# The Structure of Groups of Diffeomorphisms

## Geometry of the Euler equations

\dot v + \nabla_v v = -\nabla p
\operatorname{div}v = 0

Euler's equations describe Riemannian geodesics on

\operatorname{Diff}_{\mu}(M) = \{ \varphi\in\operatorname{Diff}(M) \mid |D\varphi| = 1 \}

## Riemannian metrics on Lie groups

V \in T_g G
g
\langle V,V\rangle_g =
?
G

## Riemannian metrics on Lie groups

V\cdot g^{-1} \in \mathfrak{g}
e
\langle V,V\rangle_g =
\langle V\cdot g^{-1},V\cdot g^{-1}\rangle_e
G

"Right-invariant" Riemannian metric determined by inner product on $$\mathfrak{g}$$

## Simple example: free rigid body

G = SO(3), \quad \mathfrak{g} = \mathfrak{so}(3)\simeq \mathbb{R}^3

Inner product: moments of inertia tensor

## Arnold's example: Euler equations

G = \operatorname{Diff}_\mu(M), \quad \mathfrak{g} = \mathfrak{X}_\mu(M) = \{ v\mid \operatorname{div}v=0 \}

Inner product:

\displaystyle\langle v,v\rangle_{L^2} = \int_M \lvert v \rvert^2 \mu

Arnold's theorem:  $$\gamma(t)\in \operatorname{Diff}_\mu(M)$$ geodesic curve $$\Rightarrow$$ vector field $$v(t) = \dot\gamma(t)\circ\gamma(t)^{-1}$$ fulfills Euler's equations

## Manifold structure of $$\operatorname{Diff}^s(M)$$

\operatorname{Diff}^s(M)
U \subset H^s
V \subset H^s

Thm [Palais, Omori, Ebin, Ebin and Marsden]

$$\operatorname{Diff}^s(M)$$ is smooth Hilbert manifold if $$s>\operatorname{dim}(M)/2+1$$

$$\operatorname{Diff}^s_\mu(M)$$ is a submanifold

## What about Lie group structure?

Thm [Ebin]

$$\operatorname{Diff}^s(M)$$ topological group if $$s>\operatorname{dim}(M)/2+1$$

Group structure:

(\varphi,\eta)\mapsto \varphi\circ \eta
\varphi\mapsto \varphi^{-1}

Failure of smoothness (illustration):
Compute  derivative of left translation $$L_\varphi(\eta) = \varphi\circ\eta$$

T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
H^s

## What about Lie group structure?

Thm [Ebin]

$$\operatorname{Diff}^s(M)$$ topological group if $$s>\operatorname{dim}(M)/2+1$$

Group structure:

(\varphi,\eta)\mapsto \varphi\circ \eta
\varphi\mapsto \varphi^{-1}

Failure of smoothness (illustration):
Compute  derivative of left translation $$L_\varphi(\eta) = \varphi\circ\eta$$

T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
H^s

## What about Lie group structure?

Thm [Ebin]

$$\operatorname{Diff}^s(M)$$ topological group if $$s>\operatorname{dim}(M)/2+1$$

Group structure:

(\varphi,\eta)\mapsto \varphi\circ \eta
\varphi\mapsto \varphi^{-1}

Failure of smoothness (illustration):
Compute  derivative of left translation $$L_\varphi(\eta) = \varphi\circ\eta$$

T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
H^s

## What about Lie group structure?

Thm [Ebin]

$$\operatorname{Diff}^s(M)$$ topological group if $$s>\operatorname{dim}(M)/2+1$$

Group structure:

(\varphi,\eta)\mapsto \varphi\circ \eta
\varphi\mapsto \varphi^{-1}

Failure of smoothness (illustration):
Compute  derivative of left translation $$L_\varphi(\eta) = \varphi\circ\eta$$

T_\eta L_\varphi\cdot V = D\varphi\circ\eta \cdot V
H^{s-1}

## "Fix" the Lie group structure

### Option 1

\operatorname{Diff}(M)
U \subset C^\infty
V \subset C^\infty

• Pros: proper Lie group structure
• Cons: no Picard iterations

### Option 2

Right translation $$R_\varphi(\eta) = \eta\circ\varphi$$

T_\eta R_\varphi\cdot V = V\circ\varphi
H^s

## "Fix" the Lie group structure

### Option 1

\operatorname{Diff}(M)
U \subset C^\infty
V \subset C^\infty

• Pros: proper Lie group structure
• Cons: no Picard iterations

### Option 2

Right translation $$R_\varphi(\eta) = \eta\circ\varphi$$

T_\eta R_\varphi\cdot V = V\circ\varphi
H^s

## "Fix" the Lie group structure

### Option 1

\operatorname{Diff}(M)
U \subset C^\infty
V \subset C^\infty

• Pros: proper Lie group structure
• Cons: no Picard iterations

### Option 2

Right translation $$R_\varphi(\eta) = \eta\circ\varphi$$

T_\eta R_\varphi\cdot V = V\circ\varphi
H^s

So, let's work only with right translation

## Simple example of how it works

\dot v - \dot v_{xx} + 3 v v_x - 2v_x v_{xx} - v v_{xxx} = 0

Camassa-Holm equation

A \dot v + v Av_x + 2 v_x A v = 0,
A = 1 - \partial_{xx}^2

### Why PDE not ODE?

Idea: [Ebin and Marsden] maybe geodesic equation on $$T\operatorname{Diff}^s(S^1)$$ is an ODE

H^{s-2}
H^{s-2}
H^{s-3}

## Camassa-Holm as geodesic equation

\operatorname{Diff}^s(S^1)
H^s(S^1)
v(t)
H^s(S^1) \ni v(t) \mapsto (\underbrace{v(t)\circ\varphi(t)}_{\dot\varphi(t)},\varphi(t))\in T\operatorname{Diff}^s(S^1)
\dot\varphi(t) = v(t)\circ\varphi(t)

## Camassa-Holm as geodesic equation

\operatorname{Diff}^s(S^1)
H^s(S^1)
\dot\varphi(t)
H^s(S^1) \ni v(t) \mapsto (\underbrace{v(t)\circ\varphi(t)}_{\dot\varphi(t)},\varphi(t))\in T\operatorname{Diff}^s(S^1)
\dot\varphi(t) = v(t)\circ\varphi(t)
\ddot\varphi = \dot v\circ\varphi + v_x\circ\varphi \, \dot\varphi
= (\dot v + v_x v)\circ\varphi = \Big( A^{-1}(-v Av_x - 2 v_x A v + A(v_x v)) \Big)\circ\varphi
= \Big( A^{-1}(-v Av_x + v A v_x + H^{s-2}\text{-terms}) \Big)\circ\varphi

## Camassa-Holm as geodesic equation

\operatorname{Diff}^s(S^1)
H^s(S^1)
\dot\varphi(t)
\dot\varphi(t) = v(t)\circ\varphi(t)
\ddot\varphi = \dot v\circ\varphi + v_x\circ\varphi \, \dot\varphi
= (\dot v + v_x v)\circ\varphi = \Big( A^{-1}(-v Av_x - 2 v_x A v + A(v_x v)) \Big)\circ\varphi
= \Big( A^{-1}(-v Av_x + v A v_x + H^{s-2}\text{-terms}) \Big)\circ\varphi

### Are we done?

No! RHS must be smooth as function of $$\varphi,\dot\varphi$$ but $$v=\dot\varphi\circ\varphi^{-1}$$

\underbrace{\phantom{gffdffd}}_{B(v,v)}

## Second piece of magic

\ddot\varphi = \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi)), \quad B:H^s \times H^s\to H^{s-2}
P: H^s \to H^{s-2}, \; \tilde P_\varphi: T_\varphi\operatorname{Diff}^s(S^1)\to T^{s-2}_\varphi\operatorname{Diff}^s(S^1)
\tilde P_\varphi V = (P(V\circ\varphi^{-1}))\circ\varphi

Geodesic equation, again

Lemma: Mapping $$T\operatorname{Diff}^s(S^1)\to T^{s-1}\operatorname{Diff}^s(S^1)$$ given by $(\varphi,\dot\varphi)\mapsto (\partial_x(\dot\varphi\circ\varphi^{-1}))\circ\varphi$ is smooth

## Second piece of magic

\ddot\varphi = \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi)), \quad B:H^s \times H^s\to H^{s-2}

Geodesic equation, again

Lemma: Mapping $$T\operatorname{Diff}^s(S^1)\to T^{s-1}\operatorname{Diff}^s(S^1)$$ given by $(\varphi,\dot\varphi)\mapsto (\partial_x(\dot\varphi\circ\varphi^{-1}))\circ\varphi$ is smooth

Proof:

\displaystyle (\partial_x (\dot\varphi\circ\varphi^{-1}))\circ\varphi = \dot\varphi_x((\partial_x\varphi^{-1})\circ\varphi) = \frac{\dot\varphi_x}{\varphi_x}

## Second piece of magic

\ddot\varphi = \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi)), \quad B:H^s \times H^s\to H^{s-2}

Geodesic equation, again

Lemma: Mapping $$T\operatorname{Diff}^s(S^1)\to T^{s-1}\operatorname{Diff}^s(S^1)$$ given by $(\varphi,\dot\varphi)\mapsto (\partial_x(\dot\varphi\circ\varphi^{-1}))\circ\varphi$ is smooth

Thm: Spray $$T\operatorname{Diff}^s(S^1)\to T^{s-2}\operatorname{Diff}^s(S^1)$$ given by $(\varphi,\dot\varphi)\mapsto \tilde A^{-1}_\varphi (\tilde B_\varphi(\dot\varphi,\dot\varphi))$ is smooth

By Klas Modin

# Groups of diffeomorphisms

Online-presentation given 2020-10 at Hebrew University Analysis Seminar.

• 1,087