nLab curvature

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 $\mathbb{R}^n$.

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 $\mathbb{R}^n$ 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 $\kappa(\gamma)$ of a smooth curve $\gamma$ at a point $p$ of a smooth curve is the (signed) inverse radius of the (oriented) circle having tangency of order 1 with the curve at the point $p$. 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 $n$-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 $p$ 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 $X$ is just a 1-form

  $$A \in \Omega^1(X) \,.$$

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

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

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

  Its curvature is the Lie-algebra valued 2-form

  $$F_A = \mathbf{d} A + [A \wedge A] \,,$$

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

• 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 $\mathfrak{g}$-valued 1-form

$$\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$$

it is useful to recall that both the deRham complex $\Omega^\bullet(X)$ as well as the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ 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 $\Pi^{inf}(X)$. This has as objects the points of $X$, as morphisms infinitesimal paths

$$x \to y$$

in $X$

as 2-morphisms infinitesimal little surfaces

$$\array{ x &\to& y_1 \\ \downarrow & \searrow & \downarrow \\ y_2 &\to& z }$$

in $X$, and so on.

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

$${*} \stackrel{e + \lambda^a t_a}{\to} {*}$$

for $e$ the neutral element of the group (the identity), for $t_a$ an element of the Lie algebra as before and for $\lambda^a$ some coefficient.

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

$$\array{ {*} &\stackrel{e + \lambda^a_1 t_a}{\to}& {*} \\ \downarrow^{\rlap{\lambda_3^a t_a}} & \searrow & \downarrow^{\rlap{\lambda_2^a t_a}} \\ {*} &\stackrel{\lambda_4^a t_a}{\to}& {*} }$$

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

$$(e + \lambda_1^a t_a) )(e + \lambda_2^b t_b) ) = e + \lambda_1^a t_a + \lambda_2^b t_b + \lambda_1^a \lambda_2^b t_a t_b$$

and the other way round yields

$$(e + \lambda_3^a t_a) )(e + \lambda_4^b t_b) ) = e + \lambda_3^a t_a + \lambda_4^b t_b + \lambda_3^a \lambda_4^b t_a t_b \,.$$

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

$$\Omega^\bullet(X) \leftarrow CE(\mathfrak{g}) : A$$

is now evidently equivalenty a morphism

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

that sends infinitesimal paths in $X$ to infinitesimal group elements of the form $e + \lambda^a t_a$:

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

If we denote by

$$v = y - x$$

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

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

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

$$A \;\; : \;\; \array{ x &\stackrel{v_1}{\to}& y_1 \\ \downarrow^{v_2} & \searrow & \downarrow \\ y_2 &\to & z } \;\;\mapsto\;\; \array{ {*} &\stackrel{e + A_x(v_1) t_a}{\to}& {*} \\ \downarrow^{\rlap{A_x(v_2)^a t_a}} & \searrow & \downarrow^{\rlap{A_{y_1}(v_2)^a t_a}} \\ {*} &\stackrel{A_{y_2}(v_1)^a t_a}{\to}& {*} } \,.$$

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

going around the top right edges, which yields

$$e + A_x^a(v_1) t_a + A^b_{y_1}(v_2) t_b + A_x^a(v_1) A^b_{y_1}(v_2) t_a t_b$$

is the same as

$$e + A_x^a(v_2) t_a + A^b_{y_2}(v_1) t_b + A_{x}^a(v_2) A^b_{y_2}(v_1) t_a t_b \,.$$

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

$$A_{y_1} = A_x + \partial_{v_1} A_x$$

and

$$A_{y_2} = A_x + \partial_{v_2} A_x \,.$$

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

$$\partial_{v_1} A^a_x(v_2) t_a + A_x^a(v_1) A^b_{x}(v_2) t_a t_b = \partial_{v_2} A^a_x(v_1) t_a + A_x^a(v_2) A^b_{x}(v_1) t_a t_b \,.$$

In other words, the expression

$$F_A(v_1,v_2) := \partial_{v_1} A^a_x(v_2) t_a - \partial_{v_2} A^a_x(v_1) t_a + \frac{1}{2} A_x^a(v_1) A^b_{x}(v_2) [t_a, t_b]$$

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 $A : \Pi^{inf}(X) \to \mathbf{B}G_{(1)}$. But then we instead have some morphism that uses the 1-forms $A^a$ to assigns data to little edges, and that uses the 2-forms $F_A^a$ 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

$$d F_A + [A \wedge F_A] = 0$$

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

$$\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$$

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

There is a canonical inclusion of graded vector spaces

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

The curvature of the $\infty$-Lie algebroid valued form $A$ is the composite

$$\Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \mathfrak{g}^*[1] : F_A \,.$$

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

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

So

$$(F_A)_n \in \Omega^n(X, \mathfrak{g}_{n-1})$$

is a $\mathfrak{g}_{n-1}$-valued $n$-form on $X$.

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 $\mathfrak{g}$ 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 $F_A$ vanish does the morphism $A : W(\mathfrak{g}) \to \Omega^\bullet(X)$ factor through the Chevalley-Eilenberg algebra $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

$$(F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right)$$

in which case we call $A$ flat.

Bianchi identity

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

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

This is the Bianchi identity.

Curvature characteristic forms

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

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

For $A$ a $\mathfrak{g}$-valued form, and $\langle - \rangle \in inv(\mathfrak{g})$, the ordinary closed $n$-form

$$\Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{}{\leftarrow} inv(\mathfrak{g}) \stackrel{\langle - \rangle}{\leftarrow} CE(b^{n-1} \mathbb{R}) : \langle F_A \rangle$$

is the corresponding curvature characteristic form.

Curved dg-algebras

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

Related concepts

• ∞-Lie algebroid valued differential forms

• connection on an ∞-bundle

• Bianchi identity

• curvature characteristic form

• Chern-Simons form

curvature in Riemannian geometry
Riemann curvature
Ricci curvature
scalar curvature
sectional curvature
p-curvature

$\,$

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map

References

General

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

Cartan structural equations and Bianchi identities

On Cartan structural equations and their Bianchi identities for curvature and torsion of Cartan moving frames and (Cartan-)connections on tangent bundles (especially in first-order formulation of gravity):

The original account:

• Élie Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales scientifiques de l’École Normale Supérieure, Sér. 3, 40 (1923) 325-412 [doi:ASENS_1923_3_40__325_0]

Historical review:

• Erhard Scholz, §2 in: E. Cartan’s attempt at bridge-building between Einstein and the Cosserats – or how translational curvature became to be known as “torsion”, The European Physics Journal H 44 (2019) 47-75 [doi:10.1140/epjh/e2018-90059-x]

Further discussion:

• Shiing-Shen Chern, p. 748 of: A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics, Second Series, 45 4 (1944) 747-752 [doi:10.2307/1969302]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §I.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.2: pdf]

• Sigurdur Helgason, §I.8 in: Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

• C. C. Briggs, A Sequence of Generalizations of Cartan’s Conservation of Torsion Theorem [arXiv:gr-qc/9908034]

• Loring Tu, §22 in: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer (2017) [ISBN:978-3-319-55082-4]

• Thoan Do, Geoff Prince, An intrinsic and exterior form of the Bianchi identities, International Journal of Geometric Methods in Modern Physics 14 01 (2017) 1750001 [doi:10.1142/S0219887817500013, arXiv:1501.01123]

• Ivo Terek Couto, Cartan Formalism and some computations [pdf, pdf]

• Cadabra, Cartan structural equations and Bianchi identity

Generalization to supergeometry (motivated by supergravity):

• Julius Wess, Bruno Zumino, p. 362 of: Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361-364 [doi:10.1016/0370-2693(77)90015-6]

• Richard Grimm, Julius Wess, Bruno Zumino, §2 in: A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys. B 152 (1979) 255-265 [doi:10.1016/0550-3213(79)90102-0]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §III.3.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch III.3: pdf]