### 数学代写|黎曼几何代写Riemannian geometry代考|MAST90029

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## 数学代写|黎曼几何代写Riemannian geometry代考|Parallel Transport

In this section we want to introduce the notion of parallel transport on a surface (along a curve), which allows us to define its main geometric invariant: the Gaussian curvature.

Definition $1.14$ Let $\gamma:[0, T] \rightarrow M$ be a smooth curve. A smooth curve of tangent vectors $\xi(t) \in T_{\nu(t)} M$ is said to be parallel if $\dot{\xi}(t) \perp T_{\gamma(t)} M$.

This notion generalizes the notion of parallelism of vectors on the plane, where it is possible to canonically identify every tangent space to $M=\mathbb{R}^{2}$ with $\mathbb{R}^{2}$ itself. ${ }^{2}$ In this case a smooth curve of tangent vectors $\xi(t) \in T_{\gamma(t)} M$ is parallel if and only if $\dot{\xi}(t)=0$.

When $M$ is the zero level of a smooth function $a: \mathbb{R}^{3} \rightarrow \mathbb{R}$, as in (1.14), we have the following description:

Proposition $1.15$ A smooth curve of tangent vectors $\xi(t)$ defined along $\gamma:[0, T] \rightarrow M$ is parallel if and only if it satisfies
$$\dot{\xi}(t)=-\frac{\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \xi(t)}{\left|\nabla_{\gamma(t)} a\right|^{2}} \nabla_{\gamma(t)} a, \quad \forall t \in[0, T] .$$
Proof As in Remark 1.7, $\xi(t) \in T_{\gamma(t)} M$ implies that $\left\langle\nabla_{\gamma(t)} a, \xi(t)\right\rangle=0$. Moreover, by assumption, $\dot{\xi}(t)=\alpha(t) \nabla_{\gamma(t)} a$ for some smooth function $\alpha$. With computations analogous to those in the proof of Proposition $1.8$ we get that
$$\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \xi(t)+\alpha(t)\left|\nabla_{\gamma(t)} a\right|^{2}=0,$$
from which the statement follows.

## 数学代写|黎曼几何代写Riemannian geometry代考|Parallel Transport and the Levi-Civita Connection

Definition 1.19 An orientation of a surface $M$ is a smooth map $v: M \rightarrow \mathbb{R}^{3}$, défineed globally on $M$, such that $v(q) \perp T_{q} M$ and $|v(q)|=1$ for every $q \in M$. Notice that if $v$ is an orientation of $M$, then $-v$ also defines an orientation of $M$.

A surface $M$ is oriented if it is given (when this exists) an orientation. On an oriented surface $M$, an orthonormal frame $\left{e_{1}, e_{2}\right}$ of $T_{q} M$ is said to be positively oriented (resp. negatively oriented) if $e_{1} \wedge e_{2}=k v(q)$ with $k>0$ (resp. $k<0$ ).
In the following we assume that $M$ is an oriented surface.
Definition $1.20$ The spherical bundle $S M$ on $M$ is the disjoint union of all unit tangent vectors to $M$ :
$$S M=\bigsqcup_{q \in M} S_{q} M, \quad S_{q} M=\left{v \in T_{q} M,|v|=1\right}$$
The spherical bundle $S M$ can be endowed with the structure of a smooth manifold of dimension 3 , and more precisely of a fiber bundle with base manifold $M$, typical fiber $S^{1}$ and canonical projection
$$\pi: S M \rightarrow M, \quad \pi(v)=q \quad \text { if } v \in T_{q} M$$

Remark $1.21$ Fix a positively oriented local orthonormal frame $\left{e_{1}(q), e_{2}(q)\right}$ on $M$. Since every vector in the fiber $S_{q} M$ has norm 1, we can write every $v \in S_{q} M$ as $v=(\cos \theta) e_{1}(q)+(\sin \theta) e_{2}(q)$ for $\theta \in S^{1}$.

The choice of such an orthonormal frame then induces coordinates $(q, \theta)$ on $S M$. Notice that the choice of a different positively oriented local orthonormal frame $\left{e_{1}^{\prime}(q), e_{2}^{\prime}(q)\right}$ induces coordinates $\left(q^{\prime}, \theta^{\prime}\right)$ on $S M$, where $q^{\prime}=q$ and $\theta^{\prime}=\theta+\phi(q)$ for $\phi \in C^{\infty}(M)$

The orientation of $M$ permits us, once a unit tangent vector is given, to define a canonical orthonormal frame.

## 数学代写|黎曼几何代写Riemannian geometry代考|Parallel Transport

$$\dot{\xi}(t)=-\frac{\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \xi(t)}{\left|\nabla_{\gamma(t)} a\right|^{2}} \nabla_{\gamma(t)} a, \quad \forall t \in[0, T] .$$

$$\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \xi(t)+\alpha(t)\left|\nabla_{\gamma(t)} a\right|^{2}=0,$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Parallel Transport and the Levi-Civita Connection

Uleft{e_{1}, e_{2} \right } } \text { 的 } T _ { q } M \text { 如果 } e _ { 1 } \wedge e _ { 2 } = k v ( q ) \text { 和 } k > 0 \text { (分别。 } k < 0 \text { )。 }

S M=\bigsqcup_{q \in M $} S_{-}{q} M, \backslash q u a d S_{-}{q} M=\backslash \mid e f t\left{v \backslash\right.$ in $T_{-}{q} M,|v|=1 \backslash$ right $}$

$$\pi: S M \rightarrow M, \quad \pi(v)=q \quad \text { if } v \in T_{q} M$$

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