In this post I discuss variational fields.

Set Up

Throughout, \((M,g)\) denotes a smooth Riemannian manifold.

Connections

The goal is to try to make sense of a directional derivative. We know how this makes sense on Euclidean space, and we’d like to generalize the important properties to Riemannian manifolds. We will denote \(\Gamma(TM)\) to be the collection of smooth vector fields on \(M\).

A connection on \(M\) is a bilinear map \(\nabla : \Gamma(TM) \times \Gamma(TM) \rightarrow \Gamma(TM)\) satisfying the following for every \(X,Y \in \Gamma(TM)\), \(f \in C^\infty(M)\):

(1) \(\nabla_{fX} Y = f \nabla_X Y,\)

(2) \(\nabla_X(fY) = X(f) \nabla_X(Y) + f \nabla_X Y.\)

A connection is said to be compatible with the metric if we have

\[X g(Y,Z) = g(\nabla_X Y, Z) + g(X, \nabla_Y Z).\]

This is analogous to the chain rule.

A connection is said to be torsion free if we have

\[\nabla_X Y - \nabla_Y X - [X,Y] = 0.\]

The Levi-Civita connection is the unique connection which is compatible with the metric and torsion free. The idea behind finding it is to use the above conditions and the non-degenerate quality of \(g\) to define a metric.

Curves

Let \(\gamma : [0,T] \rightarrow M\) be a smooth curve. We can define \(\gamma'(t) = \frac{d\gamma}{dt}(t).\) Notice this is an element in \(T_{\gamma(t)}M\) – it’s a vector. Thinking about the tangent bundle, we have that the curve \(\gamma\) induces a curve on the tangent bundle, which we’ll denote by \(\hat{\gamma} : [0,T] \rightarrow M\). This curve is defined by

\[\hat{\gamma}(t) := (\gamma(t), \gamma'(t)).\]

Recall our metric defines a norm via

\[\sqrt{g_x(v,v)} = \|v\|_x.\]

From the Levi-Civita connection, we know how to differentiate

We can define the length of a curve by

\[L(\gamma) := \int_0^T \|\gamma'(t)\|dt.\]

We can define the energy of the curve by

\[E(\gamma) := \frac{1}{2} \int_0^T \|\gamma'(t)\|_{\gamma(t)}^2 dt.\]

Parallel Transport

The idea here is that given a vector at a point, we wish to move the vector along the curve in such a way so that it remains parallel to the original vector. Let \(\gamma\) be a curve on the manifold. A vector field along \(\gamma\) is a mapping \(X : [0,T] \rightarrow TM\) so that \(X(t) \in T_{\gamma(t)}M\). The set \(\Gamma(\gamma*TM)\) will be the collection of smooth vector fields along a curve \(\gamma\). The idea now is that the Levi-Civita connection induces a connection along the curve, meaning it induces a way of differentiating along a curve. This connection is denoted by \(\frac{D}{dt}\). It satisfies the following properties:

(1) \(\frac{D}{dt}(fX) = f' X + f \frac{D}{dt}X,\)

(2) If \(X\) admits an extension to a smooth vector field \(Z\) on an open subset \(U\) of \(M\), then

\[\frac{D}{dt}X(t) = (\nabla_{\gamma(t)}Z)_{\gamma(t)}.\]

The proof for existence can be found in these notes.

A vector field is said to be parallel if \(\frac{D}{dt}X \equiv 0.\)

Induced Connection

Recall that if we have a smooth map \(\phi : N \rightarrow M\), we can define the pull back of a vector field \(X \in \Gamma(TM)\) by

\[\phi^*(X)(p) := X(\phi(p)).\]

The pushforward of a vector field \(Y \in \Gamma(TN)\) is defined by

\[\phi_*(Y)(p) = d\phi_p(Y(\phi^{-1}(p)))\]

when this makes sense.

We then denote \(\Gamma(\phi^*(TN))\) to be the set of smooth pulled back vector fields.

Given a smooth map \(\phi : N \rightarrow M\) with \(M,N\) smooth Riemannian manifolds, we get an induced connection which is a bilinear map \(\nabla^\phi : \Gamma(TN) \times \Gamma(\phi^*(TM)) \rightarrow \Gamma(\phi^*(TM))\) satisfying the following:

(1) \(\nabla^\phi_{fX}Y = f \nabla_X^\phi Y,\)

(2) \(\nabla_X^\phi(fY) = X(f) Y + \nabla_X^\phi(Y),\)

(3) If \(Y\) admits an extension to a vector field \(Z\) on an open subset \(U \subseteq M\) then

\[(\nabla_X^\phi Y)_p = (\nabla_{d\phi(X_p)}Z)_{\phi(p)}.\]

We observe that induced connections satisfy the following nice properties, similar to the Levi-Civita connection. Let \(X,Y \in \Gamma(TN)\) and $U,V \in \Gamma(\phi^*(TM)).$$ We have

(1) \(\nabla_X^\phi(\phi_*(Y)) - \nabla_Y^\phi(\phi_*(X)) - \phi_*([X,Y]) = 0,\)

(2) \(X g(U,V) = g(\nabla_X^\phi U, V) + g(U, \nabla_X^\phi V).\)

Variation

A variation of a curve \(\gamma : [0,T] \rightarrow M\) is a smooth map \(\Gamma : (-\epsilon, \epsilon) \times [0,T] \rightarrow M\) such that the following hold:

(1) \(\Gamma(s, \cdot)\) is a smooth curve,

(2) \(\Gamma(0, t) = \gamma(t)\).

This is a smooth function from a Riemannian manifold to a Riemannian manifold, so we can in particular induce a connection \(\nabla\) (not the Levi-Civita connection). We consider the following vector fields along \(H\):

(1) \(T := dH\left(\frac{d}{dt}\right),\)

(2) \(Y := dH\left(\frac{d}{ds}\right).\)

With this, we can define the variational vector field by

\[S := Y \mid_{s=0}.\]

This is a vector field defined along the curve \(\gamma\).