# Connection (principal bundle)

In differential geometry , the relationship is a concept that can be used to explain parallel transport between the fibers of a principal bundle . In physics, such relationships are used to describe fields in the Yang-Mills theories .

## definition

Be a principal bundle with the structure group . The group works through ${\ displaystyle \ pi \ colon P \ rightarrow M}$ ${\ displaystyle G}$

${\ displaystyle R \ colon P \ times G \ rightarrow P}$.

Also denote the Lie algebra of the Lie group . ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle G}$

A relationship is then a -value 1-form , which is -equivariant and whose restriction to the fibers corresponds to the Maurer-Cartan form . So the following two conditions should be met: ${\ displaystyle {\ mathfrak {g}}}$${\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}$${\ displaystyle G}$

${\ displaystyle D (R_ {g}) \ omega = \ operatorname {Ad} (g ^ {- 1}) (\ omega)}$ for all ${\ displaystyle g \ in G}$

and

${\ displaystyle \ omega (X ^ {\ sharp}) = X}$for everyone .${\ displaystyle X \ in {\ mathfrak {g}}}$

Here is defined by . denotes the differential of . is the adjoint effect and is the so-called fundamental vector field . It will go through ${\ displaystyle R_ {g} \ colon P \ to P}$${\ displaystyle R_ {g} (p) = R (p, g)}$${\ displaystyle D (R_ {g})}$${\ displaystyle R_ {g}}$${\ displaystyle \ operatorname {Ad} \ colon G \ to GL ({\ mathfrak {g}})}$${\ displaystyle X ^ {\ sharp}}$

${\ displaystyle X ^ {\ sharp} (p): = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ right | _ {t = 0} R _ {\ exp ( tX)} (p)}$ For ${\ displaystyle p \ in P}$

on defined. ${\ displaystyle P}$

## curvature

The curvature of a connected form is defined by

${\ displaystyle \ Omega = d \ omega + {\ tfrac {1} {2}} [\ omega \ wedge \ omega].}$

Here the commutator is Lie algebra valued differential forms by

${\ displaystyle [\ omega \ wedge \ eta] (v_ {1}, v_ {2}) = [\ omega (v_ {1}), \ eta (v_ {2})] - [\ omega (v_ {2 }), \ eta (v_ {1})]}$

and the outer derivative through ${\ displaystyle d \ omega}$

${\ displaystyle d \ omega (X, Y) = X (\ omega (Y)) - Y (\ omega (X)) - \ omega ([X, Y])}$

Are defined.

The curve form is -invariant and therefore defines a 2-form on . ${\ displaystyle G}$${\ displaystyle \ Omega \ in \ Omega ^ {2} (M, {\ mathfrak {g}})}$${\ displaystyle M}$

### Bianchi identity

Relationship and curvature form satisfy the equation

${\ displaystyle d \ Omega = \ left [\ Omega, \ omega \ right]}$.

## Horizontal subspaces

For a form of connection on a principal bundle , the horizontal subspaces are defined by ${\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}$${\ displaystyle G}$${\ displaystyle \ pi: P \ rightarrow M}$${\ displaystyle H_ {p}, p \ in P}$

${\ displaystyle H_ {p}: = ker (\ omega: T_ {p} P \ rightarrow {\ mathfrak {g}})}$.

The horizontal subspaces are transverse to the tangent spaces of the fibers of , and they are -invariant, i.e. H. for everyone . ${\ displaystyle \ pi}$${\ displaystyle G}$${\ displaystyle H_ {gp} = DR_ {g} (H_ {p})}$${\ displaystyle g \ in G, p \ in P}$

From the horizontal subspaces can form the connection recover (for identification of the tangent of the fiber ) by projection of along the tangent of the fiber. ${\ displaystyle {\ mathfrak {g}}}$${\ displaystyle T_ {p} P}$${\ displaystyle H_ {p}}$

## Parallel transport

For every path and every there is a path with and . (This follows from the existence and uniqueness theorem for ordinary differential equations .) ${\ displaystyle \ gamma: \ left [0.1 \ right] \ rightarrow M}$${\ displaystyle x \ in \ pi ^ {- 1} (\ gamma (0))}$${\ displaystyle {\ tilde {\ gamma}}: \ left [0,1 \ right] \ rightarrow P}$${\ displaystyle {\ tilde {\ gamma}} _ {x} (0) = x}$${\ displaystyle \ pi ({\ tilde {\ gamma}} _ {x}) = \ gamma}$

In particular, you have one through each way${\ displaystyle \ gamma: \ left [0.1 \ right] \ rightarrow M}$

${\ displaystyle P _ {\ gamma} (x) = {\ tilde {\ gamma}} _ {x} (1)}$

defined figure

${\ displaystyle P _ {\ gamma}: \ pi ^ {- 1} (\ gamma (0)) \ rightarrow \ pi ^ {- 1} (\ gamma (1))}$,

the so-called parallel transport along the way . ${\ displaystyle \ gamma}$

At one point , the holonomy group is defined as a subset of the diffeomorphisms of the fiber as follows. To a closed path with and one there is a clear elevation with and we define . The group that is for everyone is the holonomy group. ${\ displaystyle b \ in B}$${\ displaystyle F_ {b}: = \ pi ^ {- 1} (b)}$${\ displaystyle \ gamma: \ left [0,1 \ right] \ rightarrow B}$${\ displaystyle \ gamma (0) = \ gamma (1) = b}$${\ displaystyle x \ in F_ {b}}$${\ displaystyle {\ tilde {\ gamma}}}$${\ displaystyle {\ tilde {\ gamma}} (0) = x}$${\ displaystyle f _ {\ gamma} (x): = {\ tilde {\ gamma}} (1)}$${\ displaystyle f _ {\ gamma}}$${\ displaystyle \ gamma}$

## Riemannian connection

For a Riemannian manifold , the frame bundle is a principal bundle with the linear group . ${\ displaystyle M}$ ${\ displaystyle GL (n, \ mathbb {R})}$

Let be the matrix that by using a local base ${\ displaystyle A}$

${\ displaystyle A (v) = \ nabla _ {X} v}$

is defined, where the Levi-Civita context is, so is by ${\ displaystyle \ nabla}$

${\ displaystyle \ theta (X): = A}$

defines the Riemannian form of connection. It applies

${\ displaystyle \ theta \ in \ Omega ^ {1} (M, {\ mathfrak {g}} l (n, \ mathbb {R}))}$.