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# 数学代写|拓扑学代写TOPOLOGY代考|The geodesic flow and Huygens’ principle

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## 数学代写|拓扑学代写TOPOLOGY代考|The geodesic flow and Huygens’ principle

We now make a brief excursion into Riemannian geometry and give a contact geometric interpretation of the so-called geodesic flow on the tangent bundle of a Riemannian manifold. (For a comprehensive treatment of geodesic flows in the context of the theory of dynamical systems see the monograph by Paternain [204]. However, the reader should beware the possible confusion arising from the different definitions used there, which seem to render the main result of the present section tautological.) We also consider the dual flow on the cotangent bundle; here the discussion ties up with the space of contact elements and yields a simple contact geometric proof of Huygens’ principle concerning the propagation of wave fronts.

Proposition/Definition 1.5.1 Let $B$ be a manifold with a Riemannian metric $g$. There is a unique vector field $G$ on the tangent bundle $T B$ whose trajectories are of the form $t \mapsto \dot{\gamma}(t) \in T_{\gamma(t)} B \subset T B$, where $\gamma$ is a geodesic on $B$ (not necessarily of unit speed). This vector field $G$ is called the geodesic field, and its (local) flow the geodesic flow.

Note that the geodesic flow being defined globally (i.e. for all times) is equivalent to saying that $(B, g)$ is a complete Riemannian manifold.

Proof In local coordinates $\left(q_1, \ldots, q_n\right)$ on $B$, geodesics are found as solutions of the system of second-order differential equations
$$\ddot{q}k+\sum{i, j} \Gamma_{i j}^k \dot{q}i \dot{q}_j=0, k=1, \ldots, n,$$ where the $\Gamma{i j}^k$ are the Christoffel symbols of the Riemannian metric $g$, see any book on Riemannian geometry, e.g. the one by do Carmo [42]. Choose local coordinates on the tangent bundle $T B$ such that
$$\left(q_1, \ldots, q_n, v_1, \ldots, v_n\right)=\left(\sum_{j=1}^n v_j \partial_{q_j}\right)_{\left(q_1, \ldots, q_n\right)}$$

## 数学代写|拓扑学代写TOPOLOGY代考|Order of contact

In Proposition 1.5 .12 we saw that an isotropic submanifold in a $(2 n+1)-$ dimensional contact manifold has dimension at most equal to $n$. Thus, in a contact 3-manifold $(M, \xi)$ we can find curves tangent to $\xi$, but no surfaces.
The aim of the present section is to analyse this situation for 3 -manifolds a little more carefully. This will allow us to give an entirely geometric definition of the notion of contact structure, without reference to the exterior derivative of a differential 1-form. I learned this characterisation of contact structures from Jesús Gonzalo.

Consider a 2-plane field $\xi$ on a $3-$ manifold $M$ and an embedded surface $\Sigma \subset M$. Let $(u, v)$ be local coordinates on $\Sigma$ near some point $p=(0,0) \in \Sigma$. Define $\theta(u, v)$ as the angle between the tangent plane $T_{(u, v)} \Sigma$ and the plane $\xi_{(u, v)} \cdot$

Definition 1.6.1 We say that $\xi$ has contact of order at least equal to $k$ with $\Sigma$ at $p=(0,0)$ if $\theta(u, v)$ is a function of type $O\left(|(u, v)|^k\right)$ for $(u, v) \rightarrow(0,0)$, where $O$ denotes the Landau symbol. This means that $\theta(u, v) /|(u, v)|^k$ is bounded above by a constant as $(u, v) \rightarrow(0,0) . \dagger$

Thus, $\xi$ having contact of order at least 1 with $\Sigma$ at $p$ is simply saying that $\xi_p=T_p \Sigma$. Obviously, contact of order equal to $k$ is going to mean that $\theta(u, v)$ is of type $O\left(|(u, v)|^k\right)$, but not of type $O\left(|(u, v)|^{k+1}\right)$. You may want to convince yourself that the order of contact does not depend on the choice of local coordinates on $\Sigma$.

The following theorem gives a characterisation of contact structures on 3manifolds in terms of this notion of contact. I should enter the caveat that the name ‘contact structure’ does not derive from this characterisation, but rather – as mentioned before – from the space of contact elements, which (in the 3-dimensional case) has to do with tangencies of curves.

## 数学代写|拓扑学代写TOPOLOGY代考|The geodesic flow and Huygens’ principle

1.5.1设$B$为黎曼度规$g$的流形。切线束$T B$上有一个唯一的向量场$G$，其轨迹形式为$t \mapsto \dot{\gamma}(t) \in T_{\gamma(t)} B \subset T B$，其中$\gamma$是$B$上的测地线(不一定是单位速度)。这个向量场$G$称为测地线场，它的(局部)流称为测地线流。

$$\ddot{q}k+\sum{i, j} \Gamma_{i j}^k \dot{q}i \dot{q}j=0, k=1, \ldots, n,$$其中$\Gamma{i j}^k$是黎曼度规的克里斯托费尔符号$g$，请参阅任何关于黎曼几何的书，例如do Carmo[42]的书。选择切线包$T B$上的本地坐标，这样 $$\left(q_1, \ldots, q_n, v_1, \ldots, v_n\right)=\left(\sum{j=1}^n v_j \partial_{q_j}\right)_{\left(q_1, \ldots, q_n\right)}$$

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