a t_a) )(e + \lambda_4^b t_b) ) = e + \lambda_3^a t_a + \lambda_4^b t_b + \lambda_3^a \lambda_4^b t_a t_b \,.

A morphism of dg-algebras of the form we have been considering

Ω •(X)←CE(𝔤):A \Omega^\bullet(X) \leftarrow CE(\mathfrak{g}) : A

is now evidently equivalenty a morphism

A:Π inf(X)→BG (1) A : \Pi^{inf}(X) \to \mathbf{B}G_{(1)}

that sends infinitesimal paths in XX to infinitesimal group elements of the form e+λ at ae + \lambda^a t_a:

A:(x→y)↦(*→e+A a(x,y)t a*). A : (x \to y) \;\;\mapsto\;\; ({*} \stackrel{e + A^a(x,y) t_a}{\to} {*}) \,.

If we denote by

v=y−x v = y - x

the tangent vector that connects the infinitesimally close points xx and yy and write A(x,y)=A x(v)A(x,y) = A_x(v) as a function of the first point and the vector pointing away from it, then this reads

A:(x→y)↦(*→e+A x a(v)t a*). A : (x \to y) \;\;\mapsto\;\; ({*} \stackrel{e + A^a_x(v) t_a}{\to} {*}) \,.

We can now look at what this assignment AA of infinitesimal group elements to infinitesimal paths does to a little square in XX as above, with sides spanned by tangent vectors v 1v_1 and v 2v_2. We find

A:x →v 1 y 1 ↓ v 2 ↘ ↓ y 2 → z↦* →e+A x(v 1)t a * ↓ A x(v 2) at a ↘ ↓ A y 1(v 2) at a * →A y 2(v 1) at a *. A \;\; : \;\; \array{ x &\stackrel{v_1}{\to}& y_1 \\ \downarrow^{v_2} & \searrow & \downarrow \\ y_2 &\to & z } \;\;\mapsto\;\; \array{ {*} &\stackrel{e + A_x(v_1) t_a}{\to}& {*} \\ \downarrow^{\rlap{A_x(v_2)^a t_a}} & \searrow & \downarrow^{\rlap{A_{y_1}(v_2)^a t_a}} \\ {*} &\stackrel{A_{y_2}(v_1)^a t_a}{\to}& {*} } \,.

For the result on the right to qualify as a 2-morphism in BG (1)\mathbf{B}G_{(1)} we need that

going around the top right edges, which yields

e+A x a(v 1)t a+A y 1 b(v 2)t b+A x a(v 1)A y 1 b(v 2)t at b e + A_x^a(v_1) t_a + A^b_{y_1}(v_2) t_b + A_x^a(v_1) A^b_{y_1}(v_2) t_a t_b

is the same as

e+A x a(v 2)t a+A y 2 b(v 1)t b+A x a(v 2)A y 2 b(v 1)t at b. e + A_x^a(v_2) t_a + A^b_{y_2}(v_1) t_b + A_{x}^a(v_2) A^b_{y_2}(v_1) t_a t_b \,.

To express what this means as a condition at the point xx, we may Taylor expand to first order

A y 1=A x+∂ v 1A x A_{y_1} = A_x + \partial_{v_1} A_x

and

A y 2=A x+∂ v 2A x. A_{y_2} = A_x + \partial_{v_2} A_x \,.

Then some terms cancel and the above condition becomes, to second order

∂ v 1A x a(v 2)t a+A x a(v 1)A x b(v 2)t at b=∂ v 2A x a(v 1)t a+A x a(v 2)A x b(v 1)t at b. \partial_{v_1} A^a_x(v_2) t_a + A_x^a(v_1) A^b_{x}(v_2) t_a t_b = \partial_{v_2} A^a_x(v_1) t_a + A_x^a(v_2) A^b_{x}(v_1) t_a t_b \,.

In other words, the expression

F A(v 1,v 2):=∂ v 1A x a(v 2)t a−∂ v 2A x a(v 1)t a+12A x a(v 1)A x b(v 2)[t a,t b] F_A(v_1,v_2) := \partial_{v_1} A^a_x(v_2) t_a - \partial_{v_2} A^a_x(v_1) t_a + \frac{1}{2} A_x^a(v_1) A^b_{x}(v_2) [t_a, t_b]

has to vanish. This is the curvature form that we already found above by more algebraic means.

If this does not vanish, then we don’t really have a morphism A:Π inf(X)→BG (1)A : \Pi^{inf}(X) \to \mathbf{B}G_{(1)}. But then we instead have some morphism that uses the 1-forms A aA^a to assigns data to little edges, and that uses the 2-forms F A aF_A^a to assign data to little surfaces. That morphism then will respect a condition as above, but now on little cubes. That condition is the Bianchi identity

dF A+[A∧F A]=0 d F_A + [A \wedge F_A] = 0

on the curvature 2-form.

Curvature of ∞\infty-Lie algebroid-valued forms

The notion of curvature of a Lie-algebra valued 1-form discussed above generalizes to that of ∞-Lie algebroid valued differential forms.

Definition

Let 𝔤\mathfrak{g} be an ∞-Lie algebra. A 𝔤\mathfrak{g}-valued differential form on a smooth manifold XX is a morphism

Ω •(X)←W(𝔤):A \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A

of dg-algebras, where Ω •(X)\Omega^\bullet(X) is the de Rham complex and W(𝔤)W(\mathfrak{g}) is the Weil algebra.

There is a canonical inclusion of graded vector spaces

W(𝔤)←𝔤 *[1]:F (−). W(\mathfrak{g}) \leftarrow \mathfrak{g}^*[1] : F_{(-)} \,.

The curvature of the ∞\infty-Lie algebroid valued form AA is the composite

Ω •(X)←AW(𝔤)←F (−)𝔤 *[1]:F A. \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \mathfrak{g}^*[1] : F_A \,.

This consists, in general, of a tower of components: write 𝔤 n\mathfrak{g}_n for the degree nn-part of the ∞\infty-Lie algebra, then we have the further restrictions

Ω •(X)←AW(𝔤)←F (−)𝔤 k *[1]:(F A) k+1. \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \mathfrak{g}_{k}^*[1] : (F_A)_{k+1} \,.

So

(F A) n∈Ω n(X,𝔤 n−1) (F_A)_n \in \Omega^n(X, \mathfrak{g}_{n-1})

is a 𝔤 n−1\mathfrak{g}_{n-1}-valued nn-form on XX.

One may speak of the 2-form curvature, the 3-form curvature, the 4-form curvature and so on.

If instead of just an ∞\infty-Lie algebra 𝔤\mathfrak{g} we take more generally an ∞-Lie algebroid, then there is also a 1-form curvature component.

Remark In some places in the literature, the lower curvature form components have been called fake curvature (BreenMessing).

Properties

Obstruction to flatness

Precisely if the curvatures F AF_A vanish does the morphism A:W(𝔤)→Ω •(X)A : W(\mathfrak{g}) \to \Omega^\bullet(X) factor through the Chevalley-Eilenberg algebra W(𝔤)→CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}).

(F A=0)⇔( CE(𝔤) ∃A flat↙ ↑ Ω •(X) ←A W(𝔤)) (F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right)

in which case we call AA flat.

Bianchi identity

By the fact that AA is a dg-algebra homomorphism, its curvature forms satisfy

d dRF A+A(d W(𝔤)(−))=0. d_{dR} F_A + A(d_{W(\mathfrak{g})}(-)) = 0 \,.

This is the Bianchi identity.

Curvature characteristic forms

The algebra inv(𝔤)inv(\mathfrak{g}) of invariant polynomials embeds into the Weil algebra

W(𝔤)←inv(𝔤). W(\mathfrak{g}) \leftarrow inv(\mathfrak{g}) \,.

For AA a 𝔤\mathfrak{g}-valued form, and ⟨−⟩∈inv(𝔤)\langle - \rangle \in inv(\mathfrak{g}), the ordinary closed nn-form

Ω •(X)←AW(𝔤)←inv(𝔤)←⟨−⟩CE(b n−1ℝ):⟨F A⟩ \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{}{\leftarrow} inv(\mathfrak{g}) \stackrel{\langle - \rangle}{\leftarrow} CE(b^{n-1} \mathbb{R}) : \langle F_A \rangle

is the corresponding curvature characteristic form.

Curved dg-algebras

(TO ADD: Something about curved A ∞A_\infty algebras and curved dg algebras.)

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• Bianchi identity

• curvature characteristic form

• Chern-Simons form

\,

References

General

• Mikio Nakahara, Section 10.3 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)