Although Gauss was the first to study the differential geometry of surfaces in Euclidean space E3, it was not until Riemann's Habilitationsschrift of 1854 that the notion of a Riemannian space was introduced. Christoffel introduced his eponymous symbols in 1869. Tensor calculus was developed by Ricci, who published a systematic treatment with Levi-Civita in 1901. Covariant differentiation of tensors was given a geometric interpretation by Levi-Civita (1917) who introduced the notion of parallel transport on surfaces. His discovery prompted Weyl and Cartan to introduce various notions of connection, including in particular that of affine connection. Cartan's approach was rephrased in the modern language of principal bundles by Ehresmann, after which the subject rapidly took its current form following contributions by Chern, Ambrose and Singer, Kobayashi, Nomizu, Lichnerowicz and others.[citation needed]
Connections on a surface can be defined in a variety of ways. The Riemannian connection or Levi-Civita connection[9] 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 sections of the frame bundle. In the case of an embedded surface, this 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.[9] This gives an equivalent, more geometric way of describing the connection as lifting paths in the manifold to paths in the frame bundle. This formalises the classical theory of the "moving frame", favoured by French authors.[10] 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.[11]
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.[12]
For a surface M embedded in E3 (or more generally a higher-dimensional Euclidean space), there are several equivalent definitions of a vector fieldX on M:
a smooth map of M into E3 taking values in the tangent space at each point;
with a C∞(M)-valued inner product (X,Y), which encodes the Riemannian metric on M.
Since (M) is a submodule of C∞(M, E3)=C∞(M)E3, the operator XI is defined on (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. Thus for the unit normal vector np at p, uniquely defined up to a sign, and v in E3, the projection is given by P(p)(v) = v - (v · np) np.
Pointwise multiplication by P gives a C∞(M)-module map of C∞(M, E3) onto (M) . The assignment
defines an operator on (M) called the covariant derivative, satisfying the following properties
The first three properties state that 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
vanishes identically, so that the affine connection is torsion-free.
The assignment 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.
Its existence can be proved directly for a general surface by noting that the four properties imply the Koszul formula
so that depends only on the metric and is unique. On the other hand, if this is used as a definition of , it is readily checked that the four properties above are satisfied.[13]
For u an isometric embedding of M in E3, the tangent vectors and yield a matrix It is a positive-definite matrix. Its inverse is also positive-definite symmetric, with matrix . The inverse also has a unique positive-definite square root, with matrix . It is routine to check that form an orthonormal basis of the tangent space. In this case, the projection onto the tangent space is given by so that
Formulas for covariant derivative can be also be derived from local coordinates (x,y) without the use of isometric embeddings. Taking and ' as vector fields, the connection can be expressed purely in terms of the metric using the Christoffel symbols:[14]
To derive the formula, the Koszul formula can be applied with X, Y and Z set to 's; in that case all the Lie brackets commute.
The Riemann curvature tensor can be defined by covariant derivatives using the curvature operator:
Since the assignment is C∞(M)-linear in each variable, it follows that R(x,Y)p is an endomorphism at p. For X and Y linearly independent tangent vectors at p,
is independent of the choice of basis and is called the Gaussian curvature at p. The Riemann curvature tensor is given by[15][16]
To check independence of K it suffices to note that it does not change under elementary transformations sending (X,Y) to (Y,X), (λX,Y) and (X + Y,Y). That in turn relies on the fact that the operator R(X,Y) is skew-adjoint.[17] Skew-adjointness entails that (R(X,Y)Z,Z) = 0 for all Z, which follows because
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.[18]
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 kg(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 satisfies
This yields the previous rule for parallel transport along a geodesic, because in that case kg = 0, so the angle θ(t) should remain constant.[20] The existence of parallel transport follows from standard existence theorems for ordinary differential equations. The above differential equation can be rewritten in terms of the covariant derivative as
This equation shows once more that parallel transport depends only on the metric structure so is an intrinsic invariant of the surface. Parallel transport can be extended immediately to piecewise C1 curves.
When M is a surface embedded in E3, this last condition can be written in terms of the projection-valued function P as
The velocity vector of v must be normal to the surface.
Arnold has suggested[22][23] 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.[24]
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.[25] In fact 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.
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:
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 bundleF 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 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.[7] The definitions of the tangent bundle, the unit tangent bundle and the (oriented orthonormal) frame bundleF can be extended to arbitrary surfaces in the usual way.[7][15] There is a similar identification between the latter two which again become principal SO(2)-bundles. In other words:
There is also a corresponding notion of parallel transport in the setting of frame bundles:[26][27]
Every continuously differentiable curve in M can be lifted to a curve in F 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 parallelly 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 F for the Sasaki metric (see below).[28] 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).[29]
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.[30] 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's definition of a connection can be understood as a way of lifting vector fields on M to vector fields on the frame bundle F invariant under the action of the structure group K. Since parallel transport has been defined as a way of lifting piecewise C1 paths from
M to F, this automatically induces infinitesimally a way to lift vector fields or tangent vectors from M to F. At a point take a path with given tangent vector and then map it to the tangent vector of the lifted path. (For vector fields the curves can be taken to be the integral curves of a local flow.) In this way any vector field X on M can be lifted to a vector field X* on F satisfying[32]
X* is a vector field on F;
the map X ↦ X* is C∞(M)-linear;
X* is K-invariant and induces the vector field X on C∞(M) C∞(F).
Here K acts as a periodic flow on F, 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 F, the lifts X* span a two-dimensional subspace of horizontal vectors, forming a complementary subspace to the vertical vectors. The canonical Riemannian metric on F of Shigeo Sasaki is defined by making the horizontal and vertical subspaces orthogonal, giving each subspace its natural inner product.[28][33]
Horizontal vector fields admit the following characterisation:
Every K-invariant horizontal vector field on F has the form X* for a unique vector field X on M.
This "universal lift" then immediately induces lifts to vector bundles associated with F and hence allows the covariant derivative, and its generalisation to forms, to be recovered.
If σ is a representation of K on a finite-dimensional vector space V, then the associated vector bundle F xKV over M has a C∞(M)-module of sections that can be identified with
the space of all smooth functions ξ : F → V which are K-equivariant in the sense that
for all x ∈ F and g ∈ K.
The identity representation of SO(2) on R2 corresponds to the tangent bundle of M.
The covariant derivative is defined on an invariant section ξ by the formula
The connection on the frame bundle can also be described using K-invariant differential 1-forms on F.[7][34]
The orthonormal frame bundle F is a 3-manifold. One of the key facts about F is that it is (absolutely or completely) parallelizable, i.e. for n = dim F, there are n vector fields on F which form a basis at each point. As a result its Lie algebra is easy to understand; and the dual 1-forms on F have a particularly simple structure described by the Cartan structural equations discussed below.[35][36] In general it is known from Milnor & Stasheff (1974) that any orientable compact 3-manifold is parallelizable, although the proof is not elementary. For frame bundles, however, it is a straightforward consequence of the formalism of transition matrices between local trivializing charts.[37][38][39]
The space of p-forms on F is denoted Λp(F).[40] It admits a natural action of the structure group K.
Given a connection on the principal bundle F corresponding to a lift X ↦ X* of vector fields on M, there is a unique connection form ω in
,
the space of K-invariant 1-forms on F, such that[15]
for all vector fields X on M and
for the vector field A* on F corresponding to the canonical generator A of .
Conversely the lift X* is uniquely characterised by the following properties:
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.[44]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 rotation 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.
This geometric interpretation of curvature relies on a similar geometric of the Lie bracket of two vector fields on F. Let U1 and U2 be vector fields on F with corresponding local flows αt and βt.
Starting at a point A corresponding to x in F, travel along the integral curve for U1 to the point B at .
Travel from B by going along the integral curve for U2 to the point C at .
Travel from C by going along the integral curve for U1 to the point D at .
Travel from D by going along the integral curve for U2 to the point E at .
In general the end point E will differ from the starting point A. As s 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.[44]
To apply this theory, introduce vector fields U1, U2 and V on the frame bundle F which are dual to the 1-forms
θ1, θ2 and ω at each point. Thus
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:[44]
The geometrical interpretation of the Lie bracket can be applied to the last of these equations. Since ω(Ui)=0, the flows αt and βt in F 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 . So in the limit as s 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.[44]
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
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.[45]
One of the other standard approaches to curvature, through the covariant derivative , identifies the difference
as a field of endomorphisms of the tangent bundle, the Riemann curvature tensor.[15][46]
Since is induced by the lifted vector field X* on F, the use of the vector fields Ui 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 could also be added since it commutes with V and satisfies [W,U1] = U2 and [W,U2] = —U1.
Its tangent bundle T, unit tangent bundle U and oriented orthonormal frame bundle E are given by
The map sending (a,v) to (a, v, a x v) allows U and E to be identified.
Let
be the orthogonal projection onto the normal vector at a, so that
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 subgroupK of the vector (1,0,0) 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
Thus E becomes a principal bundle with structure group K. Taking the G-orbit of the point ((1,0,0),(0,1,0),(0,0,1)), 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 of SO(3) consists of all skew-symmetric real 3 x 3 matrices.[47] the adjoint action of G by conjugation on reproduces the action of G on E3. The group SU(2) has a 3-dimensional Lie algebra consisting of complex skew-hermitiantraceless 2 x 2 matrices, which is isomorphic to . 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.[48] 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 have the form exp Xt where X is a skew-symmetric matrix.
In this case the Sasaki metric agrees with this biinvariant metric on SO(3).[49][50]
The actions of G on itself, and hence on C∞(G) by left and right translation induce infinitesimal actions of on C∞(G) by vector fields
The right and left invariant vector fields are related by the formula
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.
The vector fields λ(A), λ(B), λ(C) form a basis of the tangent space at each point of G.
Similarly the left invariant vector fields ρ(A), ρ(B), ρ(C) form a basis of the tangent space at each point of G.
Let α, β, γ be the corresponding dual basis of left invariant 1-forms on G.[51] The Lie bracket relations imply the Maurer–Cartan equations
a left invariant matrix-valued 1-form on G, which satisfies the relation
The inner product on defined by
is invariant under the adjoint action. Let π be the orthogonal projection onto the subspace generated by A, i.e. onto , the Lie algebra of K. For X in , the lift of the vector field Π(X) from C∞(G/K) to C∞(G) is given by the formula
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 left invariant 1-form α is the connection form ω on G corresponding to this lift. The other two 1-forms in the Cartan structural equations are given by θ1 = β and θ2 = γ. The structural equations themselves are just the Maurer–Cartan equations. In other words;
The Cartan structural equations for SO(3)/SO(2) reduce to the Maurer–Cartan equations for the left invariant 1-forms on SO(3).
Since α is the connection form,
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 fi 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 can be accomplished by directly solving a matrix-valued ordinary differential equation ("transport equation") of the form gt = A · g where A(t) is skew-symmetric and g takes values in SO(3).[52][53][54]
In fact it is equivalent and more convenient to lift a path from SO(3)/O(2) to SO(3). Note that O(2) is the normaliser of SO(2) in SO(3) and the quotient group O(2)/SO(2), the so-called Weyl group, is a group of order 2 which acts on SO(3)/SO(2) = S2 as the antipodal map. The quotient SO(3)/O(2) is the real projective plane. It can be identified with space of rank one or rank two projections Q in M3(R). Taking Q to be a rank 2 projection and setting F = 2Q − I, a model of the surface SO(3)/O(2) is given by
matrices F satisfying F2 = I, F = FT and Tr F = 1. Taking F0= diag (–1,1,1) as base point, every F can be written in the form gF0g−1.
Given a path F(t), the ordinary differential equation , with initial condition , has a unique C1 solution g(t) with values in G, giving the lift by parallel transport of F.
If Q(t) is the corresponding path of rank 2 projections, the conditions for parallel transport are
Set A = ½FtF. Since F2 = I and F is symmetric, A is skew-symmetric and satisfies
QAQ = 0.
The unique solution g(t) of the ordinary differential equation
with initial condition g(0) = I guaranteed by the Picard–Lindelöf theorem, must have gTg constant and therefore I, since
Moreover,
since g−1Fg has derivative 0:
Hence Q = gQ0g−1. The condition QAQ=0 implies Qgtg−1Q = 0 and hence that Q0g−1gtQ0 =0.[55]
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.[56] 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 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.[57]
Embedded surfaces
When a surface M is embedded in E3, the Gauss map from MS2 extends to a SO(2)-equivariant map between the orthonormal frame bundles E SO(3). Indeed, the triad consisting of the tangent frame and the normal vector gives an element of SO(3).
Under the extended Gauss map, the connection on SO(3) induces the connection on E.
This means that the forms ω, θ1 and θ2 on E are obtained by pulling back those on SO(3); and that lifting paths from M to E can be accomplished by mapping the path to the 2-sphere, lifting the path to SO(3) and then pulling back the lift to E. Thus for embedded surfaces, the 2-sphere with the principal connection on its frame bundle provides a "universal model", the prototype for the universal bundles discussed in Narasimhan & Ramanan (1961).
In more concrete terms this allows parallel transport to be described explicitly using the transport equation. 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 is normal to the surface at c(t), i.e. P(c(t))v(t)=0.
This always has a unique solution, called the parallel transport ofv0alongc.
The existence of parallel transport can be deduced using the analytic method described for SO(3)/SO(2), which from a path into the rank two projections
Q(t) starting at Q0 produced a path g(t) in SO(3) starting at I such that
g(t) is the unique solution of the transport equation
gtg−1 = ½ FtF
with g(0) = I and F = 2Q − I. Applying this with Q(t) = P(c(t)), it follows that, 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
It therefore is exactly the parallel transport of v along the curve c.[53] 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. The condition on the velocity vector
may be rewritten in terms of the covariant derivative as[15][59]
the defining equation for parallel transport.
The kinematic way of understanding parallel transport for the sphere applies equally well to any closed surface in E3 regarded 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. As for the sphere, the usual curvature of the planar curve equals the geodesic curvature of the curve traced on the surface.
This geometric way of viewing parallel transport can also be directly expressed in the language of geometry.[60] 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.
There is another simple way of constructing the connection form ω using the embedding of M in E3.[61]
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.
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.[62][63][64] 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 the following structure equations:
(symmetry equation)
(Gauss equation)
(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).
^Arnold's method of approximation also applies to higher-dimensional Riemannian manifolds, after having given an appropriate geometric description of parallel transport along a geodesic. Parallel transport can be shown to be a continuous function on the Sobolev space of paths of finite energy, introduced in Klingenberg (1982). In this case the ordinary differential equation is solved by an integral which depends continuously on a as a varies through piecewise continuous or even just square integrable functions. The higher-dimensional case requires the transport equation gt = Ag and an extension of the analysis in Nelson (1969).
^The definition presented here is due essentially to Charles Ehresmann. However, it is different from, though related to, what is commonly called an Ehresmann connection. It is also different from, though related to, what is commonly called a Cartan connection. See Kobayashi (1957) and Sharpe (1997) for a survey of some of the various types of connections and the relations between them.
^A general connection on a principal bundle P with structure group H is described by a 1-form on P with values in invariant under the tensor product of the action of H on 1-forms and the adjoint action. For surfaces, H is Abelian and 1-dimensional, so the connection 1-form is essentially given by an invariant 1-form on P.
^The space of p-forms can be identified with the space of alternating p-fold C∞(F)-multilinear maps on the module of vector fields. For further details see Helgason (1978), pages 19–21.
^A similar argument applies to the transitive action by conjugation of SU(2) on matrices F = 2Q − I with Q a rank one projection in M2(C). This action is trivial on ± I, so passes to a transitive action of SO(3) with stabilizer subgroup SO(2), showing that these matrices provide another model for S2. This is standard material in gauge theory on SU(2); see for example Narasimhan & Ramadas (1979).
^Sharpe 1997, pp. 375–388, Appendix B: Rolling without Slipping or Twisting
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Choquet-Bruhat, Yvonne; Dewitt-Morette, Cécile; Dillard-Bleick, Margaret (1982), Analysis, manifolds and physics. Part I: Basics, North Holland, ISBN0-444-82647-5
Eisenhart, Luther P. (1947), An Introduction to Differential Geometry with Use of the Tensor Calculus, Princeton Mathematical Series, vol. 3, Princeton University Press
Ivey, Thomas A.; Landsberg, J.M. (2003), Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, ISBN0-8218-3375-8
Kobayashi, Shochichi (1957), "Theory of connections", Annali di Matematica Pura ed Applicata, Series 4, 43 (1): 119–194, doi:10.1007/BF02411907, S2CID120972987,
Kobayashi, Shoshichi; Nomizu, Katsumi (1963), Foundations of differential geometry, Vol. I, Wiley Interscience, ISBN0470496487
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Селище Френсіс-Крікангл. Francis Creek Координати 44°12′10″ пн. ш. 87°43′18″ зх. д. / 44.20280000002777854° пн. ш. 87.72190000002778731° зх. д. / 44.20280000002777854; -87.72190000002778731Координати: 44°12′10″ пн. ш. 87°43′18″ зх. д. / 44.20280000002777854° пн. ш. 87.72190000002778731° зх....
Implementation of reward in an organisation Reward management is concerned with the formulation and implementation of strategies and policies that aim to reward people fairly, equitably and consistently in accordance with their value to the organization.[1] Reward management consists of analysing and controlling employee remuneration, compensation and all of the other benefits for the employees. Reward management aims to create and efficiently operate a reward structure for an organis...
3.ª Super Copa IndiaIndia 2023 Sede India Fecha 3 de abril de 202325 de abril de 2023 Cantidad de equipos 21 Podio • Campeón• Subcampeón• Semifinalistas Odisha (1)BengaluruJamshedpurNorthEast United Partidos 31 Goles anotados 113 (3.65 por partido) Goleador Wilmar Jordán (7 goles) Diego Maurício (7 goles) La Super Copa de India 2023 fue la tercera edición de la Super Copa, la principal competición de fútbol de copa nacional en India. La competición estuvo ...
Indonesian television channel TVRI Kanal 3 redirects here. For the defunct Bulgarian TV channel, see Kanal 3. Television channel TVRI WorldCountryIndonesiaBroadcast areaNationwide, and Worldwide (planned) Southeast Asia, Middle East and Europe.AffiliatesSEA TodayHeadquartersJakarta, IndonesiaProgrammingLanguage(s)EnglishPicture format1080i HDTV (downscaled to 16:9 576i for the SDTV feed)OwnershipOwnerLPP TVRISister channelsTVRITVRI SportHistoryLaunched21 December 2010; 12 years a...
Brereton Hall, the seat of the Brereton family. Baron Brereton, of Leighlin in the County of Carlow, was a title in the Peerage of Ireland. It was created on 11 May 1624 for Sir William Brereton, of Brereton, Cheshire.[1][2] William Brereton was from an old and distinguished family in Cheshire,[3] and the family seat was Brereton Hall in Cheshire,[2] however Brereton had an estate near Old Leighlin, for which he and his heirs were absentee landlords. The first ...
Ne doit pas être confondu avec Championnat d'Italie féminin de rugby à XV 2023-2024. Championnat d'Italie de rugby à XV 2023-2024 Généralités Sport rugby à XV Organisateur(s) Federazione Italiana Rugby Édition 94e Date du 27 octobre au 26 mai Participants 9 équipes Statut des participants professionnel Site web officiel www.federugby.it Hiérarchie Hiérarchie 1re division Palmarès Tenant du titre Rovigo Delta Promu(s) en début de saison Vicenza Relégué(s) en début de saison CU...
Football tournament season 2015 Abkhazian CupTournament detailsCountry AbkhaziaDatesAugust 4,2015 - October 12, 2015Teams10Final positionsChampionsRitsa FC (1st title)Runner-upFC AfonTournament statisticsMatches played17Goals scored60 (3.53 per match)← 20142016 → The 2015 Abkhazian Cup was the 22nd edition of Abkhazian Cup organized by Football Federation of Abkhazia. The competition was held in the month of May.[1] Participating teams This ed...
Bus transit services operator in Metro Vancouver This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Coast Mountain Bus Company – news · newspapers · books · scholar...
El conocido como Puente de la Virgen es un puente de origen medieval, construido sobre el río Jabalón, se sitúa cerca de la Ermita de la Virgen de la Antigua, en Villanueva de los Infantes. El río Jabalón pasando por el puente, verano de 2016. Situación Geográfica El puente esta muy asociado a la familia Muñoz Triviño, con esta construcción Triviño se aseguraba el paso a sus vastas posesiones a la vez que le servía de la ostentación y propaganda de su poder, ya que el puente es p...
British Labour Party politician (1911–1982) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Tony Greenwood, Baron Greenwood of Rossendale – news · newspapers · books · scholar · JSTOR (September 2014) (Learn how and when to remove this template message) The Right HonourableThe Lord Greenwood of RossendaleP...
This article is about the newspaper. For the 2011 film, see The Oregonian (film). For other uses, see Oregonian. Daily newspaper published in Portland, Oregon, U.S. The OregonianTypeDaily newspaperFormatTabloid (since April 2, 2014)Owner(s)Advance Publications[1]PublisherOregonian Media Group[2][3]EditorTherese Bottomly[4]Staff writers288/75 (full-time/part-time)[5]Founded1850Headquarters1500 SW First Avenue[6]Portland, Oregon97201CirculationSun...
Lingkungan di Madinahالاحيا المدينة المنورة (bahasa Arab)Juga dikenal sebagai:LingkunganاحياKategoriPerkotaanLetakArab SaudiJumlah wilayah73 Lingkungan (hingga 2017)Penduduk1,183,205PemerintahanPemerintah Kota, dipimpin oleh Wali kotaPembagian administratifLingkungan Kota Madinah adalah kota di wilayah Provinsi Madinah, berada di bagian barat Arab Saudi, yang memiliki jumlah populasi total sebanyak 1.183.205 pada tahun 2010.[1] Madinah secara administratif terb...
TersanjungGenre Drama Roman PembuatTripar Multivision PlusDitulis oleh Deddy Armand Adi Nugroho Skenario Deddy Armand Adi Nugroho Sutradara Vasant R. Patel Revy Maghriza Pemeran Lulu Tobing Ari Wibowo Reynold Surbakti Adam Jordan Robby Sutara Robby Sugara Jeremy Thomas Feby Febiola Jihan Fahira Didi Riyadi Cut Tari Lagu pembuka Tersanjung — Retno Susanti feat. Ozy Syahputra [a] Tersanjung Diri Ini — Netta Margaretha [b] Ku Tersanjung — Retno Susanti [c] Oh! Ku Te...
2019 American filmThe Last VermeerTheatrical release posterDirected byDan FriedkinScreenplay by John Orloff Mark Fergus Hawk Ostby Based onThe Man Who Made Vermeersby Jonathan LopezProduced by Ryan Friedkin Dan Friedkin Bradley Thomas Vijay Waghmare Starring Guy Pearce Claes Bang CinematographyRemi AdefarasinEdited byVictoria BoydellMusic byJohan SöderqvistProductioncompanies TriStar Pictures Imperative Entertainment NL Film Distributed bySony Pictures ReleasingRelease dates August 31,&...
The Girl Who Heard Dragons First edition coverAuthorAnne McCaffreyCover artistMichael WhelanCountryUnited StatesLanguageEnglishGenreScience fictionPublisherTor BooksPublication dateMay 1994Media typePrint (hardback & paperback)Pages352 pp.ISBN0-312-93173-5 The Girl Who Heard Dragons is a 1994 collection of short fantasy and science fiction stories by the American-Irish author Anne McCaffrey.[1] It opens with an essay on her celebrity, or lack thereof, and includes 23 dra...
Town in New Hampshire, United StatesBoscawen, New HampshireTownHannah Duston statueLocation in Merrimack County and the state of New Hampshire.Coordinates: 43°18′54″N 71°37′15″W / 43.31500°N 71.62083°W / 43.31500; -71.62083CountryUnited StatesStateNew HampshireCountyMerrimackIncorporated1760Government • Select BoardMatthew T. Burdick, ChairLorrie J. CareyBill Bevans • Town AdministratorKatie PhelpsArea[1] • Total2...
British H-class destroyer For other ships with the same name, see HMS Harvester and HMS Handy. HMS Harvester after 1942 conversion to escort destroyer History Brazil NameJurua Ordered6 December 1937 BuilderVickers-Armstrongs, Barrow-in-Furness Laid down3 June 1938 FatePurchased by the United Kingdom, 5 September 1939 United Kingdom NameHMS Handy Launched29 September 1939 Acquired5 September 1939 Commissioned23 May 1940 RenamedHMS Harvester, January 1940 IdentificationPennant number: H19[1...
Species of flatworm Echinoplectanum chauvetorum Scientific classification Domain: Eukaryota Kingdom: Animalia Phylum: Platyhelminthes Class: Monogenea Order: Dactylogyridea Family: Diplectanidae Genus: Echinoplectanum Species: E. chauvetorum Binomial name Echinoplectanum chauvetorumJustine & Euzet, 2006 Echinoplectanum chauvetorum is a species of diplectanid monogenean parasitic on the gills of the black-saddled coralgrouper, Plectropomus laevis. It has been described in 2006. [1...