Riemannian geometry

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from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



In terms of differential forms

In Riemannian geometry, the curl or rotation of a vector field vv on an oriented 33-dimensional Riemannian manifold (X,g)(X,g) is the vector field denoted curl(v)curl(v) (or rot(v)rot(v) or ×v\Del \times v) defined by

curl(v)g 1( gd dRg(v)). curl(v) \;\coloneqq\; g^{-1} \left(\star_g d_{dR}\, g(v) \right) \,.


  1. Γ(TX)AAAAgg 1Ω 1(X)\Gamma(T X) \underoverset{\underset{g}{\longrightarrow}}{\overset{g^{-1}}{\longleftarrow}}{\phantom{AA}\simeq\phantom{AA}} \Omega^1(X) is the linear isomorphism between vector fields and differential 1-forms given by the metric tensor gg;

  2. d dR:Ω n(X)Ω n+1(X)d_{dR} \;\colon\; \Omega^n(X) \longrightarrow \Omega^{n+1}(X) is the de Rham differential

  3. g:Ω n(X)Ω dim(X)n(X)\star_g \;\colon\; \Omega^n(X) \to \Omega^{dim(X)-n}(X) is the Hodge star operator (which uses the orientation of XX).

Notice that for this to make sense it is crucial that the dimension of XX is 33, for only then is the Hodge dual of the de Rham differential of a 11-form again a 11-form; that is, n=3n = 3 is the unique solution of n(1+1)=1n - (1 + 1) = 1.

Via integration

Alternatively, the curl/rotation of a vector field v\vec v at some point xXx\in X may be defined as the limit integral formula

nrotv=lim areaS01areaS Stvdr \vec{n}\cdot rot \vec v = \lim_{area S\to 0} \frac{1}{area S} \oint_{\partial S} \vec{t}\cdot \vec v d r

where DD runs over the smooth oriented surfaces (submanifolds of dimension 22) containing the point xx and with smooth boundary D\partial D, n\vec{n} is the unit vector through the surface SS, and t\vec{t} is the unit vector tangent to the curve S\partial S. (We use the orientation of XX to determine the direction of n\vec{n} from the orientation of SS.)

This formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and discs with decreasing radius in this particular coordinate chart. One can even use πr 2\pi r^2 in place of the actual area of the disc around xx with coordinate-radius rr, to save on calculating this area, as long as the coordinate chart assigns the standard coordinates to the metric at xx.

The proof that this definition is coherent and agrees with the previous one is essentially the Kelvin–Stokes Theorem; see below for discussion.

Via cross products

More generally, if (X,g)(X,g) is a Riemannian manifold whose cotangent spaces (equivalently, tangent spaces) are smoothly equipped with a binary cross product :Ω 2(X;)Ω 1(X;)⨉\colon \Omega^2(X;\mathbb{R}) \to \Omega^1(X;\mathbb{R}), then the curl of any vector field vv is

curl(v)=g 1d dRg(v) curl(v) \;=\; g^{-1} ⨉ d_{dR} g(v)

However, this is not as general as it may appear:

  • in 00 or 11 dimension, the cross product, hence the curl, must always be 00;
  • in 33 dimensions, a smooth choice of cross product is equivalent to a smooth choice of orientation, and we recover the previous formula;
  • in 77 dimensions, if a smooth choice of cross product is possible (as on the 7-sphere), then uncountably many are possible, giving as many different notions of curl;
  • in any other number of dimensions, no binary cross product exists at all, hence no curl.

That said, there are also cross products of other arity? in other dimensions; using essentially the same formula, we can take the curl of a kk-multivector field if we have a smooth (k+1)(k+1)-ary cross product. Or if the cross product is other than vector-valued, then we can obtain a curl that is other than a vector field.

In particular, in 22 dimensions, we have the scalar curl

curl(v)=d dRg(v) curl(v) \;=\; ⨉ d_{dR} g(v)

where :Ω 2(X;)Ω 0(X;)⨉\colon \Omega^2(X;\mathbb{R}) \to \Omega^0(X;\mathbb{R}) is the volume form (or area form) on the 22-dimensional Riemannian manifold XX.


If (X,g)(X,g) is 3\mathbb{R}^3 endowed with the canonical Euclidean metric, then the curl of a vector field (v 1,v 2,v 3)=v 1 1+v 2 2+v 3 3(v^1,v^2,v^3) = v^1\partial_1 + v^2\partial_2 + v^3\partial_3 is

curl(v) 1=v 3x 2v 2x 3;curl(v) 2=v 1x 3v 3x 1;curl(v) 3=v 2x 1v 1x 2. curl(v)^1 = \frac{\partial v^3}{\partial x^2}-\frac{\partial v^2}{\partial x^3} ;\qquad curl(v)^2 = \frac{\partial v^1}{\partial x^3}-\frac{\partial v^3}{\partial x^1} ;\qquad curl(v)^3 = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2} .

This is the classical curl from vector analysis?.

If (X,g)(X,g) is 2\mathbb{R}^2 endowed with the canonical Euclidean metric, then the curl of a vector field (v 1,v 2)=v 1 1+v 2 2(v^1,v^2) = v^1\partial_1 + v^2\partial_2 is

curl(v)=v 2x 1v 1x 2. curl(v) = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2} .

Relation to the Stokes theorems

Recall that if XX is an nn-dimensional differentiable manifold, DD is a pp-dimensional submanifold with boundary, and α\alpha is a differentiable (p1)(p-1)-rank exterior differential form on a neighbourhood of DD in XX, then the generalized Stokes Theorem says that the integral of α\alpha on the boundary D\partial{D} equals the integral on SS of the de Rham differential d DRα\mathrm{d}_{DR}\alpha.

When n=3n=3, p=2p=2, and XX is equipped with an orientation and a metric, then this is equivalent to saying that the integral of a vector field vv along the boundary of an oriented surface DD in XX is equal to the integral of the vector field's curl across the surface:

(1) Dvdr= DcurlvdS. \int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \cdot \mathrm{d}\mathbf{S} .

(In particular, when XX is 3\mathbb{R}^3 with its standard orientation and metric, then this is equivalent to the classical Kelvin–Stokes Theorem.) At least, this is what it says if the curl is defined in terms of differential forms; if the curl is defined via integration instead, then (1) is immediate, and the Kelvin–Stokes Theorem says that this definition matches the other one.

When n=2n=2, p=2p=2, and XX is equipped with an orientation and a metric, then the Stokes Theorem is equivalent to saying that the integral of a vector field vv along a simple closed curve in XX is equal to the integral of the vector field's scalar curl on the region DD enclosed by the curve:

(2) Dvdr= DcurlvdA. \int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \mathrm{d}A .

(In particular, when XX is 2\mathbb{R}^2 with its standard orientation and metric, then this is equivalent to the curl-circulation form of Green's Theorem.)


In many classical applications of the curl in vector analysis?, the Riemannian structure is actually irrelevant, and the gradient can be replaced with the deRham differential d dRd_{dR}. That is, XX is treated as the 11-form g(X)g(X), its curl is treated as the 22-form d dRg(X)d_{dR} g(X), and once these identifications are made there is no need to involve gg or XX directly at all.


See also

Last revised on July 31, 2020 at 09:56:19. See the history of this page for a list of all contributions to it.