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 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.
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.
Curvature of a smooth curve at a point of a smooth curve is the (signed) inverse radius of the (oriented) circle having tangency of order 1 with the curve at the point . 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 describe the curvature, torsion and higher analogues for naturally parametrized curves in -dimensional Euclidean space.
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 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.
The curvature of a connection on a bundle measures how the connection is locally non-trivial.
The curvature in this case is the 2-form .
Its curvature is the Lie-algebra valued 2-form
where is the Lie bracket in .
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.
For the geometric interpretation of the curvature 2-form of a -valued 1-form
it is useful to recall that both the deRham complex as well as the Chevalley-Eilenberg algebra are naturally interpreted as function algebras on infinitesimal objects, as discussed at ∞-Lie algebroid.
as 2-morphisms infinitesimal little surfaces
in , and so on.
On the other hand, is the algebra of functions on the infinitesimal version of what is called the delooping groupoid of the Lie group of which is a Lie algebra. This has a single object , and a morphism is an infinitesimal group element
for the neutral element of the group (the identity), for an element of the Lie algebra as before and for some coefficient.
A 2-morphism is an infinitesimal surface bounded by such infinitesimal 1-morphisms such that going either way around the surface
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
and the other way round yields
A morphism of dg-algebras of the form we have been considering
is now evidently equivalenty a morphism
that sends infinitesimal paths in to infinitesimal group elements of the form :
If we denote by
We can now look at what this assignment of infinitesimal group elements to infinitesimal paths does to a little square in as above, with sides spanned by tangent vectors and . We find
For the result on the right to qualify as a 2-morphism in we need that
going around the top right edges, which yields
is the same as
To express what this means as a condition at the point , we may Taylor expand to first order
Then some terms cancel and the above condition becomes, to second order
In other words, the expression
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 . But then we instead have some morphism that uses the 1-forms to assigns data to little edges, and that uses the 2-forms 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
on the curvature 2-form.
There is a canonical inclusion of graded vector spaces
The curvature of the -Lie algebroid valued form is the composite
This consists, in general, of a tower of components: write for the degree -part of the -Lie algebra, then we have the further restrictions
is a -valued -form on .
One may speak of the 2-form curvature, the 3-form curvature, the 4-form curvature and so on.
If instead of just an -Lie algebra 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).
Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra .
in which case we call flat.
This is the Bianchi identity.
The algebra of invariant polynomials embeds into the Weil algebra
For a -valued form, and , the ordinary closed -form
is the corresponding curvature characteristic form.
(TO ADD: Something about curved algebras and curved dg algebras.)
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|
|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|