Introduction to geodesis

## Overview

Because the field of differential geometry is too vast, this document aims at understanding geodesics only.
In engineering, geodesics are known as the shortest path between two points on a surface. It becomes a straight line if the surface is a flat plane. On a curved surface such as sphere, geodesics are no longer straight lines but can be explained as 'naturally straight lines'. As is known well, geodesics on a sphere is a largest circle whose center point matches with the center of the sphere.

What is surprising for engineers is that in contemporary differential geometry, it seems like they are defined in completely different manner.
The classic definition, the shortest path between two points, is a definition based on variational principle, which most engineers are familiar with.
Contrary, in contemporary differential geometry, we first define covariant differentiation from "connection", then parallel vectors from covariant differentiation, parallel transportation from parallel vectors, and finally geodesics from parallel transportation, which, fairy speaking, seems far beyond within our reach.

However, all of these simply means that once "connection" is given, we can draw a geodesics. The concept "connection' itself is an abstract concept, though, the concept is reallized by connection coefficients, which are the coefficients appear in covariant differentiation. Now it turned out that the contemporay way of defining geodesics is trully practical though it's difficult to visually understand, and thus all the engineers should be familiaried.

### Connection coefficients

First, let us suppose that $x^1\cdots x^n$ represents a curvilinear coordinate parameters (curvilinear means not orthonormal) and

(1)
\begin{align} \boldsymbol{r}=\left[\begin{array}{c}f_1(x^1\cdots x^n)\\ \cdots \\ f_m(x^1\cdots x^n)\end{array}\right] \end{align}

represents a position vector that represents an n-dim surface (manifold) floating in an m-dim Euclidean space. By which, the position vector represents orthonormal coordinates.

This is a natural generalization of 2-dim surface in an 3-dim Euclidean space.
We define base vectors adjunct with the curvilinear coordinate parameters by

(2)
\begin{align} \boldsymbol{g}_i=\frac{\partial\boldsymbol{r}}{\partial x^i}=\left[\begin{array}{c} \frac{\partial f_1(x^1\cdots x^n)}{\partial x^i}\\ \cdots \\ \frac{\partial f_m(x^1\cdots x^n)}{\partial x^i}\end{array}\right]. \end{align}

With having $dx^1\cdots dx^n$, a small motion of position vector can be represented as

(3)
\begin{align} d\boldsymbol{r}=\frac{\partial\boldsymbol{r}}{\partial x^i}dx^i=dx^i\boldsymbol{g}_i\ . \end{align}

Next, we think about small change of the base vectors. This can be wrote as

(4)
\begin{align} d\boldsymbol{g}_j=\frac{\partial \boldsymbol{g}_j}{\partial x^i}dx^i. \end{align}

So far none of them are difficult. It's getting complicated from here and we need some geometric considerations.
In general, it is not guaranteed that $d\boldsymbol{g}_i$ is given as a linear combination of tangent vectors, $\boldsymbol{g}_i\cdots\boldsymbol{g}_n$. Because it contained a component that is perpendicular to the tangent plane.
So let's think about another differentiation that is similar to $d\boldsymbol{g}_i$, but truly given as a linear conbination of $\boldsymbol{g}_i\cdots\boldsymbol{g}_n$. We represent this differentiation as

(5)
\begin{align} \delta \boldsymbol{g}_1=w^1\boldsymbol{g}_1+w^2\boldsymbol{g}_2+\cdots \end{align}

Since there are n independent basis, we need to have n different $w^1\cdots w^n$. Thus, we can write

(6)
\begin{align} \delta\boldsymbol{g}_j=w_j^k\boldsymbol{g}_k \end{align}

さて、3式と同様、このような微分は$dx^1\cdots dx^n$の線形結合であると仮定します。(線形性だけは、勝手に仮定されていることは実は重要です)
するとそれぞれの$w_j^k$は一般に一次微分形式として

(7)
\begin{align} w_j^k\equiv\Gamma_{ij}^kdx^i \end{align}

と表されます。
$w_j^k$ は伝統的に接続形式と呼ばれ、また、$\Gamma_{ij}^k$は接続係数と呼ばれます。
ここまでは一般論です。山のようにある係数を整理しただけです。
さて、接続係数を総て求めるにはどうしたらよいでしょうか。一つの自然な方針として、$d\boldsymbol{g}_j$から超曲面に直交する成分を取り除く、というものがあります。別の言い方では、超曲面上へ正射影する、ともいえます。そのような操作は、

(8)
\begin{align} \delta\boldsymbol{g}_j=\left(d\boldsymbol{g}_j\cdot\boldsymbol{g}^k\right)\boldsymbol{g}_k \end{align}

と書けます。

(43)
\begin{align} da^i=\frac{da^i}{dt}dt \end{align}

です。これらを平行なベクトルの満たすべき条件$\delta a^k=0$に代入することで、

(44)
\begin{align} \delta a^k=0\Rightarrow\frac{da^k}{dt}+a^j\Gamma_{ij}^k\frac{dx^i}{dt}dt=0 \end{align}

さらに、$dt$をくくりだして

(45)
\begin{align} \left(\frac{da^k}{dt}+a^j\Gamma_{ij}^k\frac{da^i}{dt}\right)dt=0 \end{align}

$dt$の任意性より

(46)
\begin{align} \frac{da^k}{dt}+a^j\Gamma_{ij}^k\frac{dx^i}{dt}=0 \therefore \frac{da^k}{dt}=-a^j\Gamma_{ij}^k\frac{dx^i}{dt} \end{align}

これは、形式的に$dt$で両辺を除したものとして

(47)
\begin{align} \frac{\delta a^k}{dt}\equiv\frac{da^k}{dt}+a^j\Gamma_{ij}^k\frac{dx^i}{dt}=0 \end{align}

とも書かれますが、そのような演算はなくあくまで形式的なものですから、「$dt$の任意性より」とするのが正確です。
$a^k(t)$が(46)式を満足するとき、曲線$c(t)$に沿って平行であると言われます。
また、初期条件$a^k(\alpha)=a^k_0$を満たすとき、ベクトル$\boldsymbol{a}_0$の曲線$c(t)$に沿った平行移動と呼ばれます。

### 測地線

いよいよ測地線の定義です。基本的なアイデアは、ベクトルをベクトル自身にそって平行移動し、次々と繋いでゆくというものです。まず、曲線$c(t)={x^1(t),\cdots,x^n(t)}$の接ベクトルを$\boldsymbol{a}(t)=a^i\boldsymbol{g}_i$とおきます。つまり、

(48)
\begin{align} a^i\equiv\frac{dx^i}{dt} \end{align}

とします。これは速度ベクトルとも呼ばれます。
そして、$\boldsymbol{a}$$c(t)$に沿った平行移動であるとき、曲線$c(t)$を測地線と呼ぶのです。
(46)式に速度ベクトルの定義を代入すれば

(49)
\begin{align} \frac{da^k}{dt}=-\Gamma_{ij}^k\frac{dx^j}{dt}\frac{dx^i}{dt} \end{align}

もしくは

(50)
\begin{align} \frac{da^k}{dt}=-\Gamma_{ij}^k{a^j}{a^i} \end{align}

を得ます。これらが測地線の満たすべき方程式です。逆に曲線$c(t)$の速度ベクトル$a^i=\frac{dx^i}{dt}$がこの方程式を満たすとき、この曲線を測地線と呼びます。

page revision: 50, last edited: 10 Sep 2016 22:41