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**Kent E. MORRISON**

Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, USA

E-mail: [email protected]

- *Corresponding Author:
- Kent E. MORRISON

Department of Mathematics

California Polytechnic State University

San Luis Obispo, CA 93407, USA

**E-mail:**[email protected]

**Received date:** February 19, 2009; **Revised date:** March 06, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Let G be a Lie group with Lie algebra g. On the trivial principal G-bundle over g there is a natural connection whose curvature is the Lie bracket of g. The exponential map of G is given by parallel transport of this connection. If G is the dieomorphism group of a manifold M, the curvature of the natural connection is the Lie bracket of vectorelds on M. In the case that G = SO(3) the motion of a sphere rolling on a plane is given by parallel transport of a pullback of the natural connection by a map from the plane to so(3). The motion of a sphere rolling on an oriented surface in R3 can be described by a similar connection.

Samelson [6] has shown that the covariant derivative of a connection can be expressed as a Lie bracket. It is the purpose of this article to show that the Lie bracket of a Lie algebra can be expressed as the curvature form of a natural connection. Although it is plausible that this natural connection has been described before or is known as mathematical folklore, it does not appear in the standard references (e.g., Kobayashi and Nomizu [4, 5] or Sharpe [7])..

The setting for this result is the following. Let *π: P → X* be a right principal *G*-bundle with the Lie group *G* as the structure group. A connection on* P *is a smooth G-equivariant distribution of horizontal spaces in the tangent bundle *TP * complementary to the vertical tangent spaces of the bers. The curvature of a connection is a g-valued 2-form on the total space P. A good reference is Bleecker's book [2].

Let *G* be a Lie group and g its Lie algebra. Let be the total space of the trivial right principal *G*-bundle with projection P → g : The right action of *G* on *P * is given by (*x; h) g = (x; hg)* and let R_{g} denote the action of *g* on *P*; that is, R_{g} : P → P : Let 1 G be the identity element and let be the identity section of the bundle.

Let and 'be the left and right multiplication by g. The context should make it clear whether R_{g} is acting on* P* or on *G*. The adjoint action of *G* on g is the derivative at the identity of the conjugation action of *G* on itself. That is,

Finally, we recall the denition of a fundamental vectoreld on a principal *G*-bundle *P*. For be the vectoreld on *P* whose value at *p* is given by

The vectorfield is called a fundamental vectorfield. In the case that *P* is the trivial bundle it is routine to check that

**Theorem 1.1**. *There is a natural connection on the trivial bundle whose local curvature 2-form (with respect to the identity section ) is the constant g-valued 2-form on g whose value on a pair of tangent vectors is the Lie bracket*

**Proof.** We dene the connection on *P* by choosing the horizontal spaces. For (*x; g*) *P* let H_{(x;g)} be the subspace of T _{(x;g)}*P* dened by

It is easy to check that the distribution is right equivariant and that *H*_{(x;g)} is complementary to the vertical tangent space at (*x; g*).

Let α be the g-valued 1-form on *P* dening this connection. It is characterized by having the horizontal space H_{(x;g)} as the kernel of (*x;g*) and by satisfying the conditions

From these properties one can check that α is dened by

The curvature of a connection is the *g*-valued 2-form on the total space *P* dened by

Pulling α back to g by gives the local connection 1-form

The local curvature form Ω is then

Computing we see that

Hence, is a constant form, andWe evaluate[, ] on a pair of tangent vectors

Therefore, the local curvature form Ω is the constant g-valued 2-form that maps a pair of tangent vectors which are just elements of g, to the Lie bracket of and

For a Lie group homomorphism be the associated Lie algebra homomorphism. (Note that when the Lie algebras are viewed as the tangent spaces at the identities of the groups.) Then the map

is a morphism of principal bundles, which means that it commutes with the right actions of *G _{1}* and

Furthermore, it is a morphism that preserves the horizontal spaces of the natural connections dened in Theorem 1.1. More precisely, maps the horizontal subspaces in to the horizontal subspaces in ) as follows. Let

which is an element of the horizontal space at

We also note that for a composition of Lie group homomorphisms the principal bundle map and thatis the identity on the principal bundle for the identity Hence, we have proved the following theorem

**Theorem 1.2**. *There is a covariant functor F rom the category of Lie groups to the category of principal bundles with connection, which is dened on objects so that F(G) is the trivial principal bundle with its natural connection and dened on morphisms by*

Next we consider the relationship between the connection 1-forms of the two bundles.

**Proposition 1.3**.* The following diagram commutes:*

Equivalently,

**Proof**. Starting with the composition we have

**Proposition 1.4**.* The following diagram commutes:*

**Proof.** Because is linear, for all Thus, for

Also, the local curvature forms commute with the Lie algebra homomorphism

**Proposition 1.5**. Let be the local curvature 2-form for the natural connection on

**Proof**. This is simply restating the fact that is a homomorphism of Lie algebras:

**Remark 1.6**. Let *G* be the dieomorphism group of a manifold *M* with g being the space of vectorelds on *M*. Although *G* is not, strictly speaking, a Lie group, the natural connection on still makes sense, and so the curvature of this connection is given by the Lie bracket of vectorfields.

Given a smooth curve parallel transport along c is horizontal lift *(c(t); g(t))* with initial condition *g(0)* = 1. Therefore, g is a solution of the dierential equation

which simply says that is in the horizontal subspace at the point *(c(t); g(t))*

* Theorem 1.7*. Let be an element of the Lie algebra and define Then parallel transport along c is given by

**Proof**. It suces to show that satises the dierential equation with initial condition g(0) = 1. The derivative of exp is given by

**Remark 1.8**. With this result there is another way to see that the Lie bracket is the curvature, since

is the innitesimal parallel transport around the parallelogram spanned by cand which, in turn, is the curvature tensor applied to and

**Corollary 1.9**. Let and be elements of and define be the horizontal lift of c with

**Proof**. Since the value of does not matter. Therefore, let = 0. By the theorem is the horizontal lift of *c* with The right-invariance of the horizontal spaces implies that

**Remark 1.10.** Parallel transport along c is also known as the time-ordered (or path-ordered) exponential of . One of the equivalent ways to dene the time-ordered exponential of a curve is to dene it as the solution of the dierential equation with this is the dierential equation dening parallel transport along c. To understand the use of the phrase \time-ordered exponential," we use a piecewise linear approximation to c starting at c(0) and consisting of the line segments connecting and Then by repeated use of the corollary g(t) is approximated by

Therefore, *g(t)* is the limit as *n* goes to innity of this product of exponentials, which are ordered according the parameter value.

**Remark 1.11**. The holonomy subgroup (at a point x_{0} in the base space) of a connection on a principal G-bundle is the subgroup of *G* consisting of the results of parallel transporting around closed curves starting and ending at x_{0}. The null holonomy group is the subgroup resulting from transporting around null-homotopic curves. By the Ambrose-Singer theorem [1] the Lie algebra of the null holonomy group is generated by the values of the curvature tensor. Now is simply connected and so the null holonomy group is the holonomy group and its Lie algebra is the derived algebra [; ]. Assuming G is connected, the holonomy group is the derived group of G.

**The special orthogonal group**

Let G be the rotation group SO(3) with g = (3), the Lie algebra of innitesimal rotations. We identify (3) with R^{3} in the standard way so that the vector *v* in R^{3} corresponds to the innitesimal rotation with axis v that goes counterclockwise with respect to an observer with v pointing towards him. The angular velocity is the magnitude of *v*. If *v* = (*v _{1}; v_{2}; v_{3}*), then the corresponding matrix in(3) is

Then so that the map is an isomorphism of Lie algebras

Given a curve parallel transport along c is given by the curve SO(3), which is the unique solution to the dierential equation

In other words, the innitesimal rotation at t has for its axis of rotation the vector One can visualize this as a sphere of radius 1 rotating at the head of a screw (with right-hand threads) that is tunneling through space following the trajectory given by the curve c. Note that the spherical head is not rigidly attached to the screw because the axis of rotation must be free to vary.

A variation of this natural connection can be used to describe the geometry of a sphere rolling without slipping on a horizontal plane. The plane is R^{2} embedded in R^{3} as the set of points and a sphere of radius 1 sits on top of the plane. Let be a smooth curve with initial point Roll the sphere along the curve *c* until it reaches the endpoint As the sphere rolls along, the point of contact is c(t) and the conguration of the sphere is given by a curve g(t) in SO(3). At each point c(t) the innitesimal rotation is about the axis where rotation counterclockwise dened by We can formulate the dierential equation satised by g(t) as

(Briefly from which the dierential equation follows.) In order to see this dierential equation as the parallel transport equation for the curve c with initial condition g(0) = I, we dene a connection on with horizontal space at the point (x; g) in given by

The connection 1-form is dened by

Let 'be the local connection 1-form. Since

we see that is constant, i.e., Then the local curvature 2-form acts on a pair of tangent vectors

For the last step note that because is an isomorphism of Lie algebras. Also, from the geometric properties of the cross product. Finally,

The derived algebra of is because each of the basis elements is a cross product. The group SO(3) is connected, and so, by the Ambrose-Singer theorem, the holonomy subgroup is all of SO(3). In other words, any rotation of a sphere can be achieved by rolling the sphere around some closed path in the plane. An elementary proof (without the apparatus of modern dierential geometry) of this old result has recently appeared [3].

The connection just dened is actually just a pullback of the natural connection on SO(3).

**Theorem 2.1**. Define

Then the connection on associated to the rolling sphere is the pullback byof the natural connection on

**Proof.** The pullback bundle of the trivial bundle is also trivial. Let f denote the bundle map Then it suces to compute , the pullback of the connection 1-form α on to see that it is the connection 1-form of the rolling sphere. At a point

Now more generally, we consider a sphere rolling on a surface in R^{3}. We will construct a connection on the trivial SO(3)-bundle over the surface whose parallel transport describes the rotation of the sphere. However, in this generality, the connection need not be a pullback of the natural connection on Let *X* be a smooth orientable surface in R^{3} and let n be the unit normal vectoreld pointing to the side on which the sphere rolls. Let J be the automorphism of the tangent bundle *TX* that rotates each tangent space counterclockwise with axis of rotation given by the unit normal n. With the natural identication of withAs compared with the planar surface it is now more complicated to describe the innitesimal rotation at a point in the direction because of the twisting and turning of the tangent spaces of the surface.

The unit normal vectoreld is a map The derivative of n at x is a linear map Differentiating the constant function shows thatshows that Therefore, and hencealso lies in Then the vector is the axis of the innitesimal rotation of the sphere at x in the direction v. We dene a connection on the trivial bundle whose horizontal space at (x; g) is

The connection 1-form α is given by

Let be the local connection 1-form. Thus,

When the surface X is itself a sphere, it is possible to explicitly compute the local curvature form. Let X be the sphere of radius r centered at the origin.Then n(x) = x/r, Dn(x)(v) = v/r, and v + Dn(x)(v) = (1 + 1/r)v. Recall that J_{x}(v) = n(x) v, and so

In order to compute the local curvature form in coordinates we use the isomorphism between R^{3} and so(3) in order to treat as an R^{3}-valued 1-form. Hence

With coordinates

Hence,

Evaluating on a pair of tangent vectors *u; v* gives

Next we evaluate Recall that is dened so that its value on a pair of tangent vectors *u* and *v* is

Therefore, In this case the bracket operation is the cross product and so

For the last step above we use the geometry of the cross product with x=r being a unit vector normal to both u and v to conclude that. Putting the pieces together we see that

As the radius goes to innity, the curvature form approaches the curvature form for the sphere rolling on a plane|as one would expect. For the sphere rolling on the outside of a sphere of radius 1, the curvature vanishes and so the connection is at and the holonomy is trivial around any null-homotopic path and hence around any closed path because S^{2} is simply connected. Fix a basepoint There is a global section of the trivial bundle mapping to the holonomy along any path from x_{0} to x. Antipodal points take the same value as can be seen by rolling the sphere along a great circle from the north to south pole. This section is an integral surface for the horizontal distribution of the connection. It is possible to describe this map in coordinates explicitly and it is especially nice using unit quaternions, i.e., S^{3}, to represent rotations. The quaternion q denes a rotation by mapping The Lie group homomorphism has kernel {1}g. In fact, S^{3} is the universal cover of SO(3) and this homomorphism is the projection. Using (0; 0; 1) for the basepoint, the holonomy map from S^{2} to SO(3) lifts to a map to S^{3} given by This represents the counterclockwise rotation through the angle about the axis (-y; x; 0). One can easily check that rolling the unit sphere from the north pole (0; 0; 1) to (x; y; z) does indeed turn the sphere through twice the angle between the two points and with axis that is normal to (x; y; 0).

To roll the sphere on the inside of a sphere of radius r simply change the unit normal to and follow the same calculations. The local curvature form turns out to be exactly the same

Although the curvature forms are equal, the connections are not the same and parallel transport is not the same. This can be seen easily for r = 1. Rolling inside the sphere produces no movement at all; the rolling sphere stays xed. Rolling the sphere on the outside does change the conguration along nonconstant paths.

The author would like to thank Jim Delany for helpful discussions about the rolling sphere and for Mathematica code to compute holonomy for the plane and spheres.

- Ambrose W,Singer IM (1953) A theorem on holonomy. Trans. Amer. Math. Soc 75:428-443.
- Bleecker D (1981) Gauge Theory and Variational Principles. Global Analysis Pure and Applied Series
- A 1, Addison-Wesley Publishing Co. Reading, Mass.
- Johnson BD (2007) Thenonholonomy of the rolling sphere. Amer. Math. Monthly 114: 500-508.
- Kobayashi S, Nomizu K (1996) Foundations of Differential Geometry. John Wiley & Sons,New York 1.
- Kobayashi S, Nomizu K (1996) Foundations of Differential Geometry. John Wiley & Sons,New York 2.
- Samelson H (1989) Lie bracket and curvature. Enseign. Math 35: 93-97.
- Sharpe RW (1997) Differential Geometry.Cartan’s Generalization of Klein’s Erlangen Program. Graduate
- Texts in Math.166, Springer-Verlag, New York.

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