This post will be shorter than usual. I thought it might be fun to write up some neat small results that have come up in my classes. I'll start today by talking about Lie subalgebras and Lie subgroups.

Recall that we can use the group structure of a Lie group $G$ to define a product on $T_eG$ (the tangent space to the identity). We call $T_eG$ with this product structure the Lie algebra of $G$, and denote it $\g$. The Lie algebra encodes a lot of significant information about the group - the Baker-Campbell-Haussdorf formula lets us relate the group product to the Lie bracket (at least in the image of the exponential map).

A Lie subgroup $H \subseteq G$ is a Lie group $H$ along with an injective Lie group homomorphism $\iota:H \inj G$. The differential of this homomorphism gives us a map between their Lie algebras $d\iota: \h \to \g$. The image of $d\iota$ is a Lie subalgebra of $\g$ (i.e. a linear subspace which is closed under the Lie bracket). This gives us a nice way of associating Lie subalgebras of $\g$ to Lie subgroups of $G$.

A natural follow-up question to ask is whether this correspondence works the other way as well: given a Lie subalgebra $\h \subseteq \g$, does it necessarily come from a Lie subgroup $i:H \inj G$? It turns out that the answer is yes! The proof is pretty neat, and not too long, although that's largely because I'll use a powerful theorem without proof.

The general idea of the proof is fairly intuitive. We can view the subalgebra $\h \subseteq \g$ as a linear subspace of $T_eG$. Using left-multiplication, we can translate this subspace to get a subspace of $T_xG$ for all $x \in G$. Then, we can essentially "integrate up" these planes to get a submanifold which is tangent to these planes. To make this argument more formal, we will look at distributions, which are just assignments of planes to each point in a manifold.

A $k$-plane distribution on a manifold $M^n$ is a rank-$k$ subbundle of the tangent bundle $TM$. Explicitly, this means that for each point $x \in M$, we assign a $k$-dimensional subspace $\Delta_x \subseteq T_xM$, and we make these choices in a smooth way. We denote the distribution by $\Delta$. Now, suppose we have a submanifold $N \subseteq M$ such that for every $x \in N$, $\Delta_x$ is the tangent space to $N$ at $x$. In this case, we call $N$ an integral manifold of $\Delta$.

We call a distribution $\Delta$ involutive if for any vector fields $X,Y$ whose vectors all lie in $\Delta$, then the Lie bracket $[X,Y]$ also lies in $\Delta$. Frobenius' Theorem tells us that if a distribution is involutive, then we can find a unique maximal integral manifold passing through any point $x \in M$ (Frobenius' theorem is actually stronger than this, but this is enough for us). This is great for us!

We can use Frobenius' theorem to find our subgroup $H$. Suppose we have a Lie subalgebra $\h$. We can construct a distribution $\Delta$ be defining $\Delta_x := dL_x\h$. Since $\h$ is a Lie subalgebra, it is closed under the Lie bracket. So $\Delta$ is involutive. Thus, we can find a maximal integral manifold of $\Delta$ passing through the identity $e$. Suggestively, we'll call this submanifold $H$.

Now, we just need to show that $H$ is a subgroup. This sounds like it might be difficult, but there's actually a clever trick that makes it really easy!. Let $h \in H$. Consider the translated submanifold $h^{-1}H$. $h^{-1}H$ is an integral manifold of $h^{-1}\Delta$. Since we constructed $\Delta$ by left-translating a subspace of $T_eG$, $\Delta$ must be left-invariant. So $h^{-1}\Delta$ is just $\Delta$. Thus, $h^{-1}H$ is a maximal integral submanifold of $\Delta$. And since $h \in H$, $h^{-1}h = e \in h^{-1}H$. By uniqueness of maximal integral submanifolds, we conclude that $h^{-1}H = H$. Thus, $H$ is a subgroup.

These are notes for a presentation I had to give for my Riemannian Geometry class. They follow Chapters 11-17 of Milnor's book Morse Theory very closely.

Morse Functions

Let $M$ be a manifold and $f:M \to \R$ be a smooth function. A helpful example to keep in mind is the height function of a surface immersed in $\R^3$, the restriction of $f(x,y,z) = z$ to $M \subseteq \R^3$.

At a critical point, $f$ has a well-defined Hessian (second derivative)

We can give a nice coordinate-free expression for the Hessian as follows.

It's not obvious that this is bilinear, or even that it is well-defined (a priori, it could depend on the extension of $Y$). But we can show both of these facts with one neat computation. \[X \cdot (\tilde Y \cdot f)(p) - Y \cdot (\tilde X \cdot f)(p) = [\tilde X, \tilde Y] \cdot f (p) = (df)_p([\tilde X, \tilde Y]) = 0\] Therefore, $H(X,Y) = H(Y,X)$. So the Hessian is symmetric. It is clearly linear in the first argument. So the fact that it is symmetric shows that it is well-defined and bilinear.

For example, consider the height function on the torus $T^2$.

There are 4 critical points: one minimum, two saddle points, and one maximum. These correspond to the CW structure on the torus with one 0-cell, one 1-cells, and one 2-cell.

The Calculus of Variations

We want to treat the space of piecewise smooth paths on a manifold like an infinite-dimensional manifold. I won't make this idea fully formal (mostly because I don't know the full formalism), but we can use analogies with finite-dimensional manifolds to motivate some useful definitions related to this space of paths. By analogy to the finite-dimensional case, we will define tangent vectors to the space of paths.

The Path Space of a Smooth Manifold

Let $M$ be a smooth manifold and let $p$ and $q$ be two points of $M$. $p$ and $q$ are allowed to be the same point. We will denote the set of all piecewise-smooth paths from $p$ to $q$ in $M$ by $\Omega(M;p,q)$. If $p$ and $q$ are clear from context, we will just write $\Omega$. Later, we will topologize this space, but we don't need to worry about that yet.

Before defining the tangent space of $\Omega$ and critical points, we will review their definitions for finite-dimensional manifolds. Given a finite-dimensional manifold $M$, we can think of tangent vectors as the velocities of curves. Concretely, given a tangent vector $v \in T_pM$, we can always find a curve $c:(-\epsilon, \epsilon) \to M$ such that $c(0) = p$ and $c'(0) = v$. We can push forward tangent vectors along a map $\phi:M \to N$ by defining $d\phi_p (v) = (\phi \circ c)'(0)$. We say that $p$ is a critical point of $\phi$ if $d\phi_p = 0$, which is to say that $(\phi \circ c)'(0) = 0$ for all curves $c$ through $p$.

Now, we give analogous definitions on $\Omega$. To define tangent vectors on $\Omega$, we have to generalize the idea of a curve on a manifold $M$. We do so with the idea of a variation.

We can think of $\bar \alpha$ as a "smooth path" in $\Omega$. Its "velocity vector", $\dd {\bar \alpha} {u} (0)$ is the vector field $W$ along $\omega$ given by \[W(t) = \left.\ddo u \right|_{u=0} \bar \alpha(u)(t) = \left.\pd{\alpha(u,t)} u\right|_{u=0} \] Inspired by this, we define the tangent space to $\Omega$ at a path $\omega$ as follows.

We note that $\dd{\bar\alpha}{u}(0)$ is such a vector field. And given any such vector field, we can find an associated variation by setting \[\bar \alpha(u)(t) := \exp_{\omega(t)}(u W(t))\] Now that we have a definition of tangent vectors, we can define critical points.

The Energy of a Path

On Riemannian manifolds, we often want to talk about the lengths of paths. However, the length functional is kind of annoying because there's a square root involved. To get around this, we define an energy functional which is similar to length, but better behaved.

Recall that the arc-length of a curve from $a$ to $b$ is given by \[L_a^b(\omega) := \int_a^b \left\|\dd \omega t \right\|\;dt\] Using the Cauchy-Schwarz inequality, we can relate the length and energy of a curve. \[(L_a^b)^2 = \left(\int_a^b \left\|\dd \omega t\right\| \cdot 1\;dt\right)^2 \leq \left(\int_a^b \left\|\dd \omega t \right\|^2\;dt\right)\left(\int_a^b 1^2\;dt\right) = (b-a)E_a^b\] Suppose that $\gamma$ is a minimal geodesic with $\gamma(0) = p$ and $\gamma(1) = q$, and $\omega$ is any other path. Then (using the fact that $\|\dot \gamma\|$ is constant) \[E_0^1(\gamma) = \int_0^1 \|\dot\gamma\|^2\;dt = \|\dot\gamma\|^2 = L_0^1(\gamma)^2 \leq L_0^1(\omega)^2 \leq E_0^1(\omega)\] We can only have $L(\gamma)^2 = L(\omega)^2$ if $\omega$ is a reparameterization of a minimal geodesic from $p$ to $q$. And we can only have $L(\omega)^2 = E(\omega)$ if $\omega$ is parameterized proportional to arclength. Thus, we conclude that $E(\gamma) \leq E(\omega)$ with equality iff $\omega$ is a minimal geodesic. That means that the minima of the energy functional are the minimal geodesics from $p$ to $q$.

Now that we understand the minima of the energy functional, we turn to the critical points.

The Hessian of the Energy Functional at a Critical Path

To do Morse Theory, we need to talk about the Hessian of this functional. The Hessian will be a bilinear functional \[E_{**}:T\Omega_\gamma \times T\Omega_\gamma \to \R\] Note that we only define the Hessian at critical points of $E$ (that is, geodesics).

It's not obvious from this definition that this is actually well defined (i.e. that it depends only on $W_1$ and $W_2$, and not on the particular variation $\bar \alpha$ that you pick). It turns out that it is well-defined. We can see this using the second variation formula.

I won't prove this here, but it's a pretty straightforward computation given the first variation formula, and some other identities, which can be found in Milnor or Lee.

Jacobi Fields and the Null Space of $E_{**}$

Recall that a Jacobi field $J$ is determined by its initial conditions \[J(0), \frac{DJ}{dt}(0) \in TM_{\gamma(0)}\]

The proof is a fairly straightforward computation using the second variation formula.

The Morse Index Theorem

The proof is pretty involved, so we split it up into steps.

Thus, the index of $E_{**}$ equals the index of $E_{**}$ restricted to $T\Omega_\gamma(t_0, \ldots, t_k)$. This shows our claim that the index is finite, since $T\Omega_\gamma(t_0, \ldots, t_k)$ is finite-dimensional.

Now, we will prove the formula for the index. Let $\gamma_\tau$ be the restriction of $\gamma$ to $[0,\tau]$, and let $\lambda(\tau)$ be the index of the associated Hessian $(E_0^\tau)_{**}$. We are going to show a formula for $\lambda(1)$.

This finishes our proof of the Morse Index Theorem.

A Finite-Dimensional Approxmination to $\Omega^C$

Finally, we will put a topology on $\Omega$. Let $\rho$ denote $M$'s topological metric which is induced by its Riemannian metric.

For convenience, we will write the manifold of broken geodesics $\Int \Omega(t_0, \ldots, t_k)^c$ as $B$. Let $E':B \to \R$ be the restriction of the energy functional to $B$.

The Topology of the Full Path Space

The topology we put on $\Omega$ is kind of weird. A more natural topology for this space is the so-called "compact open topology", in which a sequence of functions converges whenever it converges uniformly on every compact subset of the domain. An equivalent description of this topology is that it is induced by the metric \[d^*(\omega, \omega') = \max_t \rho(\omega(t), \omega'(t))\]

I will describe the proof here. Let $a_0 < a_1 < \cdots$ be a sequence of real numbers which are not critical values of the energy functional $E$. Pick the numbers so that each interval $(a_i, a_{i+1})$ contains exactly one critical value. Now, consider the sequence \[\Omega^{a_0} \subset \Omega^{a_1} \subset \Omega^{a_2} \subset \cdots\]

We see that each $\Omega^{a_{i+1}}$ is homotopic to $\Omega^{a_i}$ with a finite number of cells attached, corresponding to the finitely many geodesics in $E^{-1}((a_i, a_{i+1}))$. So we can construct a sequence of $CW$ complexes \[K_0 \subset K_1 \subset K_2 \subset \cdots\] such that for each $i$, we have a homotopy equivalence $\Omega^{a_i} \to K_i$. We can take a direct limit to get a map $f:\Omega \to K$. Clearly $f$ induces isomorphisms of homotopy groups in every dimension. Since $\Omega$ is homotopy equivalent to a CW complex, it follows by Whitehead's theorem that $f$ is a homotopy equivalence.