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Members

• Professor
• Academy research fellow
• Postdoctoral researcher Henri Hänninen
• Postdoctoral researcher Keijo Mönkkönen
• Postdoctoral researcher Hjørdis Schlüter
• PhD student David Johansson
• PhD student Antti Kykkänen
• PhD student Janne Nurminen
• PhD student Antonio Pop Gorea
• Previous members

Research

Examples of research topics in the field are given on the research highlights webpage of the .

Recent publications

Recent publications of the group may be found on the TUTKA database and on the . The most up-to-date information is available on the members' personal web pages.

Selected publications

• J. Ilmavirta: Coherent quantum tomography.
  SIAM Journal on Mathematical Analysis 48 (2016), no. 5, 3039–3064.
• G. Paternain, M. Salo and G. Uhlmann: Spectral rigidity and invariant distributions on Anosov surfaces.
  Journal of Differential Geometry 98 (2014), no. 1, 147-181.
• G. Paternain, M. Salo and G. Uhlmann: Tensor tomography on surfaces.
  Inventiones Mathematicae 193 (2013), no. 1, 229-247.
• C. Kenig, M. Salo and G. Uhlmann: Inverse problems for the anisotropic Maxwell equations.
  Duke Mathematical Journal 157 (2011), no. 2, 369-419.
• D. Dos Santos Ferreira, C. Kenig, M. Salo and G. Uhlmann: Limiting Carleman weights and anisotropic inverse problems. Inventiones Mathematicae 178 (2009), no. 1, p. 119-171.

Links

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme / ERC grant agreement no 307023 and the Horizon 2020 research and innovation programme / ERC grant agreement no 770924.

In Spring 2018 the inverse problems reading group will meet on Tuesdays at 14-15 in room MaD245. We will go through recent research articles that are of interest to the audience. The idea is that each participant presents an article written by others than him/herself.

Schedule

30.1.

Mikko Salo: presentation of possible articles

6.2.

No reading group

13.2.

Mikko Salo: Calderón problem with disjoint data (following )

20.2.

Jere Lehtonen: X-ray transforms on AH manifolds (following )

27.2.

No reading group

6.3.

Jere Lehtonen: X-ray transforms on AH manifolds (following )

13.3.

Giovanni Covi: Calderón problem for L^p potentials in 2D (following )

20.3.

Leyter Potenciano: Stability in the Calderón problem with partial data (following )

27.3.

No reading group (Easter break)

3.4.

Tommi Brander (DTU): Calderón's problem for variable exponent p(x)-Laplacian in one dimension

10.4.

No reading group

17.4.

Joonas Ilmavirta: Invisibility, X-ray tomography, and boundary regularity

24.4.

Cristobal Meronno (UAM): Shape identification and corner scattering

1.5.

No reading group

8.5.

Tony Liimatainen: The Calderón problem on transversally anisotropic manifolds (following )

15.5.

Keijo Mönkkönen: Abel transforms with low regularity with applications to X-ray tomography (following )

 Below are a few possible recent articles listed by topic (other papers can certainly be included).

Calderón type problems in R^n

Full data: , . Partial data: , , . Two-dimensional case: , , . Fractional equations: see references in . . .

Geometric inverse problems

Calderón type problems: ,, , , . Hyperbolic problems: , , . Boundary rigidity: , , . X-ray transforms: ,, , , .

Inverse scattering problems

, , , .

Current events

(12-14 December, Oulu)

(4-5 January 2018, Joensuu)

Reading group fall 2017

The reading group will discuss microlocal analysis and its applications. References for the basic theory (roughly in ascending order of difficulty):

• J. Wunsch:
• M. Wong: An introduction to pseudo-differential operators
• X. Saint-Raymond: Elementary introduction to the theory of pseudodifferential operators
• N. Lerner:
• G. Eskin: Lectures on linear partial differential equations
• A. Grigis, J. Sjöstrand: Microlocal analysis for differential operators - an introduction
• M. Shubin: Pseudodifferential operators and spectral theory
• M. Taylor: Partial differential equations, vol. II
• L. Hörmander: The analysis of linear partial differential operators, vol. III

References for the applications:

• DN map as a pseudodifferential operator: the flat case, (CPAM 1989),

(Please note that the notes below are not in final form and may be somewhat rough.)

26.09. Introduction to microlocal analysis (Mikko Salo)

10.10. Pseudodifferential symbols (Jesse Railo)

24.10. Pseudodifferential operators (Jere Lehtonen)

31.10. Elliptic pseudodifferential operators (Valter Pohjola)

07.11. The Dirichlet-to-Neumann map as a pseudodifferential operator (Tony Liimatainen)

14.11. The normal operator of the geodesic X-ray transform (Joonas Ilmavirta): and rough notes

21.11. L^2 boundedness of pseudodifferential operators (Giovanni Covi)

Reading group 2016-2017

(Please note that the notes below are not in final form and may be somewhat rough.)

10.10. Geodesic flows, notes 1: symplectic geometry

17.10. Geodesic flows, notes 2: geodesic flow as Hamilton flow

31.10. Geodesic flows, notes 3: symplectic and volume-preserving maps

07.11. Geodesic flows, notes 4: the Sasaki metric

14.11. Geodesic flows, notes 5: symplectic isoperimetric inequalities

21.11. Geodesic flows, notes 6: the Sasaki metric in local coordinates

06.02. Geodesic flows, notes 7: recap

13.02. Geodesic flows, notes 8: derivatives on the unit sphere bundle

20.02. Geodesic flows, notes 9: commutator formulas

06.03. Fractional Calderón problem, notes 1: motivation (slides from a colloquium talk)

14.03. Fractional Calderón problem, notes 2: introduction (slides from a seminar talk)

03.04. (section 2 of the paper in the link)

10.04.  (section 2 of the paper in the link)

18.04.  (section 2 of the paper in the link)

02.05.  (section 2 of the paper in the link)

Geodesic flows: notes 1

Geodesic flows, notes 1: symplectic geometry

Our first topic is to understand geodesics, first defined as curves \(\gamma(t)\), in terms of a Hamilton flow in the cotangent bundle. Hamilton flows belong to the realm of symplectic geometry. This is an area of mathematics which comes up in a variety of contexts, such as:

1. Historically, symplectic geometry arose from the study of classical mechanics. Consider the \(N\)-body problem, where \(N\) celestial objects (planets, stars etc) interact with each other gravitationally. If the objects have mass \(m_i\), position \(q_i(t) \in \mathbb{R}^3\), and momentum (or velocity) \(p_i(t) = \dot{q}_i(t) \in \mathbb{R}^3\), by Newton's laws one has the equations \( m_i \ddot{q}_i(t) = \sum_{j \neq i} \frac{G m_i m_j}{|q_i - q_j|^2}\) where \(G\) is the gravitational constant. These can be rewritten as the Hamilton equations \begin{align*} \dot{q}(t) &= \nabla_p H(q(t),p(t)), \\ \dot{p}(t) &= -\nabla_q H(q(t),p(t)) \end{align*} where \(H\) is the Hamilton function \(H = T + U\), with \(T\) the kinetic energy and \(U\) the self-potential energy (source: Wikipedia). The underlying symplectic structure makes it possible to consider these equations in many different (symplectic) coordinate systems, while still retaining the basic properties.

2. The geodesic flow on a Riemannian manifold \((M,g)\) can be understood as a Hamilton flow on the cotangent space \(T^* M\). This explains the facts that geodesics have constant speed, that geodesic flow preserves volume, and that the geodesic vector field on \(SM\) is divergence free and formally skew-adjoint.

3. If \(A(x,D)\) is a linear differential operator in an open set \(\Omega \subset \mathbb{R}^n\), so \(A(x,D) u = \sum_{|\alpha| \leq m} a_{\alpha}(x) D^{\alpha} u\) where \(a_{\alpha} \in C^{\infty}(\Omega)\), the principal symbol of \(A(x,D)\) is

\begin{align*} a(x,\xi) = \sum_{|\alpha| = m} a_{\alpha}(x) \xi^{\alpha}, \qquad (x,\xi) \in \Omega \times \mathbb{R}^n. \end{align*}

It turns out that under a change of coordinates, the principal symbol of \(A\) transforms like an invariant function on the cotangent bundle \(T^* \Omega\). Moreover, if \(A\) and \(B\) are two differential operators, their commutator \([A,B] = AB - BA\) has principal symbol \(\{a,b\}\) where \(\{\,\cdot\,, \,\cdot\,\}\) is the Poisson bracket on \(T^* \Omega\): \begin{align*} \{ a, b \}(x,\xi) = \nabla_{\xi} a \cdot \nabla_x b - \nabla_x a \cdot \nabla_{\xi} b. \end{align*} These basic facts indicate that symplectic geometry has a fundamental role in the study of linear partial differential equations.

4. The billiard map in the context of the broken geodesic ray transform preserves the related symplectic structure and hence volumes.

Symplectic manifolds

We proceed to explain some basic ideas in the setting of a general \(2n\)-dimensional symplectic manifold.

Definition. A symplectic manifold is a pair \((N, \sigma)\) where \(N\) is an \(2n\)-dimensional smooth manifold, and \(\sigma\) is symplectic form, that is, a closed \(2\)-form on \(N\) which is nondegenerate in the sense that for any \(\rho \in N\), the map \(I_{\rho}: T_{\rho} N \to T^*_{\rho} N, I_{\rho}(s) = \sigma(s, \,\cdot\,)\) is bijective.

Example 1. The space \(\mathbb{R}^{2n}\) has a standard symplectic structure given by the \(2\)-form \(\sigma = dx_1 \wedge \,dx_{n+1} + \ldots + dx_n \wedge \,dx_{2n}\).

Example 2. More generally, if \(M\) is an \(n\)-dimensional \(C^{\infty}\) manifold, then \(N = T^* M\) becomes a symplectic manifold as follows: if \(\pi: T^* M \to M\) is the natural projection, there is a \(1\)-form \(\lambda\) on \(N\) (called the Liouville form) defined by \(\lambda_{\rho} = \pi^* \rho, \rho \in T^* M\). Then \(\sigma = d\lambda\) is a closed \(2\)-form. If \(x\) are local coordinates on \(M\), and if \((x,\xi)\) are associated local coordinates (called canonical coordinates) on \(T^* M\), then in these local coordinates \begin{align*} \lambda &= \xi_j \,dx^j, \\ \sigma &= d\xi_j \wedge \,dx^j. \end{align*} It follows that \(\sigma\) is nondegenerate and hence a symplectic form.

It is a theorem of Darboux that if \((N,\sigma)\) is a symplectic manifold, then near any point of \(N\) there are local coordinates \((x,\xi)\) so that \(\sigma = d\xi_j \wedge \,dx^j\) in these coordinates. Hence every symplectic manifold is locally symplectically equivalent to \(\mathbb{R}^{2n}\), and all obstructions to having a symplectic structure are global in nature: for instance if \(N\) is a closed manifold having a symplectic form \(\sigma\), then the de Rham cohomology groups \(H^2(N)\) and \(H^{2n}(N)\) are nontrivial (this follows since the \(n\)-fold wedge product of \(\sigma\) is non vanishing, more about this later).

Hamilton flows

Definition. Let \((N,\sigma)\) be a symplectic manifold. Given any function \(f \in C^{\infty}(N)\), the Hamilton vector field of \(f\) is the vector field \(H_f\) on \(N\) defined by \[ H_f = I^{-1}(df) \] where \(df\) is the exterior derivative of \(f\) (a \(1\)-form on \(N\)), and \(I\) is the isomorphism \(TN \to T^* N\) given by the nondegenerate \(2\)-form \(\sigma\).

Example 1. Let \(M\) be an \(n\)-manifold, and let \(\sigma\) be the standard symplectic form on \(T^* M\). If \((x,\xi)\) are canonical local coordinates, then (using that \(\alpha \wedge \beta(v, w) = \alpha(v) \beta(w) - \alpha(w) \beta(v)\) for \(1\)-forms \(\alpha, \beta\)) \[ I_{x,\xi}(s^j \partial_{x_j} + t^j \partial_{\xi_j})(\tilde{s}^j \partial_{x_j} + \tilde{t}^j \partial_{\xi_j}) = (d\xi_i \wedge \,dx^i)(s^j \partial_{x_j} + t^j \partial_{\xi_j}, \tilde{s}^k \partial_{x_k} + \tilde{t}^k \partial_{\xi_k}) = \sum_{i=1}^n (t^i \tilde{s}^i - s^i \tilde{t}^i) \] which gives that \begin{align*} I(s^j \partial_{x_j} + t^j \partial_{\xi_j}) &= t^j \,dx_j - s^j \,d\xi_j \\ H_f &= \partial_{\xi_j} f \partial_{x_j} - \partial_{x_j} f \partial_{\xi_j}. \end{align*}

Example 2. In particular, in \( \mathbb{R}^{2n} \) one has \(I(s,t)=(t,-s)\), \(s, t \in \mathbb{R}^n\), and \[ H_f = \nabla_{\xi} f \cdot \nabla_x - \nabla_x f \cdot \nabla_{\xi}. \]

Definition. Let \((N,\sigma)\) be a symplectic manifold, and let \(f \in C^{\infty}(N)\). Denote by \(\varphi_t\) the flow on \(N\) induced by \(H_f\), that is, \[ \varphi_t: \rho(0) \mapsto \rho(t) \text{ where } \dot{\rho}(t) = H_f(\rho(t)). \]

We will later see that any Hamilton flow map is symplectic (\((\varphi_t)^* \sigma = \sigma\)) and consequently volume-preserving.

Geodesic flows, notes 2: geodesic flow as Hamilton flow

There are at least two ways to view the (co-)geodesic flow on a Riemannian manifold as a Hamilton flow.

Method 1: direct computation. Let \((M,g)\) be a Riemannian manifold, and recall the geodesic equation for a curve \(x(t)\), \[ \ddot{x}^l(t) + \Gamma_{jk}^l(x(t)) \dot{x}^j(t) \dot{x}^k(t) = 0. \] Consider the Hamilton function \[ f: T^* M \to \mathbb{R}, \ \ f(x,\xi) = \frac{1}{2} |\xi|_g^2 = \frac{1}{2} g^{jk}(x) \xi_j \xi_k. \] In canonical local coordinates \((x,\xi)\) on \(T^* M\), the Hamilton equations are given by \begin{align*} \dot{x}(t) &= \nabla_{\xi} f(x(t),\xi(t)), \\ \dot{\xi}(t) &= -\nabla_x f(x(t), \xi(t)). \end{align*} For \(f\) given above, this reduces to \begin{align*} \dot{x}^l &= g^{lk} \xi_k, \\ \dot{\xi}_k &= -\frac{1}{2} \frac{\partial g^{ij}}{\partial x_k} \xi_i \xi_j. \end{align*} Using that \( g_{jk} g^{kl} = \delta_j^l\), we have \( g_{jk} \partial_{x_m} g^{kl} = - g^{kl} \partial_{x_m} g_{jk} \) and multiplying by \(g^{ij}\) gives \( \partial_{x_m} g^{il} = - g^{ij} g^{kl} \partial_{x_m} g_{jk} \). Using that \(\xi_k = g_{km} \dot{x}^m\), the Hamilton equations are equivalent with \begin{align*} \dot{x}^l &= g^{lk} \xi_k, \\ \dot{\xi}_k &= \frac{1}{2} \partial_{x_k} g_{pq} \dot{x}^p \dot{x}^q. \end{align*}If \((x(t),\xi(t))\) satisfy the Hamilton equations, then \(x(t)\) satisfies the geodesic equation: \begin{align*} \ddot{x}^l = g^{lm} \dot{\xi}_m + \partial_p g^{lk} \dot{x}^p \xi_k = \frac{1}{2} g^{lm} \left[ \partial_m g_{pq}  - 2 \partial_p g_{mq} \right] \dot{x}^p \dot{x}^q = -\Gamma_{pq}^l \dot{x}^p \dot{x}^q. \end{align*} Conversely, if \(x(t)\) solves the geodesic equation and if  \(\xi_k(t) = g_{kl}(x(t)) \dot{x}^l(t)\), a similar computation shows that \((x(t),\xi(t))\) solves the Hamilton system with Hamilton function \(f(x,\xi) = \frac{1}{2} g^{jk}(x) \xi_j \xi_k\).

Method 2: Lagrangian/Hamiltonian formalism. Geodesics \(x(t)\) minimize length at least locally, and thus they are obtained as stationary points of the action integral \( I(x) = \int_a^b L(x(t), \dot{x(t)}) \,dt \) where \(L(x,v) = g_{jk}(x) v^j v^k \) is the Lagrangian function (arc length). Equivalently, one could take the kinetic energy \( T(x,v) = \frac{1}{2} g_{jk}(x) v^j v^k\) as the Lagrangian. The geodesic equation is the Euler-Lagrange equation of the action integral. There is a general procedure for converting such Euler-Lagrange equations into an equivalent Hamilton system where the Hamilton function \(E\) is the Legende transform of \(T\), \[ E(x,\xi) = \xi(v) - T(x,v) \] where \(v^j = g^{jk} \xi_k\). For geodesics, if \( T(x,v) = \frac{1}{2} g_{jk}(x) v^j v^k\), the Legendre transform is exactly \(E(x,\xi) = \frac{1}{2} g^{jk}(x) \xi_j \xi_k\). See M.E. Taylor, Partial differential equations vol. 1, section 1.12.

Notes 3:

\(\newcommand{\qed}{\square}\)

Let \((N,\sigma)\) be a symplectic manifold, let \(f \in C^{\infty}(N)\), and let \(H_f\) be the Hamilton vector field related to \(f\) defined by \[ \sigma(H_f, Y) = df(Y) \] or equivalently by the formula \( i_{H_f} \sigma = df \) where \(i_X\) is the interior product by a vector field \(X\), so \(i_X\) takes \(k\)-forms to \((k-1)\)-forms.

Recall the Hamilton flow \[ \varphi_t: N \to N, \rho(0) \mapsto \rho(t) \] where \(\rho(t)\) is a curve in \(N\) satisfying \(\dot{\rho}(t) = H_f(\rho(t))\). (Of course \(\varphi_t\) may only be defined in a subset of \(N\), but we will ignore this point.)

Recall that the Lie derivative is defined by \( \mathcal{L}_X \omega = \frac{d}{dt} \varphi_t^* \omega \Big|_{t=0}\) where \(\varphi_t\) is the flow of \(X\).

Lemma. \(\mathcal{L}_{X} \omega = 0\) if and only if \( \varphi_t^* \sigma = \sigma\).

Proof. If \( \varphi_t^* \sigma = \sigma\), then clearly \(\mathcal{L}_{X} \omega = 0\). Conversely, if \(\mathcal{L}_{X} \omega = 0\), it is enough to show that \(\varphi_t^* \omega\) is constant in \(t\) (this uses that \(\varphi_0^* \omega = \omega\)). Now, for any \(s\), one has \( \varphi_{t+s} = \varphi_t \circ \varphi_s \), and thus \[ \frac{d}{dt} \varphi_t^* \omega \Big|_{t=s} = \frac{d}{dt} \varphi_t^* (\varphi_s^* \omega) \Big|_{t=0} = \mathcal{L}_X (\varphi_s^* \omega). \] Now it follows from the definition of Lie derivative that \(\mathcal{L}_X (\varphi_s^* \omega)\) = \varphi_s^* (\mathcal{L}_X \omega)\), which vanishes by assumption. \(\qed\)

Theorem. \(\varphi_t\) is a symplectic map, that is, \(\varphi_t^* \sigma = \sigma\).

Proof. Using Cartan's magic formula, \[ \mathcal{L}_{H_f} \sigma = d(i_{H_f} \sigma) + i_{H_f}(d\sigma) = d(d\sigma) + i_{H_f}(d\sigma) = 0 \] since \(d^2 = 0\) and since \(\sigma\) is closed. \(\qed\)

Definition. If \((N,\sigma)\) is a symplectic manifold, the \(n\)-form \(\sigma^n = \sigma \wedge \ldots \wedge \sigma\) (\(n\) times) is called the (Liouville/symplectic) volume form of \((N,\sigma)\).

Clearly \(\sigma^n\) is a \(2n\)-form. It is also nonvanishing at any point, which follows from a routine calculation using the fact that \(\sigma\) is nondegenerate.

Theorem. (Hamilton flow preserves volume) \(\varphi_t^* (\sigma^n) = \sigma^n.\)

The previous theorem follows from the fact that \(\varphi_t^* \sigma = \sigma \), and from the basic differential geometry fact that \(F^*(\omega_1 \wedge \ldots \wedge \omega_n) = (F^* \omega_1) \wedge \ldots \wedge (F^* \omega_n)\). More generally, we say that a smooth map \(F: N \to N\) is symplectic if \(F^* \sigma = \sigma\), and it follows that any symplectic map preserves volumes. In particular, one has the inclusions \[ \{ \text{Hamilton flow maps} \} \subset \{ \text{symplectic maps} \} \subset \{ \text{volume-preserving maps} \}. \]

Let \(f \in C^{\infty}(N)\) as above, and let \(S\) be a level set of \(f\). Let \(X\) be any vector field on \(N\) which satisfies \(Xf|_S = 1\). Define \[ \sigma_S = j^* (i_X (\sigma^n)) \] where \(j: S \to N\) is the natural inclusion.

Example. Let \(N = T^* M\) with canonical symplectic form \(\sigma\). Let \(f(x,\xi) = \frac{1}{2} g^{jk}(x) \xi_j \xi_k\) be the Hamilton function which generates geodesic flow. The level set \(S = \{ f = 1/2 \}\) is just the unit cosphere bundle \(S^* M\).

Lemma. \(\sigma_S\) is independent of the choice of \(X\) (as long as \(Xf|_S = 1\)).

Proof. One has \(\sigma_S - \tilde{\sigma}_S = j^*( i_{X_1 - X_2} \sigma^n )\), where \((X_1-X_2)f = 0\) so \(X_1-X_2\) is tangent to \(S\). \(\qed\)

Note that the Hamilton flow maps satisfy \(\varphi_t: S \to S\), since the Hamilton function \(f\) is constant along its Hamilton flow. The maps \(\varphi_t\) also preserve the volume form of \(S\):

Lemma. \(\varphi_t^* (\sigma_S) = \sigma_S\).

Proof. TBD.

Geodesic flows, notes 4: the Sasaki metric

Largely based on Gabriel Paternain's book "Geodesic Flows".

Vertical and horizontal bundles

Let $\pi\colon TM\to M$ be the canonical projection. $ \newcommand{\vali}{(-\varepsilon,\varepsilon)} \newcommand{\id}{\operatorname{id}} \newcommand{\im}{\operatorname{im}} \newcommand{\Der}[1]{\frac{d}{d#1}} \newcommand{\Sa}{{\text{Sasaki}}} \newcommand{\ip}[2]{\left\langle#1,#2\right\rangle} \newcommand{\R}{{\mathbb R}} $

Definition. The vertical subbundle of $TTM$ is defined so that the fiber $V(\theta)$ at $\theta=(x,v)\in TM$ is the kernel of the differential of the projection. That is, $$V(\theta)=\ker(d_\theta\pi).$$ In other words, the vertical subbundle is given by tangents of curves $\sigma\colon\vali\to TM$ of the form $\sigma(t)=(x,v+tw)$ for $w\in T_xM$.

Definition. We define a connection map $K\colon TTM\to TM$ as follows. Let $\xi\in T_\theta TM$ and let $\sigma\colon\vali\to TM$ be a curve starting at $\theta$ in direction $\xi$. We write $\sigma(t)=(\alpha(t),Z(t))$, so that $\alpha$ is a curve on $M$ and $Z$ a vector field along it. We define $$K_\theta\xi=(\nabla_\alpha Z)(0).$$ Roughly, $K_\theta\xi$ is the covariant derivative of the second component of $\xi$ in the direction of the first component.

Definition. The horizontal bundle is given by the fibers $H(\theta)=\ker(K_\theta)$.

Discussion. Elements of $TTM$ describe directions of motion on $TM$. The vertical bundle corresponds to "vertical motion", motion only in the fiber, keeping the base fixed. The horizontal bundle corresponds to "horizontal motion", motion only in the base, keeping the tangent vector fixed. The vertical bundle is natural on a differentiable manifold. However, the horizontal bundle is based on a Riemannian metric; the meaning of "constant tangent vector" requires a connection.

Once we know that the dimensions of the fibers $V(\theta)$ and $H(\theta)$ are constant, we have really produced two subbundles of $TTM$.

Lifts and relations

We introduce a lift and use it to find relations between the horizontal and vertical bundle.

Definition. We define a horizontal lift $L_\theta\colon T_xM\to T_\theta TM$ as follows. Let $w\in T_xM$, and let $\alpha\colon\vali\to M$ be such that $\dot\alpha(0)=w$. Let $Z(t)$ be the parallel transport of $v$ along $\alpha$. Define a curve $\sigma\colon\vali\to TM$ by $\sigma(t)=(\alpha(t),Z(t))$. We define $$L_\theta w=\dot\sigma(0)\in T_\theta TM.$$ Roughly, $L_\theta(w)=(w,$parallel transport of $v$ in direction of $w)$.

Lemma.

1. $K_\theta\colon T_\theta TM\to T_xM$ and $L_\theta\colon T_xM\to T_\theta TM$ are well defined linear maps.
2. $\ker(K_\theta)=\im(L_\theta)$.
3. $d_\theta\pi\circ L_\theta=\id_{T_xM}$.
4. $d_\theta\pi|_{H(\theta)}\colon H(\theta)\to T_xM$ is a linear isomorphism.
5. $K_\theta|_{V(\theta)}\colon V(\theta)\to T_xM$ is a linear isomorphism.

Proof.

1. Clear.
2. Since $Z$ in the construction of $L_\theta$ was parallel, we have $\nabla_\alpha Z=0$. Thus $K_\theta\circ L_\theta=0$, which implies $\im(L_\theta)\subset\ker(K_\theta)$. For the other direction, let $\xi\in\ker(K_\theta)\subset T_\theta TM$. Let $\sigma\colon\vali\to TM$ be such that $\sigma(0)=\theta$, $\dot\sigma(0)=\xi$, and $\sigma(t)=(\alpha(t),Z(t))$. We can make changes to $Z$ as long as we leave its first order Taylor polynomial at $t=0$ intact. Since $\nabla_\alpha Z(0)=K_\theta\xi=0$, we change $Z$ so that $\nabla_\alpha Z(t)=0$ for all $t$. Now $Z$ is like in the definition of $L_\theta$, so $\xi=\dot\sigma(0)=L_\theta(\dot\alpha(0))\in\im(L_\theta)$.
3. Let $\alpha$ and $\sigma$ be as in the definition of $L_\theta$. Recall that $\sigma(0)=\theta$. We have $$ d_\theta\pi(L_\theta w) = d_\theta\pi(\dot\sigma(0)) = \Der{t}\pi(\sigma(t))|_{t=0} = \Der{t}\alpha(t)|_{t=0} = w. $$
4. It follows from the previous points that $L_\theta$ is a linear isomorphism $T_xM\to H(\theta)$. Therefore $d_\theta\pi|_{H(\theta)}$ is also a linear isomorphism.
5. For any $w\in T_xM$, let $\sigma_w(t)=(x,v+tw)\in T_xM$. By definition, $V(\theta)=\{\dot\sigma_w(0);w\in T_xM\}$. Since the base is fixed along these curves, $\pi\circ\sigma_w(t)=x$ for all $t$ and $w$, the covariant derivative is just the usual derivative. Thus $$K_\theta(\dot\sigma_w(0))=\Der{t}(v+tw)|_{t=0}=w.\quad\square$$

Corollary. $T_\theta TM=H(\theta)\oplus V(\theta)$ and $j_\theta\colon T_\theta TM\to T_xM\times T_xM$, $j_\theta(\eta)=(d_\theta\pi\eta,K_\theta\eta)$, is a linear isomorphism.

Definition. We define an inner product on $T_\theta TM$ so that $H(\theta)\perp V(\theta)$ and the maps $d_\theta\pi$ and $K_\theta$ are isometries. This is the Sasaki metric.

Alternatively, we may define the Sasaki metric by the formula $$ \ip{\eta}{\xi}_\Sa = \ip{(d_\theta\pi)\eta}{(d_\theta\pi)\xi} + \ip{K_\theta\eta}{K_\theta\xi}. $$ It is also common to leave $j_\theta$ implicit, so that $\eta=j_\theta(\eta)=(\eta_h,\eta_v)$. In this notation $$ \ip{\eta}{\xi}_\Sa = \ip{\eta_h}{\xi_h} + \ip{\eta_v}{\xi_v}. $$

In this notation the geodesic vector field is particularly simple: $X(x,v)=(v,0)$. This means that all movement is on the base, not on the fiber. (Geodesics parallel transport their tangents, so tangent vectors are "constant".)

Symplectic structure

This section contains no proofs due to expected lack of time in the seminar.

On $T^*M$ we have the canonical symplectic form $\omega=dx^i\wedge dp_i$. The Riemannian metric gives a natural diffeomorphism $g\colon TM\to T^*M$. Its differential gives a diffeomorphism $dg\colon TTM\to TT^*M$. We can pull back the symplectic structure from $T^*M$ to $TM$ using the metric. We define a 2-form $\Omega$ on $TM$ so that $$ \Omega_\theta(\eta,\xi) = \omega(dg\eta,dg\xi) $$ for all $\eta,\xi\in T_\theta TM$. This can be expressed neatly with the Sasaki metric: $$ \Omega_\theta(\eta,\xi) = \ip{\eta_h}{\xi_v}-\ip{\xi_h}{\eta_v} = \ip{J\eta}{\xi}_\Sa , $$ where $J\eta=J(\eta_h,\eta_v)=(-\eta_v,\eta_h)$. This $J$ is clearly an isometry of of the Sasaki metric and $J^2=-\id$.

Using this symplectic structure and the Hamiltonian $H(x,v)=\frac12g_{ij}(x)v^iv^j$, the Hamiltonian vector field turns out to be exactly the geodesic vector field. The Hamilton flow is the geodesic flow. This implies that the symplectic form $\Omega$ is preserved. The volume form corresponding to $\Omega$ coincides with Sasaki volume, so it is preserved, too.

Splitting of TSM

Perhaps this section should be discussed later when we discuss operators on $SM$.

Let $\iota\colon SM\to TM$ be the inclusion. The pullback bundle $\iota^*TTM$ is a bundle over $SM$ with the same fibers as $TTM$. $TSM$ is a subbundle of $\iota^*TTM$. We have a splitting on the fibers of $\iota^*TTM$ to horizontal and vertical directions, and we want to use this to find a natural splitting of $TSM$.

Let $\theta\in SM$. Now $\iota^*TTM|_\theta=H(\theta)\oplus V(\theta)$. We can define $H'(\theta)\subset H(\theta)$ as $H'(\theta)=(X(\theta))^\perp$. (Recall that $X(\theta)\in H(\theta)$.) We can define $V'(\theta)\subset V(\theta)$ as $V'(\theta)=K_\theta^{-1}(v^\perp)$. Now $T_\theta SM=\R X\oplus H'(\theta)\oplus V'(\theta)$, where the splitting is orthogonal. This allows us to decompose the gradient of a function $u\in C^\infty(SM)$ in a way that will be useful for the analysis of ray transforms.

Geodesian flows, notes 5: symplectic isoperimetric inequalities

If \(\Omega \subset \mathbb{R}^n\) is a bounded open set with smooth boundary, the isoperimetric inequality states that \[ \mathrm{Vol}_n(\Omega)^{\frac{n-1}{n}} \leq \frac{1}{n \omega_n^{1/n}} \mathrm{Vol}_{n-1}(\partial \Omega) \] with equality iff \(\Omega\) is a ball (here \(\omega_n = \mathrm{Vol}_n(B_1)\)). This inequality is equivalent with the (Gagliardo-Nirenberg-)Sobolev inequality \(W^{1,1}(\mathbb{R}^n) \subset L^{\frac{n}{n-1}}(\mathbb{R}^n)\) with sharp constant, \[ \lVert u \rVert_{L^{\frac{n}{n-1}}} \leq \frac{1}{n \omega_n^{1/n}} \lVert \nabla u \rVert_{L^1}. \] (One direction follows by applying the Sobolev inequality to a smoothed version of \(\chi_{\Omega}\), and the other direction involves the coarea formula and writing the \(L^{\frac{n}{n-1}}\) norm of a function in terms of super-level sets, see e.g. .)

Isoperimetric inequalities have been studied in many different geometries besides \(\mathbb{R}^n\) (sphere, Gaussian measure, Riemannian, ...). We intend to discuss a rather different looking variant, called a symplectic isoperimetric inequality, based on (the notions that appear will not be explained at this time):

Theorem. One has \[ d(L)^{n/2} \leq c_n \mathrm{Vol}_n(L)\] where \(L\) is a (\(n\)-dimensional) Lagrangian manifold in \(\mathbb{R}^{2n}\) and \(d(L)\) is the so called displacement energy of \(L\).

(The displacement energy \(d\) is a "special capacity" on open subsets of a symplectic manifold and it satisfies \(d(A) > 0\) for nonempty open \(A \subset \mathbb{R}^{2n}\), \(d(A) \leq d(B)\) if \(A \subset B\), and \(d(F(A)) = d(A)\) if \(F\) is a symplectic diffeomorphism. See Hofer-Zehnder, Symplectic invariants and Hamiltonian dynamics.)

Why is such an inequality called isoperimetric? This is related to celebrated work of Gromov on symplectic geometry in the 1980s. In particular, Gromov proved that if \(L\) is any closed Lagrangian manifold in \(\mathbb{R}^{2n} = \mathbb{C}^n\), there is a holomorphic map \(f: \overline{\mathbb{D}} \to \mathbb{C}^n\) (where \(\mathbb{D}\) is the unit disc in \(\mathbb{C}\)) which maps \(\partial \mathbb{D}\) into \(L\). A subsequent result of Chekanov shows (presumably, I did not have access to the paper) that such a map can be chosen so that \[ \mathrm{Area}(f(\overline{\mathbb{D}})) \leq d_n \mathrm{Vol}_n(L)^{2/n}. \] If \(n=1\), any \(1\)-manifold \(L\) in \(\mathbb{R}^2\) is Lagrangian, and if \(L\) is a smooth curve bounding a simply connected domain, presumably this is a version of the standard isoperimetric inequality in \(\mathbb{R}^2\). This is related to the (vague) idea of symplectization introduced by Arnold, which means in particular that for a statement related to submanifolds of a smooth manifold, there should be an analogous symplectic statement phrased in terms of Lagrangian manifolds.

Let us work out a consequence of the symplectic isoperimetric inequality, which implies a version of the Aleksandrov-Bakelman-Pucci maximum principle. This is an important tool in the Krylov-Safonov estimates (De Giorgi-Nash-Moser for nondivergence form equations), and these are fundamental for the regularity theory of fully nonlinear elliptic PDE.

Theorem. (ABP) If \(\Omega \subset \mathbb{R}^n\) is a bounded smooth domain, and if \(u \in C^2(\Omega) \cap C(\overline{\Omega})\), then \[ \sup_{\Omega} u \leq \sup_{\partial \Omega} u + C \left[ \int_{\Omega} |\det(D^2 u)| \,dx \right]^{1/n} \] where \(D^2 u = (\partial_{jk} u)_{j,k=1}^n\) is the Hessian. (It is important for applications that the integral can actually be taken over the subset of \(\Omega\) where \(u\) touches its concave envelope, but we will not worry about this.)

We derive this from the following variant of the symplectic isoperimetric inequality, also proved by Viterbo:

Theorem. If \(K \subset \mathbb{R}^n\) is compact, and if \(L\) is a Lagrangian manifold contained in \(T^* K\), then \[ \gamma(L)^n \leq c_{n,K} \mathrm{Vol}_n(L) \] where \(\gamma(L)\) is another symplectic invariant.

Let \(u \in C^2_c(\Omega)\), and define \[ L = \{ (x, \nabla u(x)) \,;\, x \in \Omega \}. \] This turns out to be a Lagrangian manifold in \(\mathbb{R}^{2n}\). To compute its (usual Riemannian) volume, we note that the tangent vectors of \(L\) are of the form \(v_j = (e_j, \partial_j \nabla u)\) and the metric on \(L\) (induced by the Euclidean metric on \(\mathbb{R}^{2n}\) is given by \[ g_{jk} = v_j \cdot v_k = \delta_{jk} + \partial_j \nabla u \cdot \partial_k \nabla u = \delta_{jk} + \sum_{l=1}^n (D^2 u)_{jl} (D^2 u)_{lk}. \] Thus \(g = \mathrm{I} + (D^2 u)^2\), and \[ \mathrm{Vol}_n(L) = \int_{\Omega} \,dV_L(x) = \int_{\Omega} |\det(g_{jk}(x))|^{1/2} \,dx = \int_{\Omega} \sqrt{\det(\mathrm{I} + (D^2 u)^2)} \,dx. \] It also turns out that for this \(L\), one has \(\gamma(L) = \max_{\Omega} u - \min_{\Omega} u\). Thus the symplectic isoperimetric inequality gives \[ (\max_{\Omega} u - \min_{\Omega} u)^n \leq c_{n,\Omega} \int_{\Omega} \sqrt{\det(\mathrm{I} + (D^2 u)^2)} \,dx. \] Since \(u|_{\partial \Omega} = 0\) we have \(\max u \geq 0\) and \(\min u \leq 0\), and thus replacing \(u\) by \(\lambda u\) and letting \(\lambda \to \infty\) gives that \[ \lVert u \rVert_{L^{\infty}} \leq C_{n,\Omega} \lVert \det(D^2 u) \rVert_{L^1}^{1/n}, \qquad u \in C^2_c(\Omega). \]

(A related paper on . Some references for ABP: Caffarelli-Cabre, Taylor PDE 3 (section 14.13), Gilbarg-Trudinger (section 9.1), .)

Geodesic flows, notes 6: geometry of the sphere bundle

Here we use the definitons and notations of notes 4.

Computations in local coordinates

Given a local coordinate chart \( (U,x) \) on \(M\), there are corresponding coordinates \((x,y)\) for \(TM\) and \((x,y,X,Y)\) for \(TTM\), so that if \(\xi \in T_{(x,y)}TM\), then $$ \xi = X^j \partial_{x^j} + Y^k \partial_{y^k}. $$

Definition.The vector field \(\delta_{x^j} \in T_{(x,y)}TM\) is defined as $$ \delta_{x^j} := \partial_{x^j} - \Gamma^l_{jk} y^k \partial_{y^l}, $$ where the \( \Gamma^l_{jk} \) are the Christoffel symbols.

Lemma 1. The vectors \(\{\delta_{x^j}\}_{j=1}^n\) are a basis for the subspace \(H(\theta)\), where \(\theta =(x_0,y_0)\).

Proof. We will show that $$ K_\theta(\delta_{x^j}) = 0, \quad \text{ and } \quad d_\theta\pi(\delta_{x^j}) = \partial_{x^j}. $$ The claim follows from this, since now \(\delta_{x^j} \in \ker(K_\theta)=H(\theta)\), and since \(d\pi_\theta \colon H(\theta) \to T_x M\) is an isomorphism, so that a basis vector maps to a basis vector.

To compute \(K_\theta\) we pick a curve \(\sigma \colon (-\epsilon,\epsilon) \to TM \), \( \sigma(t)=(x(t),y(t)) \), s.t. $$ \sigma(0) = (x_0,y_0), \quad \quad \dot \sigma(0) = (\dot x(0), \dot y(0)) = \delta_{x^j}. $$ Notice that \( \dot x(0)= e_j = (0,..,1,0,..,0) \) and \( \dot y(0) = (- \Gamma^1_{jk} y^k,..,-\Gamma^n_{jk} y^k) \).

Recall that the covariant derivative of the vector field \( Z(t) = Z^j(t) \partial_{x^j} \) along a curve \( \alpha(t) \) is defined as $$ \nabla_{\dot{\alpha}} Z = (\dot{Z}^k(t) + \dot{\alpha}^i Z^a \Gamma^k_{ia}) \partial_k. $$

And that by definition $$ K_\theta \delta_{x_j} = (\nabla_{\dot x} y) |_{t=0}. $$ Since \(\dot x(0)= e_j\), we have that \[ K_\theta \delta_{x_j} = ( \dot{y}^k + y^a \Gamma^k_{ja} ) \partial_{x^k} |_{t=0}. \] On the other hand we know that \(\dot y(0)= (- \Gamma^1_{jk} y^k,..,-\Gamma^n_{jk} y^k)\). Substituting this in the expression for \(K_\theta \delta_{x^j}\), gives then that $$ K_\theta \delta_{x_j} = ( - \Gamma^k_{ja} y^a + y^a \Gamma^k_{ja} ) \partial_{x^k} |_{t=0} = 0. $$ We thus see that \(\delta_{x^j} \in H(\theta)\).

It remains to show that \(d_\theta \pi(\delta_{x^j}) = \partial_{x^j}\). Since \(d_\theta \pi\) is a linear map, we need only show that $$ d_\theta \pi(\partial_{x^j}) = \partial_{x^j}, \quad d_\theta \pi(\partial_{y^k}) = 0. $$ This holds because firstly we have that $$ (d_\theta \pi(\partial_{x^j})) f = \partial_{x^j}(f\circ \pi) = \partial_{x^j} f, $$ And secondly that $$ d_\theta \pi(\partial_{y^k}) = \partial_{y^k}(f\circ \pi) = 0. \quad\square$$

Lemma 2. The vectors \(\{\partial_{y^k}\}_{k=1}^n\) are a basis for the subspace \(V(\theta)\), \(\theta =(x_0,y_0)\).

Proof. By similar but simpler computations as in Lemma 1 one sees that $$ d_\theta\pi(\partial_{y^k}) = 0 \quad \text{ and } \quad K_\theta(\partial_{y^k}) = \partial_{x^k}. $$ It follows that \(\partial_{x^k} \in \ker(d_\theta\pi)=V(\theta)\). Moreover, we had that \(K_\theta \colon V(\theta) \to T_x M\) is an isomorphism, so that basis vectors are mapped to basis vectors.

Recall that the Sasaki metric was defined by the formula $$ \langle \eta, \xi \rangle_{\text{Sasaki}} := \langle (d_\theta\pi)\eta, (d_\theta\pi)\xi \rangle + \langle K_\theta\eta, K_\theta\xi \rangle, $$ for \(\xi,\eta \in T_\theta TM\).

Given vectors \(\xi,\eta \in T_\theta TM\) and writing them in the basis given by Lemma 1 and Lemma 2, i.e. $$ \xi = X^i \delta_{x^i} + Y^k \partial_{y^k}, \quad \eta = \tilde X^i \delta_{x^i} + \tilde Y^k \partial_{y^k}, $$

we get that $$\langle \xi, \eta \rangle_{\text{Sasaki}} = \langle X^i \partial_{x^i}, \tilde X^i \partial_{x^i} \rangle + \langle Y^k \partial_{x^k}, \tilde Y^k \partial_{x^k} \rangle = g_{jk} X^j \tilde X^k + g_jk Y^j \tilde Y^k. $$

Geodesic flows, notes 7: recap

The section numbers correspond to the numbering of previous blog entries. $ \newcommand{\vali}{(-\varepsilon,\varepsilon)} \newcommand{\id}{\operatorname{id}} \newcommand{\im}{\operatorname{im}} \newcommand{\Der}[1]{\frac{d}{d#1}} \newcommand{\Sa}{{\text{Sasaki}}} \newcommand{\ip}[2]{\left\langle#1,#2\right\rangle} \newcommand{\R}{{\mathbb R}} $

0. The goal

The goal is to show that a function on a suitable Riemannian manifold $(M,g)$ is uniquely determined by its integrals over all maximal geodesics. To this end, we must understand the geometry of geodesics in great detail. The geodesic X-ray transform takes a function $f\colon M\to\R$ to a function $If\colon\Gamma\to\R$, where $\Gamma$ is the set of all geodesics on $M$. This $I$ is a linear integral transform, and it is known as the geodesic X-ray transform. The question is whether this transform is injective.

1. Symplectic manifolds and Hamilton flows

Definition. A symplectic manifold is a pair \((N, \sigma)\) where \(N\) is an \(2n\)-dimensional smooth manifold, and \(\sigma\) is symplectic form, that is, a closed \(2\)-form on \(N\) which is nondegenerate in the sense that for any \(\rho \in N\), the map \(I_{\rho}\colon T_{\rho} N \to T^*_{\rho} N, I_{\rho}(s) = \sigma(s, \,\cdot\,)\) is bijective.

Example 1. The space \(\mathbb{R}^{2n}\) has a standard symplectic structure given by the \(2\)-form \(\sigma = dx_1 \wedge \,dx_{n+1} + \ldots + dx_n \wedge \,dx_{2n}\).

Example 2. More generally, if \(M\) is an \(n\)-dimensional \(C^{\infty}\) manifold, then \(N = T^* M\) becomes a symplectic manifold as follows: if \(\pi\colon T^* M \to M\) is the natural projection, there is a \(1\)-form \(\lambda\) on \(N\) (called the Liouville form) defined by \(\lambda_{\rho} = \pi^* \rho, \rho \in T^* M\). Then \(\sigma = d\lambda\) is a closed \(2\)-form. If \(x\) are local coordinates on \(M\), and if \((x,\xi)\) are associated local coordinates (called canonical coordinates) on \(T^* M\), then in these local coordinates \begin{align*} \lambda &= \xi_j \,dx^j, \\ \sigma &= d\xi_j \wedge \,dx^j. \end{align*} It follows that \(\sigma\) is nondegenerate and hence a symplectic form.

A Riemannian metric is an isomorphism $TM\to T^*M$, so it can be used to give a natural symplectic structure on the tangent bundle of a Riemannian manifold.

Definition. Let \((N,\sigma)\) be a symplectic manifold. Given any function \(f \in C^{\infty}(N)\), the Hamilton vector field of \(f\) is the vector field \(H_f\) on \(N\) defined by \[ H_f = I^{-1}(df) \] where \(df\) is the exterior derivative of \(f\) (a \(1\)-form on \(N\)), and \(I\) is the isomorphism \(TN \to T^* N\) given by the nondegenerate \(2\)-form \(\sigma\).

Example 2. In \( \mathbb{R}^{2n} \) one has \(I(s,t)=(t,-s)\), \(s, t \in \mathbb{R}^n\), and \[ H_f = \nabla_{\xi} f \cdot \nabla_x - \nabla_x f \cdot \nabla_{\xi}. \]

Definition. Let \((N,\sigma)\) be a symplectic manifold, and let \(f \in C^{\infty}(N)\). Denote by \(\varphi_t\) the flow on \(N\) induced by \(H_f\), that is, \[ \varphi_t\colon \rho(0) \mapsto \rho(t) \text{ where } \dot{\rho}(t) = H_f(\rho(t)). \]

Any Hamilton flow map is symplectic (\((\varphi_t)^* \sigma = \sigma\)) and consequently volume-preserving.

2. The geodesic flow

The geodesic flow on a Riemannian manifold $(M,g)$ is a dynamical system on $T^*M$ (or $TM$, the two are naturally isomorphic via the Riemann metric). A geodesic is uniquely determined by its initial position and velocity. The tangent bundle $T^*M$ is a symplectic manifold, and the geodesic flow can be realized as a Hamilton flow. It is given by the Hamilton function $$ f\colon T^* M \to \mathbb{R}, \ \ f(x,\xi) = \frac{1}{2} |\xi|_{g^{-1}}^2 = \frac{1}{2} g^{jk}(x) \xi_j \xi_k. $$ The Hamiltonian equation of motion becomes exactly the geodesic equation. One can also see the geodesic flow from the Langrangian point of view, or geometrically via local length minimization.

4. The Sasaki metric

The tangent bundle of a smooth manifold is a smooth manifold (of double dimension). There is a canonical Riemannian metric on the tangent bundle of a Riemannian manifold. This is the Sasaki metric.

The tangent bundle describes possible directions of motion on $M$; each point on $TM$ contains a point $x\in M$ and a vector $v\in T_xM$. Similarly, $TTM$ describes the directions of motion on $TM$. It is natural to split motion in two components: motion within a fiber (vertically) or motion of the base point only (horizontally). This division is most clear when $M=\R^n$; then $TM=\R^{2n}$ and $TTM=\R^{4n}$.

For any $\theta=(x,v)\in TM$, we split $T_\theta TM=H(\theta)\oplus V(\theta)$. Note that if $\dim(M)=n$, then $\dim(H(\theta))=\dim(V(\theta))=\dim(T_xM)=n$. It turns out that there are natural isomorphisms $H(\theta)\to T_xM$ and $V(\theta)\to T_xM$.

Let $\pi\colon TM\to M$ be the canonical projection. The vertical fiber is then $V(\theta)=\ker(d_\theta\pi)$ — there is no movement in the base.

There is a connection map $K\colon TTM\to TM$. A point $\theta\in TTM$ describes (to first order) a curve on $TM$. This is a curve on $M$ and a vector field along it. The covariant derivative of this vector field along this curve is $K(\theta)\in T_xM$. The horizontal fiber is then $H(\theta)=\ker(K_\theta)$ — there is no movement in the fiber (parallel transport).

The natural isomorphisms are $d_\theta\pi|_{H(\theta)}\colon H(\theta)\to T_xM$ and $K_\theta|_{V(\theta)}\colon V(\theta)\to T_xM$. The Sasaki metric is obtained by declaring these to be isometries (inherit the metric from $T_xM$) and $H(\theta)\perp V(\theta)$. We can split any vector $TTM\ni\eta=(\eta_h,\eta_v)$. In this notation $$ \ip{\eta}{\xi}_\Sa = \ip{\eta_h}{\xi_h} + \ip{\eta_v}{\xi_v}. $$

6. Coordinate representations of the Sasaki metric

For any coordinates on $M$, there are corresponding coordinates on $TM$ given by the coordinate functions and their differentials. If $x$ denotes the coordinates on $M$, let $(x,y)$ be the corresponding coordinates on $TM$. Let also $(x,y,X,Y)$ be the corresponding coordinates on $TTM$.

The vectors \(\{\delta_{x^j}=\partial_{x^j} - \Gamma^l_{jk} y^k \partial_{y^l}\}_{j=1}^n\) are a basis for the subspace \(H(\theta)\), where \(\theta =(x_0,y_0)\). The vectors \(\{\partial_{y^k}\}_{k=1}^n\) are a basis for the subspace \(V(\theta)\), \(\theta =(x_0,y_0)\).

The operators $d\pi$ and $K$ can be described in these coordinates: \begin{align*} K_\theta(\delta_{x^j}) &= 0,\\ d_\theta\pi(\delta_{x^j}) &= \partial_{x^j},\\ d_\theta\pi(\partial_{y^k}) &= 0,\\ K_\theta(\partial_{y^k}) &= \partial_{x^k}. \end{align*}

Given vectors \(\xi,\eta \in T_\theta TM\) and writing them in the basis given by $\delta_{x^j}$ and $\partial_{x^j}$, i.e. $$ \xi = X^i \delta_{x^i} + Y^k \partial_{y^k}, \quad \eta = \tilde X^i \delta_{x^i} + \tilde Y^k \partial_{y^k}, $$ we get that $$\langle \xi, \eta \rangle_{\text{Sasaki}} = \langle X^i \partial_{x^i}, \tilde X^i \partial_{x^i} \rangle + \langle Y^k \partial_{x^k}, \tilde Y^k \partial_{x^k} \rangle = g_{jk} X^j \tilde X^k + g_jk Y^j \tilde Y^k. $$

Notes 8:

1. Unit sphere bundle

Let \((M,g)\) be a smooth \(n\)-dimensional Riemannian manifold. If \(x\) are local coordinates on \(M\), there are associated local coordinates \((x,y)\) on \( T M \) and corresponding local coordinates \((x,y,X,Y)\) on \(T ( T M )\), so that 

\[ T_{(x,y)} (T M) = \{ X^j \delta_{x_j} + Y^k \partial_{y_k} \,;\, X, Y \in \mathbb{R}^n \} \]

where \( \delta_{x_j} = \partial_{x_j} - \Gamma_{jk}^l y^k \partial_{y_l}\). Recall that \( \delta_{x_j} \) and \( \partial_{y_k} \) span the horizontal and vertical subspaces of \(T(T M)\), respectively. The unit sphere bundle is 

\[ SM = \{ (x,v) \in T M \,;\, \lvert v \rvert_g = 1 \}. \]

This is a smooth \((2n-1)\)-dimensional submanifold of \(T M\), since \(S M = f^{-1}(1) \) where \(f: T M \to \mathbb{R}, \ f(x,y) = \lvert y \rvert_g^2 = g_{jk}(x) y^j y^k\) is smooth.

Let us determine \(T(SM)\) as a submanifold of \(T(T M)\). If \((x,v) \in SM\) and \( \eta(t) = (\gamma(t), Z(t)) \) is a horizontal curve through \((x,v)\) on \(T(T M)\), then \(f(\eta(t)) \equiv 1\) since \(\lvert Z(t) \rvert \equiv 1\) by properties of parallel transport. Thus all horizontal vectors at \((x,v)\) are in \(T(SM)\) since \(df\) is orthogonal to them. If \(\eta(t) = (x, w(t))\) is a vertical curve through \((x,v)\), then \( \partial_t (f(\eta(t)))|_{t=0}\) is zero if \(\dot{w}(0)\) is orthogonal to \(v\), but is nonzero otherwise. It follows that 

\[ T(SM) = \{ X^j \delta_{x_j} + Y^k \partial_{y_k} \in T( T M ) \,;\, (x,y) \in SM, \ g_{jk} y^j Y^k = 0 \}. \]

2. Euclidean plane

Let \(M = \mathbb{R}^2\) and let \(g\) be the Euclidean metric. Then 

\[ SM = \{ (x,v) \,;\, x \in \mathbb{R}^2,\ |v| = 1 \} = \mathbb{R}^2 \times S^1. \]

Thus \(SM\) is \(3\)-dimensional, and for each \((x,v) \in SM\) the tangent space \(T_{(x,v)} (SM)\) is spanned by three vectors. There is a natural basis related to geodesics. Define the geodesic flow 

\[ \varphi_t: SM \to SM, \ \varphi_t(x,v) = (x+tv, v). \]

This is indeed a flow on \(SM\), since \(\varphi_0 = \mathrm{id}\) and \(\varphi_t \circ \varphi_s = \varphi_{t+s}\). Since \(\varphi_t\) is smooth with respect to \(t\), there is a smooth vector field on \(SM\) that acts as the infinitesimal generator of the flow. This is the geodesic vector field \(X\), acting on functions \(u \in C^{\infty}(SM)\) by 

\[ Xu(x,v) = \frac{\partial}{\partial t} u(\varphi_t(x,v)) \Big|_{t=0} = \frac{\partial}{\partial t} u(x+tv, v) \Big|_{t=0} = v \cdot \nabla_x u(x,v). \]

The vector field \(X = v^j \partial_{x_j} = v^j \delta_{x_j}\) is horizontal. There is a related horizontal vector field \(X_{\perp}\) on \(SM\), 

\[ X_{\perp} u(x,v) = v_{\perp} \cdot \nabla_x u(x,v) \]

where \(v_{\perp}\) is rotation of \(v\) by \(90^{\circ}\) clockwise. To define the third vector field in the basis, we identify \(S^1\) with \(\mathbb{R}/2\pi \mathbb{Z}\) by \( \theta \mapsto v_{\theta} = (\cos \theta, \sin \theta)\), and define the vertical vector field \(V\) by 

\[ V u(x,v_{\theta}) = \frac{\partial}{\partial \theta} u(x,v_{\theta}) = v_{\theta}^{\perp} \cdot \nabla_y \tilde{u}(x,v_{\theta}) \]

where \(v_{\theta}^{\perp}\) is the rotation of \(v_{\theta}\) by \(90^{\circ}\) counterclockwise, and \(\tilde{u}(x,y) = u(x,y/|y|)\) is the homogeneous (of degree \(0\) in \(y\)) extension of \(u\) from \(SM\) to \(T M \setminus \{0\} \). It follows that 

\[ T_{(x,v)}(SM) = \mathrm{span}\{ X|_{(x,v)}, X_{\perp}|_{(x,v)}, V|_{(x,v)} \}. \]

Moreover, the vector fields \(X, X_{\perp}, V\) are orthonormal with respect to the Sasaki metric, since \(\{ \partial_{x_1}, \partial_{x_2}, \partial_{y_1}, \partial_{y_2}\}\) is an orthonormal basis of \(T(T M)\) in the Sasaki metric. Thus \(\{X, X_{\perp}, V\}\) provides a global orthonormal frame on \(T(SM)\). If \(u \in C^{\infty}(SM)\), the metric gradient of \(u\) with respect to Sasaki metric is given by 

\[ \nabla_{SM} u = \underbrace{(Xu) X + (X_{\perp} u) X_{\perp}}_{\mathrm{horizontal}} + \underbrace{(Vu) V}_{\mathrm{vertical}}. \]

This shows that the vector fields \(\{X, X_{\perp}, V\}\) generate all possible first order derivatives on \(SM\).

3. Two-dimensional Riemannian manifolds

Let now \((M,g)\) be an oriented two-dimensional Riemannian manifold. We wish to find counterparts for the \(X, X_{\perp}, V\) vector fields in this setting. Let \(\varphi_t: SM \to SM\) be the geodesic flow, defined (possibly on a subdomain) by 

\[ \varphi_t(x,v) = (\gamma_{x,v}(t), \dot{\gamma}_{x,v}(t)) \]

where \(\gamma_{x,v}(t)\) is the geodesic through \((x,v)\). This is a flow, and the geodesic vector field \(X\) is defined to be its infinitesimal generator. In local coordinates, if \(\tilde{u}(x,y) = u(x,y/|y|_g)\) is the \(0\)-homogeneous extension, 

\[ Xu(x,v) = \frac{\partial}{\partial t} \tilde{u}(\gamma(t), \dot{\gamma}(t)) \Big|_{t=0} = (\partial_{x_j} u) \dot{\gamma}^j + (\partial_{y_k} \tilde{u}) \ddot{\gamma}^k \Big|_{t=0} = v^j \partial_{x_j} u - \Gamma^{k}_{ab} v^a v^b \partial_{y_k} \tilde{u} = v^j \delta_{x_j} \tilde{u}. \]

It follows that \(X = v^j \delta_j\), where \(\delta_j\) is the vector field on \(SM\) corresponding to \(\delta_{x_j}\):

\[ \delta_j u = \delta_{x_j} (u \circ p)|_{SM} \]

where \(p: T M \setminus \{0\} \to SM, p(x,y) = (x,y/|y|_g)\). Now \(X\) is horizontal, and we obtain another horizontal vector field \(X_{\perp}\) by 

\[ X_{\perp} = (v_{\perp})^j \delta_j \]

where \(v_{\perp}\) is the rotation of \(v\) by \(90^{\circ}\) clockwise with respect to \(g\). Finally, since \((M,g)\) is oriented we may define a flow \(r_t: SM \to SM\) so that \(r_t(x,v) = (x,v_t)\) where \(v_t\) is the rotation of \(v\) by \(t\) radians counterclockwise. The infinitesimal generator of \(r_t\) is denoted by \(V\), and in local coordinates 

\[ V = (v^{\perp})^k \partial_k \]

where \(v^{\perp}\) is the rotation of \(v\) by \(90^{\circ}\) counterclockwise, and \(\partial_k\) is the vector field on \(SM\) corresponding to \(\partial_{y_k}\) by 

\[ \partial_k u = \partial_{y_k} (u \circ p)|_{SM}. \]

Again, \(\{X, X_{\perp}, V\}\) provides a global orthonormal frame of \(SM\) with respect to Sasaki metric, and 

\[ \nabla_{SM} u = \underbrace{(Xu) X + (X_{\perp} u) X_{\perp}}_{\mathrm{horizontal}} + \underbrace{(Vu) V}_{\mathrm{vertical}} \]

so \(\{X, X_{\perp}, V\}\) generate all possible first order derivatives on \(SM\).

4. General Riemannian manifolds

Let now \((M,g)\) be an \(n\)-dimensional Riemannian manifold. The geodesic flow \(\varphi_t: SM \to SM\) is defined as before, and the geodesic vector field \(X\) is defined to be its infinitesimal generator. In local coordinates 

\[ X = v^j \delta_j \]

where \(\delta_j\) is the vector field on \(SM\) corresponding to \(\delta_{x_j}\). Thus \(X\) is a horizontal vector field with unit length. The horizontal part of \(T(SM)\) has dimension \(n\) and the vertical part has dimension \(n-1\), so for \(n \geq 3\) the vector fields \(X_{\perp}\) and \(V\) need to be replaced by suitable gradients.

If \(u \in C^{\infty}(SM)\), we define the vertical gradient \(\overset{\tt{v}}{\nabla} u\) as the vector field in \(T M\) corresponding to the vertical part of \(\nabla_{SM} u\) (the gradient with respect to Sasaki metric on \(SM\)). In previous notation, 

\[ \overset{\tt{v}}{\nabla} u = K(\mathrm{proj}_V(\nabla_{SM} u)). \]

We also define the horizontal gradient \( \overset{\tt{h}}{\nabla} u \) as the vector field on \(T M\) corresponding to the horizontal part of \(\nabla_{SM} u\) minus the part in the direction of \(X\). More precisely, 

\[ \overset{\tt{h}}{\nabla} u = d\pi(\mathrm{proj}_H(\nabla_{SM} u - (Xu) X)). \]

The horizontal and vertical gradients are maps 

\[ \overset{\tt{h}}{\nabla}, \overset{\tt{v}}{\nabla}: C^{\infty}(SM) \to \mathcal{Z} \]

where 

\[ \mathcal{Z} = \{ Z \in C^{\infty}(SM, T M) \,;\, Z(x,v) \in T_x M, \ \langle Z(x,v), v \rangle = 0 \}. \]

In local coordinates one has 

\[ \overset{\tt{h}}{\nabla} u = (\delta^j u - (v^k \delta_k u) v^j) \partial_{x_j}, \]

\[ \overset{\tt{v}}{\nabla} u = (\partial^k u) \partial_{x_k} \]

where we raise and lower indices with respect to \(g\). As before, with natural identifications, 

\[ \nabla_{SM} u = ((Xu)X, \overset{\tt{h}}{\nabla} u, \overset{\tt{v}}{\nabla} u) \]

so the three operators \(\{ X, \overset{\tt{h}}{\nabla}, \overset{\tt{v}}{\nabla}\}\) generate all first order derivatives on \(SM\). If \(n=2\), the horizontal and vertical gradients reduce to \(X_{\perp}\) and \(V\) :

\[ \overset{\tt{h}}{\nabla} u = -(X_{\perp} u) v^{\perp}, \]

\[ \overset{\tt{v}}{\nabla} u = (Vu) v^{\perp}. \]

Notes 9:

1. Review

Let \((M,g)\) be a smooth \(n\)-dimensional Riemannian manifold. If \(x\) are local coordinates on \(M\), there are associated local coordinates \((x,y)\) on \( T M \) and corresponding local coordinates \((x,y,X,Y)\) on \(T ( T M )\), so that 

\[ T_{(x,y)} (T M) = \{ X^j \delta_{x_j} + Y^k \partial_{y_k} \,;\, X, Y \in \mathbb{R}^n \} \]

where \( \delta_{x_j} = \partial_{x_j} - \Gamma_{jk}^l y^k \partial_{y_l}\). Now \( \delta_{x_j} \) and \( \partial_{y_k} \) span the horizontal and vertical subspaces of \(T(T M)\), respectively. Define the corresponding vector fields on \(SM\), 

\[ \delta_j u = \delta_{x_j} (u \circ p)|_{SM}, \]

\[ \partial_k u = \partial_{y_k} (u \circ p)|_{SM} \]

where \(p: T M \setminus \{0\} \to SM, p(x,y) = (x,y/|y|_g)\).

Recall the geodesic vector field \(X\) and the horizontal and vertical gradients 

\[ \overset{\tt{h}}{\nabla}, \overset{\tt{v}}{\nabla}: C^{\infty}(SM) \to \mathcal{Z} \]

where 

\[ \mathcal{Z} = \{ Z \in C^{\infty}(SM, T M) \,;\, Z(x,v) \in T_x M, \ \langle Z(x,v), v \rangle = 0 \}. \]

In local coordinates one has 

\[ X = v^j \delta_j, \]

\[ \overset{\tt{h}}{\nabla} u = (\delta^j u - (v^k \delta_k u) v^j) \partial_{x_j}, \]

\[ \overset{\tt{v}}{\nabla} u = (\partial^k u) \partial_{x_k} \]

where we raise and lower indices with respect to \(g\). The main point is that, with natural identifications, 

\[ \nabla_{SM} u = ((Xu)X, \overset{\tt{h}}{\nabla} u, \overset{\tt{v}}{\nabla} u) \]

so the three operators \(\{ X, \overset{\tt{h}}{\nabla}, \overset{\tt{v}}{\nabla}\}\) generate all first order derivatives on \(SM\).

2. Commutator formulas

We will prove the commutator formulas 

\[ [X, \overset{\tt{v}}{\nabla}] = - \overset{\tt{h}}{\nabla}, \]

\[ [X, \overset{\tt{h}}{\nabla}] = - R \overset{\tt{v}}{\nabla}. \]

Here, \(X\) is defined naturally on \(\mathcal{Z}\) via the geodesic flow \(\varphi_t\) and the covariant derivative:

\[ XZ(x,v) = D_t (Z(\varphi_t(x,v))) |_{t=0} = (XZ^j) \partial_{x_j} + \Gamma_{jk}^l v^j Z^k \partial_{x_l}. \]

3. Adjoints

We wish to compute the adjoints of \(\{X, \overset{\tt{h}}{\nabla}, \overset{\tt{h}}{\nabla} \}\) in natural \(L^2\) inner products.

If \((N,g)\) is any manifold with smooth boundary, recall the integration by parts formulas 

\[ \int_N \mathrm{div}(W) \,dV = \int_{\partial N} \langle W, \nu \rangle \,dS, \]

\[ \int_N (Wu) w \,dV = -\int_N u(Ww) \,dV - \int_N u w \mathrm{div}(W) \,dV + \int_{\partial N} uw \langle W, \nu \rangle \,dS \]

where \(\mathrm{div}\) is the metric divergence, \(W\) is a vector field, and \(u, w\) are functions on \(N\).

Recall that the geodesic vector field \(X\) arises from a Hamilton flow, hence it preserves the natural volume (also on \(SM\)). Thus \(\mathrm{div}(X) = 0\) by the basic identity connecting the Lie derivative, volume form and metric divergence:

\[ (\mathrm{div}(X)) \,dV_g = \mathcal{L}_X \,dV_g = 0. \]

It follows that the adjoint of \(X\) in the \(L^2(SM)\) inner product is \(-X\):

\[ (Xu, w)_{L^2(SM)} = -(Xu, w)_{L^2(SM)}, \qquad u, w \in C^{\infty}(SM), w|_{\partial(SM)} = 0. \]

The formal adjoints of \(\overset{\tt{h}}{\nabla}\) and \(\overset{\tt{v}}{\nabla}\) will be given by the horizontal and vertical divergences, defined by 

\[ \overset{\tt{h}}{\mathrm{div}}, \overset{\tt{v}}{\mathrm{div}}: \mathcal{Z} \to C^{\infty}(SM), \]

\[ \overset{\tt{h}}{\mathrm{div}} Z = (\partial_j + \Gamma_j) Z^j, \qquad \Gamma_j = \Gamma_{jk}^k, \]

\[ \overset{\tt{v}}{\mathrm{div}} Z = \partial_j Z^j. \]

One has 

\[ (\overset{\tt{h}}{\nabla} u, Z) = -(u, \overset{\tt{h}}{\mathrm{div}} Z), \]

\[ (\overset{\tt{v}}{\nabla} u, Z) = -(u, \overset{\tt{v}}{\mathrm{div}} Z) \]

whenever one of \(u\) and \(Z\) vanishes on \(\partial(SM)\). These adjoint formulas follow from 

\[ (\delta_j u, w) = -(u, (\delta_j + \Gamma_j) w), \]

\[ (\partial_j u, w) = -(u, (\partial_j - (n-1) v_j) w) \]

which can be proved by local coordinate computations.

All this gives rise to the third important commutator formula 

\[ \overset{\tt{h}}{\mathrm{div}} \overset{\tt{v}}{\nabla} - \overset{\tt{v}}{\mathrm{div}} \overset{\tt{h}}{\nabla} = (n-1) X. \]