1. 6. Connections for Riemannian Manifolds and Gauge Theories
6.1 Introduction
6.2 Parallelism on Curved Surfaces
6.3 The Covariant Derivative
6.4 Components: Covariant Derivatives of the Basis
6.5 Torsion
6.6 Geodesics
6.7 Normal Coordinates
6.8 Riemann Tensor
6.9 Geometric Interpretation of the Riemann Tensor
6.10 Flat Spaces
6.11 Compatibility of the Connection with Volume- Measure or the Metric
6.12 Metric Connections
6.13 The Affine Connection and the Equivalence Principle
6.14 Connections & Gauge Theories: The Example of Electromagnetism
6.15 Bibliography

2. 6.1. Introduction Affine connection → Shape & curvature.
Gauge connection : Gauge theory.
Connections are not part of the differential structure of the manifold.
Amount of added structure:
Volume element < Connection < Metric

3. 6.2. Parallelism on Curved Surfaces
There is no intrinsic parallelism on a manifold.
Example: Parallelism on S2.
Parallel transport = Moving a vector along a curve without changing its direction
Direction of V at C depends on the route of parallel transport.
→ Absolute parallelism is meaningless.
Affine connection defines parallel transport.

4. 6.3. The Covariant Derivative
Let C be a curve on M with tangent
At point P, pick a vector
An affine connection then allows us to define a vector field V along C by parallel transport.
The covariant derivative U along U is defined s.t.
 V is parallel transported along C.
Let W be a vector defined everywhere on C. Then
where W (P →Q ) is W(P) parallel-transported to Q = C(λ+δλ ).

5. Reminder:
Lie dragging W along U requires the congruences of U & W around C.
→ LUW requires U & W be defined in neighborhood of C.
Parallel transporting W along U requires only values of U & W on C.
→ UW requires only U & W be defined on C.
Compatibility with the differential structure requires the covariant derivative to be a derivation (it satisfies the Leibniz rule) and additive in U.
Thus
A, B = tensors
Setting
we have

6. Under a change of parametrization λ → μ :
With
we have
Combining with the additivity
we have
where f, g are functions.
UW is a vector → the gradient W is a (11) tensor s.t.
(see Ex 6.1)
Caution:  itself is not a tensor since its not linear:

7. 6.4. Components: Covariant Derivatives of the Basis
Any tensor can be expressed as a linear combination of basis tensors.
The basis tensors for V are
= vector
Γkj i = Affine connection coefficients.
= Christoffel symbols for a metric connection
Thus, →
where

8. The parallel transport of V is then given by
Ex. 6.6

9. 6.5. Torsion A connection is symmetric iff (Ex. 6.8 )
In a coordinate basis, a connection is symmetric iff
The torsion T is defined by
→ T = 0 for symmetric connections
T is a (12) tensor (Ex.6.9)
The symmetric part of Γ is defined as
Torsion is usually neglected in most theories.
Ex.6.11

10. 6.6 Geodesics A geodesic parallel transport its own tangent U, i.e.,
( Geodesic eq. )
Setting
[ Geodesic = x i (λ) ]
we get
and
The geodesic eq. is invariant under the linear transform λ→ a λ + b.
λ is therefore an affine parameter. (see Ex.6.12)
Only symmetric part of Γ contributes to the geodesic eq.
→ Geodesics are independent of torsion.

11. Geometric effects of torsion :
Let U be the tangent at P of a geodesic C.
Let RP be the (n1)-D subspace of TP(M) consisting of vectors lin. indep. of U.
Construct a geodesic through P with tangent ξ RP .
Using Γ(S) , parallel transport U along ξ a small parameter distance εto point Q, i.e., (S) ξ U = 0.
Construct another geodesic C  with tangent U through Q.
C  will be roughly parallel to C.
A congruence of geodesics ‘parallel’ to U can be constructed around P in this manner.
ξ can now be transported along U in 2 ways:
Parallel transport
Lie dragging

12. By design:
Since
we have
By definition (§6.5), the torsion T is given by
U i ξj  both sides gives →
If ξ is parallel transported along U, →
since
i.e., the parallel transported ξ is ‘twisted’ by the torsion along the geodesics.

13. 6.7. Normal Coordinates
Each vector UTP(M) defines a unique geodesic CU (λ) with tangent U at P.
A point Q near P can be associated with the unique vector UTP(M) that moves P to Q by a parallel-transport of distance Δλ = 1 along CU (λ) .
The normal coordinates of Q , with P as the origin, are defined as the components { U j } of U wrt some fixed basis of TP(M) .
Thus, a normal coordinate system is a 1-1 map from M to TP(M)  Rn.
Since geodesics can cross in a curved manifold, different normal coordinate patches are required to cover it.
The map from TP(M) to M is called the exponential map. It is well-defined even when the geodesics cross.
A manifold is geodesically complete if the exponential map is defined for all UTP(M) and all PM.
Useful property: Γijk |P = 0 in normal coordinates.

14. in normal coordinates
Proof:
Let
Normal coordinates of Q a distance λ from P along geodesic CU(λ) are
so that →
 Q on CU(λ)
Geodesic eq. for CU(λ) in arbitrary coordinates is
 wrt normal coordinates {xi} ,
on CU(λ).
i.e.
 Q on CU(λ)
Since this must be satisfied by arbitrary U(P), we must have
Reminder:
In general,
for Q  P.

15. 6.8. Riemann Tensor The Riemann tensor R is defined by
Its components are or
R is a (13) tensor because it is a multiplicative operator containing no differential operations on its arguments :
f = function
( Ex.6.13 )
In coordinate basis:
( Ex.6.14a )
In non-coordinate basis with
where

16. → Rijkl is anti-symmetric in k & l, i.e.,
Also
Ex.6.14(c)
Bianchi identities:
In coordinate basis:
The number of independent components of Rijkl in an n-D manifold is
Ex.6.14(d)
Caution: Other definitions (with different signs & index orderings) of R exist.

17. 6.9. Geometric Interpretation of the Riemann Tensor
The parallel transport of A along U = d/dλ from P (0) to Q (λ) is
for λ → 0
for finite λ
Let V = d/dμ with [ U,V ] = 0 → λ & μ are good coordinates for a 2-D subspace.

18. since [ U,V ] = 0 →
λμ = ‘area’ of loop

19. Geodesic Deviation
Consider the congruence of geodesics CU defined by
Let ξ be a vector field obtained by Lie dragging ξ|P along U, i.e.,
(c.f. Ex.6.11) 
since
since

20. where
i.e., or
since
Geodesic deviation equation

21. 6.10. Flat Spaces
Definition: A manifold is flat if Euclid’s axiom of parallelism holds, i.e.,
The extensions of two parallel line segments never meet.
Hence
where U is any geodesics & ξ is Lie dragged by U.
The sufficient condition for this to hold is R = 0,
i.e., R is a measure of the curvature of the manifold.
Properties of a flat space:
Parallel transport is path-independent so that there is a global parallelism.
All TP(M) can be made identical (not merely isomorphic).
M can be identified with any TP(M).
Exponentiation can be extended throughout any simply-connected regions.
Ex.6.16 : Polar coordinates in En with R  0

22. 6.11. Compatibility of the Connection with Volume- Measure or the Metric
Compatibility issues arises when Γ & g or τ co-exist.
E.g., there are 2 ways to define the divergence of a vector field :
via covariant derivative
via volume n-form
Compatibility requires
 V
Ex 6.17a
which is satisfied iff or
Ex 6.17b
E.g., inner product should be invariant under parallel transport :
g & Γ compatible iff
Ex 6.18
i.e.,
metric connection

23. Ex 6.20 :
Show that
 If V is a Killing vector,

24. 6.12. Metric Connections   Ex 6.21-2 : In normal coordinates →
In which case, the number of independent components in R is

25. Ricci tensor :
Ex 6.23
Ricci scalar :
Bianchi’s identities →
Weyl tensor :
Every contraction between the indices of Cijkl vanishes.
Einstein tensor :
Empty space :
6 independent eqs
A geodesic is an extremum of arc length
Ex 6.24

26. 6.13. The Affine Connection and the Equivalence Principle
Γijk = 0 for flat space in Cartesian coordinates.
Γijk  0 for flat space in curvilinear coordinates.
Principle of minimal coupling ( between physical fields & curvature of spacetime)
= Strong principle of equivalence :
Laws of physics take the same form in curved spacetime as in flat spacetime with curvilinear coordinates.

27. 6.14. Connections & Gauge Theories: The Example of Electromagnetism
For an introduction to gauge theories, see Chaps 8 & 12 of
I.D.Lawrie, “A unified grand tour of theoretical physics”, 2nd ed., IoP (2002)
Basic feature of gauge theories : Invariance under a group of gauge transformations.
E.g., electromagnetism:
Variables: 1-form A
Gauge transformations: A → A + d f

28. Consider a neutral scalar particle with mass m governed by
Klein-Gordon eq.
with
Conserved probability current density
If  is a solution, so is
, where  is a constant.
i.e., the system is invariant under the gauge transformation
Special relativity: Lorentz transformations (flat spacetime + Cartesian coord).
Generalization to curvilinear coord introduces an affine connection.
Relaxation to non-flat connections → gravitational effects (general relativity)
Restriction to  = constant is equivalent to flat space + Cartesian coord.
Non-constant  → EM forces.

29. General gauge transformation:
Since e i  is a point on the unit circle in the complex plane,
the gauge transformation is a representation of the group U(1) on .
The geometric structure is a fibre bundle ( called U(1)-bundle ) with
base manifold M = Minkowski spacetime,
and typical fibre = U(1) = unit circle in C.
A gauge transformation is a cross-section of the U(1)-bundle. 
i.e.
is not invariant under the general gauge transformation.

30. Remedy is to introduce a gauge-covariant derivative D s.t.
&
is invariant under the general gauge transformation.
This is accomplished by a 1-form connection A s.t.
and 
so that
Thus
K.G. eq in an EM field with canonical momentum

31. Affine connection: preserves parallelism.
Connection A : preserves phase of gradient under gauge transformation.
Curvature introduced by an affine connection:
Curvature introduced by A:
Faraday tensor → or
Gauge transformation: 