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Bernhard Riemann (1826-1866)

Élie Cartan (1869-1951)

The classical approach of Gauss was the standard elementary approach which predated the emergence of the concepts of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century and of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century. The notion of connection, covariant derivative and parallel transport gave a more conceptual and uniform way of understanding curvature, which not only allowed generalisations to higher dimensional manifolds but also provided an important tool for defining new geometric invariants, called characteristic classes. The approach using covariant derivatives and connections is nowadays the one adopted in more advanced textbooks.

Connections on a surface can be defined in a variety of ways. The Riemannian connection or Levi-Civita connection is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on the frame bundle: in the case of an embedded surface, the lift is very simply described in terms of orthogonal projection. Indeed the vector bundles associated with the frame bundle are all sub-bundles of trivial bundles that extend to the ambient Euclidean space; a first order differential operator can always be applied to a section of a trivial bundle, in particular to a section of the original sub-bundle, although the resulting section might no longer be a section of the sub-bundle. This can be corrected by projecting orthogonally.

The Riemannian connection can also be characterized abstractly independently of an embedding. The equations of geodesics are easy to write in terms of the Riemannian connection, which can be locally expressed in terms of the Christoffel symbols. Along a curve in the surface, the connection defines a first order differential equation in the frame bundle. The monodromy of this equation defines parallel transport for the connection, a notion introduced in this context by Levi-Civita. This gives an equivalent more geometric way of describing the connection in terms of lifting paths in the manifold to paths in the frame bundle. This formalised the classical theory of the "moving frame", favoured by French authors. Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields.

The approach of Cartan, using connection 1-forms on the frame bundle of M, gives a third way to understand the Riemannian connection, which is particularly easy to describe for an embedded surface. Thanks to a result of Kobayashi (1956), later generalized by Narasimhan & Ramanan (1961), the Riemannian connection on a surface embedded in Euclidean space E³ is just the pullback under the Gauss map of the Riemannian connection on S². Using the identification of S² with the homogeneous space SO(3)/SO(2), the connection 1-form is just a component of the Maurer-Cartan 1-form on SO(3). In other words everything reduces to understanding the 2-sphere properly.

Covariant derivative

Main article: covariant derivative

A vector field on the torus

For a surface M embedded in E³ (or more generally a higher dimensional Euclidean space), there are several equivalent definitions of a vector field X on M:

• a smooth map of M into E³ taking values in the tangent space at each point;
• the velocity vector of a local flow on M;
• a first order differential operator without constant term in any local chart on M;
• a derivation of C^∞(M).

The last condition means that the assignment f $\mapsto$ Xf on C^∞(M) satisfies the Leibniz rule

$X(fg)=(Xf)g+f(Xg).$

The space of all vector fields ${\mathcal {X}}$(M) forms a module over C^∞(M), closed under the Lie bracket

$[X,Y]f=X(Yf)-Y(Xf)$

with a C^∞(M)-valued inner product (X,Y), which encodes the Riemannian metric on M.

Since ${\mathcal {X}}$(M) is a submodule of C^∞(M, E³)=C^∞(M)$\otimes$ E³, the operator X$\otimes$ I is defined on ${\mathcal {X}}$(M), taking values in C^∞(M, E³).

Let P be the smooth map from M into M₃(R) such that P(p) is the orthogonal projection of E³ onto the tangent space at p.

Pointwise multiplication by P gives a C^∞(M)-module map of C^∞(M, E³) onto ${\mathcal {X}}$(M) . The assignment

$\nabla _{X}Y=P((X\otimes I)Y)$

defines an operator $\nabla _{X}$ on ${\mathcal {X}}$(M) called the covariant derivative, satisfying the following properties

1. $\nabla _{X}$ is C^∞(M)-linear in X
2. $\nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y$ (Leibniz rule for derivation of a module)
3. $X(Y,Z)=(\nabla _{X}Y,Z)+(Y,\nabla _{X}Z)$ (compatibility with the metric)
4. $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$ (symmetry property).

The first three properties state that $\nabla$ is an affine connection compatible with the metric, sometimes also called a hermitian or metric connection. The last symmetry property says that the torsion tensor

$T(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$

vanishes identically, so that the affine connection is torsion-free.

The assignment $\nabla$ is uniquely determined by these four conditions and is called the Riemannian connection or Levi-Civita connection.

Although the Riemannian connection was defined using an embedding in Euclidean space, this uniqueness property means that it is in fact an intrinsic invariant of the surface.

It existence can be proved directly for a general surface by noting that the four properties imply

$2(\nabla _{X}Y,Z)=X\cdot (Y,Z)+Y\cdot (X,Z)-Z\cdot (X,Y)+([X,Y],Z)+([Z,X],Y)+(X,[Z,Y]),$

so that $\nabla _{X}Y$ depends only on the metric and is unique. On the other hand if this is used as a definition of $\nabla _{X}Y$, it is readily checked that the four properties above are satisfied.

Equivalently, in local coordinates (x,y) with basis tangent vectors e_{1=$\partial _{x}$ and e2 = $\partial _{y}$, the connection $\nabla$ can be expressed purely in terms of the metric using the Christoffel symbols:}

$\nabla _{{\mathbf {e} }_{i}}{\mathbf {e} }_{j}=\sum _{k}\Gamma _{ij}^{k}{\mathbf {e} }_{k}.$

If c(t) is a path in M, then the Euler equations for c to be a geodesic can be written more compactly as

$\nabla _{\dot {c}}{\dot {c}}=0.$

Parallel transport

See also: parallel transport

Parallel transport of a vector around a geodesic triangle on the sphere. The length of the transported vector and the angle it makes with each side remain constant.

Given a curve in the Euclidean plane and a vector at the starting point, the vector can be transported along the curve by requiring the moving vector to remain parallel to the original one and of the same length, i.e. it should remain constant along the curve. If the curve is closed, the vector will be unchanged when the starting point is reached again. This is well known not to be possible on a general surface, the sphere being the most familiar case. In fact it is not usually possible to identify simultaneously or "parallelize" all the tangent planes of such a surface: the only parallelizable closed surfaces are those homeomorphic to a torus.

Parallel transport can always be defined along curves on a surface using only the metric on the surface. Thus tangent planes along a curve can be identified using the intrinsic geometry, even when the surface itself is not parallelizable.

Parallel transport along geodesics, the "straight lines" of the surface, is easy to define. A vector in the tangent plane is transported along a geodesic as the unique vector field with constant length and making a constant angle with the velocity vector of the geodesic.

For a general curve, its geodesic curvature measures how far the curve departs from being a geodesics; it is defined as the rate at which the curve's velocity vector rotates in the surface. In turn the geodesic curvature determines how vectors in the tangent planes along the curve should rotate during parallel transport.

A vector field v(t) along a unit speed curve c(t), with geodesic curvature k_g(t), is said to be parallel along the curve if

• it has constant length
• the angle θ(t) that it makes with the velocity vector ${\dot {c}}(t)$ satisfies

${\dot {\theta }}(t)=-k_{g}(t)$

This yields the previous rule for parallel transport along a geodesic, because in that case k_g = 0, so the angle θ(t) should remain constant. The existence of parallel transport follows from standard existence theorems for ordinary differential equations.

Arnold has suggested that since parallel transport on a geodesic segment is easy to describe, parallel transport on an arbitrary C¹ curve could be constructed as a limit of parallel transport on an approximating family of piecewise geodesic curves.

This equation shows once more that parallel transport depends only on the metric structure so is an intrinsic invariant of the surface; it is another way of writing the ordinary differential equation involving the geodesic curvature of c. Parallel transport can be extended immediately to piecewise C¹ curves.

The covariant derivative can in turn be recovered from parallel transport. In fact $\nabla _{X}Y$ can be calculated at a point p, by taking a curve c through p with tangent X, using parallel transport to view the restriction of Y to c as a function in the tangent space at p and then taking the derivative.

Frame bundle

See also: Connection (principal bundle)

Let M be a surface embedded in E³. The orientation on the surface means that an "outward pointing" normal unit vector n is defined at each point of the surface and hence a determinant can be defined on tangent vectors v and w at that point:

$\mathrm {det} ({\mathbf {v} },{\mathbf {w} })=({\mathbf {v} }\times {\mathbf {w} })\cdot {\mathbf {n} },$

using the usual scalar triple product on E³ (itself a determinant).

An ordered basis or frame v, w in the tangent space is said to be oriented if det(v, w) is positive.

• The tangent bundle of M consists of pairs (p, v) in M x E₃ such that v lies in the tangent plane to M at p.

• The frame bundle E of M consists of triples (p, e₁, e₂) with an e₁, e₂ an oriented orthonormal basis of the tangent plane at p.

• The circle bundle of M consists of pairs (p, v) with ||v|| = 1. It is identical to the frame bundle because for each unit tangent vector v there is a unique unique tangent vector w with det(v, w) = 1.

Since the group of rotations in the plane SO(2) acts simply transitively on oriented orthonormal frames in the plane, it follows that it also acts on the frame or circle bundles of M. The definitions of the tangent bundle, the unit tangent bundle and the (oriented orthonormal) frame bundle E can be extended to arbitrary surfaces in the usual way. There is a similar identification between the latter two which again become principal SO(2)-bundles. In other words:

The frame bundle is a principal bundle with structure group SO(2).

There is also a corresponding notion of parallel transport in the setting of frame bundles:

Every continuously differentiable curve in M can be lifted to a curve in E in such a way that the tangent vector field of the lifted curve is the lift of the tangent vector field of the original curve.

This statement means that any frame on a curve can be parallely transported along the curve. This is precisely the idea of "moving frames". Since any unit tangent vector can be completed uniquely to an oriented frame, parallel transport of tangent vectors implies (and is equivalent to) parallel transport of frames. The lift of a geodesic in M turns out to be a geodesic in E for the Sasaki metric (see below). Moreover the Gauss map of M into S² induces a natural map between the associated frame bundles which is equivariant for the actions of SO(2).

Cartan's idea of introducing the frame bundle as a central object was the natural culmination of the theory of moving frames, developed in France by Darboux and Goursat. It also echoed parallel developments in Albert Einstein's theory of relativity. Objects appearing in the formulas of Gauss, such as the Christoffel symbols, can be given a natural geometric interpretation in this framework. Unlike the more intuitive normal bundle, easily visualised as a tubular neighbourhood of an embedded surface in E³, the frame bundle is an intrinsic invariant that can be defined independently of an embedding. When there is an embedding, it can also be visualised as a subbundle of the Euclidean frame bundle E³ x SO(3), itself a submanifold of E³ x M₃(R).

Cartan-Ehresmann connection

See also: connection form and Cartan connection

.

The theory of connections according to Élie Cartan, and later Charles Ehresmann, revolves around:

• a principal bundle E;
• the exterior differential calculus of differential forms on E.

All "natural" vector bundles associated with the manifold M, such as the tangent bundle, the cotangent bundle or the exterior bundles, can be constructed from the frame bundle using the representation theory of the structure group K = SO(2), a compact matrix group.

Cartan's definition of a connection can be understood as a way of lifting vector fields on M to vector fields on the frame bundle E invariant under the action of the structure group K. This "universal lift" then immediately induces lifts to vector bundles associated with E and hence allows the covariant derivative, and its generalisation to forms, to be recovered. If X is a vector field on M, the lift X* is required to have the following properties:

• X* is a vector field on E;
• the map X $\mapsto$ X* is C^∞(M)-linear;
• X* is K-invariant and induces the vector field X on C^∞(M) $\subset$ C^∞(E).

Here K acts as a periodic flow on E, so the canonical generator A of its Lie algebra acts as the corresponding vector field, called the vertical vector field A*. It follows from the above conditions that, in the tangent space of an arbitrary point in E, the lifts X* span a two-dimensional subspace of horizontal vectors, forming a complementary subspace to the vertical vectors. This becomes an orthogonal complement for the canonical Riemannian metric on E defined by Shigeo Sasaki.

Horizontal vector fields admit the following characterisation:

• Every K-invariant horizontal vector field on E has the form X* for a unique vector field X on M.

If σ is a representation of K on a finite-dimensional vector space V, then the associated vector bundle E X_K V over M has a C^∞(M)-module of sections that can be identified with

$C^{\infty }(E,V)^{K},$

the space of all smooth functions ξ : E → V which are K-equivariant in the sense that

$\xi (x\cdot g)=\sigma (g^{-1})\xi (x)$

for all x ∈ E and g ∈ K.

The identity representation of SO(2) on R² corresponds to the tangent bundle of M.

The covariant derivative $\nabla _{X}$ is defined on an invariant section ξ by the formula

$\nabla _{X}\xi =(X^{*}\otimes I)\xi .$

The connection on the frame bundle can also be described using K-invariant differential 1-forms on E.

The frame bundle E is a 3-manifold. The space of p-forms on E is denoted Λ^p(E). It admits a natural action of the structure group K.

Given a connection on the principal bundle E corresponding to a lift X $\mapsto$ X* of vector fields on M, there is a unique connection form ω in

$\Lambda ^{1}(E)^{K}$,

the space of K-invariant 1-forms on E, such that

$\omega (X^{*})=0$

for all vector fields X on M and

$\omega (A^{*})=1,$

for the vector field A* on E corresponding to the canonical generator A of ${\mathfrak {k}}$.

Conversely the lift X* is uniquely characterised by the following properties:

• X* is K-invariant and induces X on M;
• ω(X*)=0.

Cartan structural equations

See also: Curvature form

On the frame bundle E of a surface M there are three canonical 1-forms:

• The connection form ω, invariant under the structure group K = SO(2)
• Two tautologous 1-forms θ₁ and θ₂, transforming according to the basis vectors of the identity representation of K

If π: E $\rightarrow$ M is the nature projection, the 1-forms θ₁ and θ₂ are defined by

$\theta _{i}(Y)=(d\pi (Y),e_{i})$

where Y is a vector field on E and e₁, e₂ are the tangent vectors to M of the orthonormal frame.

These 1-forms satisfy the following structural equations, due in this formulation to Cartan:

$d\theta _{1}=\omega \wedge \theta _{2},\,\,d\theta _{2}=-\omega \wedge \theta _{1}$

(First structural equations)

$d\omega =-(K\circ \pi )\theta _{1}\wedge \theta _{2}$

(Second structural equation)

where K is the Gaussian curvature on M.

Holonomy and curvature

Main article: Holonomy

Parallel transport in the frame bundle can be used to show that the Gaussian curvature of a surface M measures the amount of rotation obtained by translating vectors around small curves in M. Holonomy is exactly the phenomenon that occurs when a tangent vector (or orthonormal frame) is parallelly transported around a closed curve. The vector reached when the loop is closed will be a rotation of the original vector, i.e. it will correspond to an element of the rotaion group SO(2), in other words an angle modulo 2π. This is the holonomy of the loop, because the angle does not depend on the choice of starting vector.

Geometric interpretation of the Lie bracket of two vector fields

This geometric interpretation of curvature relies on a similar geometric of the Lie bracket of two vector fields on E. Let U₁ and U₂ be vector fields on E with corresponding local flows α_t and β_t.

• Starting at a point A corresponding to x in E, travel ${\sqrt {s}}$ along the integral curve for U₁ to the point B at $\alpha _{\sqrt {s}}(x)$.

• Travel from B by going ${\sqrt {s}}$ along the integral curve for U₂ to the point C at $\beta _{\sqrt {s}}\alpha _{\sqrt {s}}(x)$.

• Travel from C by going $-{\sqrt {s}}$ along the integral curve for U₁ to the point D at $\alpha _{-{\sqrt {s}}}\beta _{\sqrt {s}}\alpha _{\sqrt {s}}(x)$.

• Travel from D by going $-{\sqrt {s}}$ along the integral curve for U₂ to the point E at $\beta _{-{\sqrt {s}}}\alpha _{-{\sqrt {s}}}\beta _{\sqrt {s}}\alpha _{\sqrt {s}}(x)$.

In general the end point E will differ from the starting point A. As s $\rightarrow$ 0, the end point E will trace out a curve through A. The Lie bracket [U₁,U₂] at x is precisely the tangent vector to this curve at A.

To apply this theory, introduce vector fields U₁, U₂ and V on the frame bundle E which are dual to the 1-forms θ₁, θ₂ and ω at each point. Thus

$\omega (U_{i})=0,\,\theta _{i}(V)=0,\,\omega (V)=1,\,\theta _{i}(U_{j})=\delta _{ij}.$

Moreover V is invariant under K and U₁, U₂ transform according to the identity representation of K.

The structural equations of Cartan imply the following Lie bracket relations:

$[V,U_{1}]=U_{2},\,\,\,\,[V,U_{2}]=-U_{1},\,\,\,\,[U_{1},U_{2}]=(K\circ \pi )V$

The geometrical interpretation of the Lie bracket can be applied to the last of these equations. Since ω(U_i)=0, the flows α_t and β_t in E are lifts by parallel transport of their projections in M.

Informally the idea is as follows. The starting point A and end point E essentially differ by an element of SO(2), that is an angle of rotation. The area enclosed by the projected path in M is approximately ${\sqrt {s}}\cdot {\sqrt {s}}=s$. So in the limit as s $\rightarrow$ 0, the angle of rotation divided by this area tends to the coefficient of V, i.e. the curvature.

This reasoning is made precise in the following result.

Let f be a diffeomorphism of an open disc in the plane into M and let Δ be a triangle in this disc. Then the holonomy angle of the loop formed by the image under f of the perimeter of the triangle is given by the integral of the Gauss curvature of the image under f of the inside of the triangle.

In symbols, the holonomy angle mod 2π is given by

$\theta =\int _{f(\Delta )}K$

where the integral is with respect to the area form on M.

This result implies the relation between Gaussian curvature because as the triangle shrinks in size to a point, the ratio of this angle to the area tends to the Gaussian curvature at the point. The result can be proved by a combination of Stokes's theorem and Cartan's structural equations and can in turn be used to obtain a generalisation of Gauss's theorem on geodesics triangles to more general triangles.

One of the other standard approaches to curvature, through the covariant derivative $\nabla _{X}$, identifies the difference

$R(X,Y)=\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]}$

as a field of endomorphisms of the tangent bundle, the Riemann curvature tensor. Since $\nabla _{X}$ is induced by the lifted vector field X* on E, the use of the vector fields U_i and V and their Lie brackets is more or less equivalent to this approach. The vertical vector field W=A* corresponding to the canonical generator A of ${\mathfrak {k}}$ could also be added since it commutes with V and satisfies [W,U₁] = U₂ and [W,U₂] = —U₁.

Example: the 2-sphere

The differential geometry of the 2-sphere can be approached from three different points of view:

• analytic geometry, since the 2-sphere is a submanifold of E³;
• group theory, since the compact matrix group SO(3) acts transitively on the 2-sphere as a continuous group of symmetries;
• classical mechanics, since a rigid 2-sphere can roll on a plane.

S² can be identifed with the unit sphere in E³

$S^{2}=\{a\in E^{3}\colon \|a\|=1\}.$

Its tangent bundle T, unit tangent bundle U and oriented orthonormal frame bundle E are given by

$T=\{(a,v)\colon \|a\|=1,\,a\cdot v=0\},$

$U=\{(a,v)\colon \|a\|=1,\,\|v\|=1,\,a\cdot v=0\},$

$E=\{(a,e_{1},e_{2})\colon (e_{1}\times e_{2})\cdot a=1,\,\|a\|=1,\,\|e_{i}\|=1\}.$

The map sending (a,v) to (a, v, a x v) allows U and E to be identified.

Let

$Q(a)v=(v\cdot a)a$

be the orthogonal projection onto the normal vector at a, so that

$P(a)=I-Q(a)$

is the orthogonal projection onto the tangent space at a.

The group G = SO(3) acts by rotation on E³ leaving S² invariant. The stabilizer subgroup K of the vector (0,0,1) in E₃ may be identified with SO(2) and hence

S² may be identified with SO(3)/SO(2).

This action extends to an action on T, U and E by making G act on each component. G acts transitively on S² and simply transitively on U and E.

The action of SO(3) on E commutes with the action of SO(2) on E that rotates frames

$(e_{1},e_{2})\mapsto (\cos \theta \,e_{1}-\sin \theta \,e_{2},\sin \theta \,e_{1}+\cos \theta \,e_{2}).$

Thus E becomes a principal bundle with structure group K. Taking the G-orbit of the point ((0,0,1),(1,0,0),(0,1,0)), the space E may be identified with G. Under this identification the actions of G and K on E become left and right translation. In other words:

The oriented orthonormal frame bundle of S² may be identified with SO(3).

The Lie algebra ${\mathfrak {g}}$ of SO(3) consists of all skew-symmetric real 3 x 3 matrices. the adjoint action of G by conjugation on ${\mathfrak {g}}$ reproduces the action of G on E³. The group SU(2) has a 3-dimensional Lie algebra consisting of complex skew-hermitian traceless 2 x 2 matrices, which is isomorphic to ${\mathfrak {g}}$. The adjoint action of SU(2) factors through its centre, the matrices ± I. Under these identifications, SU(2) is exhibited as a double cover of SO(3), so that SO(3) = SU(2) / ± I. On the other hand SU(2) is diffeomorphic to the 3-sphere and under this identification the standard Riemannian metric on the 3-sphere becomes the essentially unique biinvariant Riemannian metric on SU(2). Under the quotient by ± I, SO(3) can be identified with the real projective space of dimension 3 and itself has an essentially unique biinvariant Riemannian metric. The geometric exponential map for this metric at I coincides with the usual exponential function on matrices and thus the geodesics through I are have the form exp Xt where X is a skew-symmetric matrix.

The actions of G on itself, and hence on C^∞(G) by left and right translation induce infinitesimal actions of ${\mathfrak {g}}$ on C^∞(G) by vector fields

$\lambda (X)f(g)={d \over dt}f(e^{-Xt}g)|_{t=0},\,\,\rho (X)f(g)={d \over dt}f(ge^{Xt})|_{t=0}.$

The right and left invariant vector fields are related by the formula

$\lambda (X)f(g)=-\rho (g^{-1}Xg)f(g).$

The vector fields λ(X) and ρ(X) commute with right and left translation and give all right and left invariant vector fields on G. Since C^∞(S²) = C^∞(G/K) can be identified with C^∞(G)^K, the function invariant under right translation by K, the operators λ(X) also induces vector fields Π(X) on S².

Let A, B, C be the standard basis of ${\mathfrak {g}}$ given by

$A={\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}},\,\,B={\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix}},\,\,C={\begin{pmatrix}-1&0&0\\0&0&0\\1&0&0\end{pmatrix}}.$

Their Lie brackets [X,Y] = XY – YX are given by

$[A,B]=C,\,\,[B,C]=A,\,\,[C,A]=B.$

The vector fields λ(A), λ(B), λ(C) form a basis of the tangent space at each point of G. Let α, β, γ be the corresponding dual basis of right invariant 1-forms on G. The Lie bracket relations imply the Maurer-Cartan equations

$d\alpha =\beta \wedge \gamma ,\,\,d\beta =\gamma \wedge \alpha ,\,\,d\gamma =\alpha \wedge \beta .$

These are also the corresponding components of the Maurer-Cartan form

$\omega _{G}=(dg)g^{-1},$

a right invariant matrix-valued 1-form on G, which satisfies the relation

$d\omega _{G}=dg(g^{-1}dg\,g^{-1})=\omega _{G}\wedge \omega _{G}.$

The inner product on ${\mathfrak {g}}$ defined by

$(X,Y)=\mathrm {Tr} \,XY^{T}$

is invariant under the adjoint action. Let π be the orthogonal projection onto the subspace generated by A, i.e. onto ${\mathfrak {k}}$, the Lie algebra of K. For X in ${\mathfrak {g}}$, the lift of the vector field Π(X) from C^∞(G/K) to C^∞(G) is given by the formula

$\Pi (X)^{*}f(g)=-\rho (\pi (g^{-1}Xg))f(g)$

This lift is G-equivariant on vector fields of the form Π(X) and has a unique extension to more general vector fields on G / K.

The right invariant 1-form α is the connection form on G corresponding to this lift. Thus

• vertical vector fields on G are those of the form f · λ(A) with f in C^∞(G);
• horizontal vector fields on G are those of the form f₁ · λ(B) + f₂ · λ(C) with f_i in C^∞(G).

The existence of the basis vector fields λ(A), λ(B), λ(C) shows that SO(3) is parallelizable. This is not true for SO(3)/SO(2) by the hairy ball theorem: S² does not admit any nowhere vanishing vector fields.

Parallel transport in the frame bundle amounts to lifting a path from SO(3)/SO(2) to SO(3). It is accomplished using a matrix-valued ordinary differential equation called the transport equation as follows:

• the space of matrices F satisfying F² = I, F = F* and Tr F = –1 can be identified with S² by choosing a base point F₀ and using the transitive action by conjugation of SO(3);
• a path F(t) starting at F₀ is lifted to the path g(t) in SO(3) satisfying $g_{t}g^{-1}=F_{t}F/2$ and g(0) = I.

Given a path F(t), the ordinary differential equation ${\dot {g}}g^{-1}={\dot {F}}F/2$, with initial condition $g(0)=I$, has a unique C¹ solution g(t) with values in G, giving the lift by parallel transport of F.

Set A = ½ F_t F. Since F² = I and F is symmetric, A is skew-symmetric.

The unique solution g(t) of the ordinary differential equation

${\dot {g}}=Ag$

with initial condition g(0) = I guaranteed by the Picard-Lindelöf theorem, must have g^Tg constant and therefore I, since

${d \over dt}(g^{T}g)={\dot {g}}^{T}g+g^{T}{\dot {g}}=g^{T}(A^{T}+A)g=0.$

Moreover

$F(t)=g(t)F(0)g(t)^{-1}\,$

since g^-1Fg has derivative 0:

${d \over dt}(g^{-1}Fg)=-g^{-1}{\dot {g}}g^{-1}Fg+g^{-1}{\dot {F}}g+g^{-1}F{\dot {g}}=g^{-1}(-{\dot {g}}g^{-1}F+{\dot {F}}+F{\dot {g}}g^{-1})g=0.$

Thus the gauge transformation g trivialises the path F(t).

There is another kinematic way of understanding parallel transport and geodesic curvature in terms of "rolling without slipping or twisting". Although well known to differential geometers since the early part of the twentieth century, it has also been applied to problems in engineering and robotics. Consider the 2-sphere as a rigid body in three dimensional space rolling without slipping or twisting on a horizontal plane. The point of contact will describe a curve in the plane and on the surface. At each point of contact the different tangent planes of the sphere can be identified with the horizontal plane itself and hence with one another.

• The usual curvature of the planar curve is the the geodesic curvature of the curve traced on the sphere.
• This identification of the tangent planes along the curve corresponds to parallel transport.

This is particularly easy to visualize for a sphere: it is exactly the way a marble can be rolled along a perfectly flat table top.

The roles of the plane and the sphere can be reversed to provide an alternative but equivalent point of view. The sphere is regarded as fixed and the plane has to roll without slipping or twisting along the given curve on the sphere.

Embedded surfaces

When the surface M is embedded in E³ (or a higher dimensional Euclidean space), parallel transport can be described in yet another way. Parallel transport along a curve c(t), with t taking values in [0,1], starting from a tangent from a tangent vector v₀ also amounts to finding a map v(t) from [0,1] to R³ such that

• v(t) is a tangent vector to M at c(t) with v(0) = v₀
• the velocity vector ${\dot {v}}(t)$ is normal to the surface at c(t)

This always has a unique solution, called the parallel transport of v₀ along c.

Analytically, the existence of parallel transport can be deduced from the transport equation

g_t g^–1 = ½ F_tF

on G = SO(3) with F(t)=2P(c(t)) – I. The transformation g trivialises the tangent bundle along c. Given a tangent vector v₀ in the tangent space to M at c(0), the vector v(t)=g(t)v₀ lies in the tangent space to M at c(t) and satisfies the equation

$P(c(t)){\dot {v}}(t)=0.$

It therefore is exactly the parallel transport of v along the curve c. In this case the length of the vector v(t) is constant. More generally if another initial tangent vector u₀ is taken instead of v₀, the inner product (v(t),u(t)) is constant. The tangent spaces along the curve c(t) are thus canonically identified as inner product spaces by parallel transport so that parallel transport gives an isometry between the tangent planes.

Parallel transport is related to the covariant derivative $\nabla$, because the condition on the velocity vector ${\dot {v}}(t)$ may be rewritten in terms of the covariant derivative as

$\nabla _{\dot {c}}v=0$

the fundamental defining equation for parallel transport.

There is another kinematic way of understanding parallel transport and geodesic curvature in terms of "rolling without slipping or twisting". Although well known to differential geometers since the early part of the twentieth century, it has also been applied to problems in engineering and robotics. Consider the surface as a rigid body in three dimensional space rolling without slipping or twisting on a horizontal plane. The point of contact will describe a curve in the plane and on the surface. At each point of contact the different tangent planes of the surface can be identified with the horizontal plane itself and hence with one another.

• The usual curvature of the planar curve is the the geodesic curvature of the curve traced on the surface.
• This identification of the tangent planes along the curve corresponds to parallel transport.

This is particularly easy to visualize for a sphere: it is exactly the way a marble can be rolled along a perfectly flat table top.

The roles of the plane and the surface can reversed to provide an alternative but equivalent point of view. The surface is regarded as fixed and the plane has to roll without slipping or twisting along the given curve in the surface.

This geometric way of viewing parallel transport can also be directly expressed in the language of geometry. The envelope of the tangent planes to M along a curve c is a surface with vanishing Gaussian curvature, which by Minding's theorem, must be locally isometric to the Euclidean plane. This identification allows parallel transport to be defined, because in the Euclidean plane all tangent planes are identified with the space itself.

Such a lift was constructed above for the special case of the 2-sphere S² = SO(3) / SO(2).

In 1956 Kobayashi proved that:

• The invariant connection on the frame bundle of S² corresponds to the Riemannian connection on S².
• Under the Gauss map, the connection it induces on the frame bundle of M corresponds to the Riemannian connection on M.

In this case the Sasaki metric is a multiple of the unique biinvariant metric on SO(3), inherited from the standard metric on its double cover SU(2) = S³.

The pullback of the connection form on SO(3) under the Gauss map gives the connection form on the frame bundle of M corresponding to the Riemannian connection.

There is another simple way of constructing the connection form ω using the embedding of M in E³.

The tangent vectors e₁ and e₂ of a frame on M define smooth functions from E with values in R³, so each gives a 3-vector of functions and in particular de₁ is a 3-vector of 1-forms on E.

The connection form is given by

$\omega =de_{1}\cdot e_{2}$

taking the usual scalar product on 3-vectors.

Gauss-Codazzi equations

See also: Gauss-Codazzi equations

When M is embedded in E³, two other 1-forms ψ and χ can be defined on the frame bundle E using the shape operator. Indeed the Gauss map induces a K-equivariant map of E into SO(3), the frame bundle of S² = SO(3)/SO(2). The form ω is the pullback of one of the three right invariant Maurer-Cartan forms on SO(3). The 1-forms ψ and χ are defined to be the pullbacks of the other two.

These 1-forms satisfy further structure equations:

$\psi \wedge \theta _{1}+\chi \wedge \theta _{2}=0$

(symmetry equation)

$d\omega =\psi \wedge \chi$

(Gauss equation)

$d\psi =\chi \wedge \omega ,\,\,d\chi =\omega \wedge \psi$

(Codazzi equations)

The Gauss–Codazzi equations for χ, ψ and ω follow immediately from the Maurer-Cartan equations for the three right invariant 1-forms on SO(3).

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