按：這篇筆記是系列筆記的第四篇，第一部分有4節(jié)，每節(jié)對(duì)應(yīng)1-2篇筆記。

筆記的方式，是引用一段個(gè)人覺得比較有亮點(diǎn)的英文原文，再給一段簡(jiǎn)化的中文說(shuō)明，不采用中文版的翻譯，不自行做直接翻譯，只說(shuō)明要點(diǎn)。因?yàn)椴豢赡艽蠖未蠖蔚厝ヒ?，必然?huì)有語(yǔ)境的丟失，會(huì)做一些補(bǔ)充說(shuō)明，以“按：”開始。對(duì)中文版翻譯進(jìn)行更正或調(diào)整的說(shuō)明，以“注：”開始。偶爾也會(huì)插入自己的議論，以“評(píng)：”開始。

前三篇筆記為：

• 《普林斯頓數(shù)學(xué)指引》讀書筆記——I.1 數(shù)學(xué)是關(guān)于什么的
• 《普林斯頓數(shù)學(xué)指引》讀書筆記——I.2 數(shù)學(xué)的語(yǔ)言和語(yǔ)法
• 《普林斯頓數(shù)學(xué)指引》讀書筆記——I.3 一些基本的數(shù)學(xué)定義（上）

5 Basic Concepts of Mathematical Analysis（數(shù)學(xué)分析的基本概念）

The sequence 1/2, 2/3, 3/4, 4/5. . . . In a sense, the numbers in this sequence approach 2, since each one is closer to 2 than the one before, but it is clear that this is not what we mean. What matters is not so much that we get closer and closer, but that we get arbitrarily close, and the only number that is approached in this stronger sense is the obvious “l(fā)imit,” 1.

注：中文版誤將此段中的2更正為1，而原書作者的意思其實(shí)是故意拿一個(gè)2來(lái)說(shuō)明極限的真正含義。

這個(gè)序列的確一個(gè)比一個(gè)靠近2，因此某種意義上也是在趨近2。然而我們真正用“趨近”來(lái)表示的意思應(yīng)該不僅僅是我們能越靠越近，而是我們能靠得任意地近。在這個(gè)更強(qiáng)的意義下，唯一在被趨近的數(shù)值，顯然就是極限1。

The notion of limit applies much more generally than just to real numbers. If you have any collection of mathematical objects and can say what you mean by the distance between any two of those objects, then you can talk of a sequence of those objects having a limit. Two objects are now called δ-close if the distance between them is less than δ, rather than the difference. (The idea of distance is discussed further in metric spaces [III.58].) For example, a sequence of points in space can have a limit, as can a sequence of functions. (In the second case it is less obvious how to define distance—there are many natural ways to do it.) A further example comes in the theory of fractals (see dynamics [IV.15]): the very complicated shapes that appear there are best defined as limits of simpler ones.

極限的概念可應(yīng)用在遠(yuǎn)比實(shí)數(shù)廣泛的領(lǐng)域。如果我們有任何一族數(shù)學(xué)對(duì)象，而且能定義任意兩個(gè)對(duì)象間的距離，那么我們就可以談?wù)撨@些對(duì)象的一個(gè)序列有沒有極限。如果兩個(gè)對(duì)象的距離（而不是差）小于δ，那么它們可被稱為“δ-逼近的”。（距離的概念將在度量空間[III.58]里進(jìn)一步討論）。

例如，空間中的點(diǎn)的序列可以擁有一個(gè)極限，函數(shù)的序列也可以。（函數(shù)間的距離的定義方式?jīng)]有那么顯然，不過(guò)存在很多自然的方式來(lái)對(duì)其進(jìn)行定義。）一個(gè)更進(jìn)一步的例子來(lái)自分形的理論（見動(dòng)力學(xué)[IV.15]）：在里面出現(xiàn)的復(fù)雜圖形，最好是定義為較簡(jiǎn)單的圖形的極限。

We say that f is continuous at a if

This says that however accurate you wish f(x) to be as an estimate for f (a), you can achieve this accuracy if you are prepared to make x a sufficiently good approximation to a. The function f is said to be continuous if it is continuous at every a. Roughly speaking, what this means is that f has no “sudden jumps.” (It also rules out certain kinds of very rapid oscillations that would also make accurate estimates difficult.)

我們說(shuō)函數(shù)在a處連續(xù)，如果

上面的公式其實(shí)就是說(shuō)，f(x)作為對(duì)f(a)的估算，無(wú)論我們希望這個(gè)估算多么精確，對(duì)應(yīng)的精度都是可以達(dá)到的，只要我們準(zhǔn)備好讓x是a的一個(gè)足夠好的近似。如果一個(gè)函數(shù)在每個(gè)點(diǎn)a處都連續(xù)，它就是連續(xù)函數(shù)。粗略地說(shuō)，就是它沒有突然的跳躍（這也就排除了會(huì)使得精確的估計(jì)變得困難的某一類急速震蕩）。

Continuous functions are functions that preserve the structure provided by convergent sequences and their limits.

連續(xù)函數(shù)就是可以保持由收斂序列及其極限所提供的結(jié)構(gòu)的函數(shù)。

Heat takes time to travel through a medium, so although the temperature at some distant point (x',y',z') will eventually affect the temperature at (x,y,z), the way the temperature is changing right now (that is, at time t) will be affected only by the temperatures of points very close to (x,y,z): if points in the immediate neighborhood of (x,y,z) are hotter, on average, than (x,y,z) itself, then we expect the temperature at (x,y,z) to be increasing, and if they are colder then we expect it to be decreasing.

熱需要時(shí)間來(lái)在介質(zhì)中傳導(dǎo)，所以雖然在很遠(yuǎn)處的點(diǎn) (x',y',z') 的溫度最終會(huì)影響到點(diǎn)(x,y,z)，此刻（在時(shí)間t）溫度改變的方式卻僅被緊挨著點(diǎn)(x,y,z)的點(diǎn)的溫度所影響：如果在(x,y,z)的極近鄰域的點(diǎn)平均比點(diǎn)(x,y,z)自己更熱，那么我們就會(huì)預(yù)期點(diǎn)(x,y,z)的溫度會(huì)上升，如果平均更冷，那我們預(yù)期就會(huì)下降。

The symbol Δ, defined by

, is known as the Laplacian. What information does Δf give us about a function f ? The answer is that it captures the idea in the last paragraph: it tells us how the value of f at (x,y,z) compares with the average value of f in a small neighborhood of (x,y,z).

定義為

的符號(hào)Δ，稱為拉普拉斯算子。Δf向我們提供了關(guān)于函數(shù)f的哪些信息？它抓住了上一段所描述的思想：f在點(diǎn)(x,y,z)的值與點(diǎn)(x,y,z)的極小領(lǐng)域的平均值相比如何。

A second equation of great importance is the Laplace equation, Δf = 0. Intuitively speaking, this says of a function f that its value at a point (x,y,z) is always equal to the average value at the immediately surrounding points.

第二個(gè)特別重要的方程式拉普拉斯方程，即Δf = 0。直觀地看，這是在說(shuō)，一個(gè)函數(shù)f在點(diǎn)(x,y,z)的值，總是等于緊挨著該點(diǎn)的點(diǎn)的平均值。

For two or more variables, a function has more flexibility—it can lie above the tangent lines in some directions and below it in others. As a result, one can impose a variety of boundary conditions on f (that is, specifications of the values f takes on the boundaries of certain regions), and there is a much wider and more interesting class of solutions.

在二元或多元的情況下，一個(gè)函數(shù)可以有更大的靈活性：它可以在某些方向上高于切線，而在其他方向上低于切線。結(jié)果是，我們可以對(duì)f賦予多種邊值條件（即在特定區(qū)域的邊界上指定f的值），從而也就有了更廣泛和更有趣的各類解。

is shorthand for

The operation

is called the d'Alembertian, after d'Alembert [VI.19], who was the first to formulate the wave equation.

是

的簡(jiǎn)寫。

算子

評(píng)：包含這兩個(gè)符號(hào)，不是因?yàn)檫@是新知識(shí)，只是在我少年時(shí)代初次接觸這個(gè)三角形、正方形還有另外一個(gè)倒三角形時(shí)，對(duì)數(shù)學(xué)符號(hào)升起了某種神秘崇高的感覺，至今看到這幾個(gè)符號(hào)依然能喚起那時(shí)的感覺。我如此喜歡這種用幼兒時(shí)代就接觸的符號(hào)來(lái)濃縮中學(xué)時(shí)代才能理解的知識(shí)的方式。

We have been at pains to distinguish integration from antidifferentiation, but a famous theorem, known as the fundamental theorem of calculus, asserts that the two procedures do, in fact, give the same answer, at least when the function in question has certain continuity properties that all “sensible” functions have. So it is usually legitimate to regard integration as the opposite of differentiation. More precisely, if f is continuous and F(x) is defined to be

for some a, then F can be differentiated and F'(x) = f(x). That is, if you integrate a continuous function and differentiate it again, you get back to where you started.

我們花了不少功夫來(lái)把積分和逆微分區(qū)分開來(lái)，但是有一個(gè)稱為微積分基本定理的著名定理斷言這兩個(gè)程序事實(shí)上會(huì)給出相同的答案，至少當(dāng)所考察的函數(shù)具有所有“合理”的函數(shù)一定會(huì)具有的某些連續(xù)性時(shí)是這樣的。因此，通常都認(rèn)為把積分看成微分的逆運(yùn)算是合法的。確切些說(shuō)，如果f是連續(xù)的，而F(x)可以對(duì)于某個(gè)常數(shù)a定義為

評(píng)：這段話對(duì)于學(xué)過(guò)高等數(shù)學(xué)的同學(xué)可能感覺平淡無(wú)奇，不過(guò)雙語(yǔ)對(duì)照閱讀起來(lái)，還是有一些新的感覺，所以摘錄出來(lái)。

These facts begin to suggest that complex differentiability is a much stronger condition than real differentiability and that we should expect holomorphic functions to have interesting properties.

這些事實(shí)（按：上面討論了柯西-黎曼方程）開始揭示，復(fù)可微是一個(gè)遠(yuǎn)比實(shí)可微要強(qiáng)得多的條件，我們也可以期待全純函數(shù)會(huì)具備許多有趣的屬性。

It is not necessary for the function f to be defined on the whole of C for Cauchy’s theorem to be valid: everything remains true if we restrict attention to a simply connected domain, which means an open set with no holes in it. If there are holes, then two path integrals may differ if the paths go around the holes in different ways. Thus, path integrals have a close connection with the topology of subsets of the plane, an observation that has many ramifications throughout modern geometry. For more on topology, see section 6.4 of this article and Algebraic Topology [IV.10].

為了使柯西定理成立，并不需要函數(shù)定義在整個(gè)復(fù)數(shù)平面C上，如果限制函數(shù)定義在整個(gè)復(fù)數(shù)平面的一個(gè)單連通區(qū)域，即沒有洞的開集合上，則一切依然成立。如果區(qū)域里有洞，則兩條有相同起點(diǎn)和終點(diǎn)的路徑積分可能不一樣，如果這兩條路徑以不同的方式環(huán)繞洞。因此，路徑積分與平面的子集合的拓?fù)鋵W(xué)有密切的關(guān)系，這一點(diǎn)觀察，在整個(gè)現(xiàn)代幾何學(xué)里非常多的引申與影響。關(guān)于拓?fù)鋵W(xué)，可以進(jìn)一步參看代數(shù)拓?fù)?/a> 這一條目。

在第五節(jié)的最后，中文版討論了Liouville's theorem，而英文電子版缺失。這個(gè)定理是說(shuō)：如果函數(shù)f是定義在整個(gè)復(fù)平面上的全純函數(shù)，而且函數(shù)f是有界的（即存在一個(gè)常數(shù)C，使得對(duì)于每一個(gè)復(fù)數(shù)z都有

6 What Is Geometry? （什么是幾何學(xué)）

However, if you have not met the advanced concepts and have no idea what modern geometry is like, then you will get much more out of this book if you understand two basic ideas: the relationship between geometry and symmetry, and the notion of a manifold.

如果你還沒有見過(guò)一些進(jìn)階的概念，并且對(duì)于現(xiàn)代幾何學(xué)是什么樣的一無(wú)所知，那么你只要理解兩個(gè)基本的概念（幾何學(xué)與對(duì)稱之間的關(guān)系，以及流形的概念），就能從這本書收獲更多。

評(píng)：點(diǎn)出了現(xiàn)代幾何學(xué)的這兩個(gè)最為核心的概念的關(guān)鍵性。這段話在中文版中，錯(cuò)誤地將“進(jìn)階”翻譯為“高深”，并且將進(jìn)階概念與一無(wú)所知的并列關(guān)系翻譯成假設(shè)關(guān)系。

Broadly speaking, geometry is the part of mathematics that involves the sort of language that one would conventionally regard as geometrical, with words such as “point,” “l(fā)ine,” “plane,” “space,” “curve,” “sphere,” “cube,” “distance,” and “angle” playing a prominent role. However, there is a more sophisticated view, first advocated by klein [VI.56], which regards transformations as the true subject matter of geometry. So, to the above list one should add words like “reflection,” “rotation,” “translation,” “stretch,” “shear,” and “projection,” together with slightly more nebulous concepts such as “angle-preserving map” or “continuous deformation.”

一般來(lái)說(shuō)，幾何學(xué)就是數(shù)學(xué)里涉及我們通常會(huì)按照慣例視為幾何語(yǔ)言的部分，如“點(diǎn)”、“直線”、“平面”、“空間”、“曲線”、“球”、“立方體”、“距離”，還有“角度”這樣的詞匯扮演了突出的角色。然而，還存在一種更為深刻的觀點(diǎn)，最初為克萊因所主張，認(rèn)為變換才是這門科學(xué)的真正的主題。所以除了上面列舉的這些詞以外，還要加上“反射”、“旋轉(zhuǎn)”、“平移”、“拉伸”、“剪切”、“投影”，以及還有稍微有些朦朧的概念，例如“保角映射”或者“連續(xù)變形”。

These can be thought of in two different ways. One is that they are the transformations of the plane, or of space, or more generally of R^n for some n, that preserve distance.

可以有兩種方式來(lái)看待剛性變換，其一是將它們看作對(duì)平面或三維空間或者更一般的R^n空間，所做的保持距離不變的變換。

Every such transformation can be realized as a combination of rotations, reflections, and translations, and this gives us a more concrete way to think about the group.

每一個(gè)這樣的變換都可以用旋轉(zhuǎn)、反射和平移的復(fù)合來(lái)實(shí)現(xiàn)。給了我們一種更具體的方式來(lái)想象群。

Since linear maps include stretches and shears, they preserve neither distance nor angle, so these are not concepts of affine geometry.

因?yàn)榫€性映射中還包含了拉伸和剪切，它們既不能保持距離，也不能保持角度，所以距離和角度都不是仿射幾何學(xué)的概念。

Although angles in general are not preserved by linear maps, angles of zero are.

雖然線性映射一般并不保持角度不變，但是為零的角度卻會(huì)被它們保持。

The idea that the geometry associated with a group of transformations “studies the concepts that are preserved by all the transformations” can be made more precise using the notion of equivalence relations [I.2 §2.3].

通過(guò)等價(jià)關(guān)系的概念，可以將與變換群相關(guān)聯(lián)的幾何“研究的是被所有的這些變換所保持的概念”這個(gè)思想表達(dá)得更確切。

Topology can be thought of as the geometry that arises when we use a particularly generous notion of equivalence, saying that two shapes are equivalent, or homeomorphic, to use the technical term, if each can be “continuously deformed” into the other.

拓?fù)鋵W(xué)可以認(rèn)為是當(dāng)我們使用特別寬松的等價(jià)概念時(shí)自然涌現(xiàn)的幾何學(xué)，其中我們說(shuō)兩個(gè)圖形是等價(jià)的，或者用技術(shù)的術(shù)語(yǔ)來(lái)說(shuō)是同胚的，只要它們均可連續(xù)變形為另外一個(gè)。

The appropriate group of transformations is SO(3): the group of all rotations about some axis that goes through the origin. (One could allow reflections as well and take O(3).)

球面幾何學(xué)中適合表達(dá)n維球面S^n的變換群 SO(3)，它是所有以經(jīng)過(guò)原點(diǎn)的直線為軸的旋轉(zhuǎn)。我們也可以選擇還包含了反射的群O(3)。

The group of transformations that produces hyperbolic geometry is called PSL(2,R), the projective special linear group in two dimensions.

產(chǎn)生雙曲幾何學(xué)的變換群是二維的射影特殊線性群，叫做PSL(2,R)。

To get from this group to the geometry one must first interpret it as a group of transformations of some two dimensional set of points. Once we have done this, we have what is called a model of two-dimensional hyperbolic geometry.

為了從這個(gè)群中得出對(duì)應(yīng)的幾何學(xué)，我們必須先把它理解為，某個(gè)2維點(diǎn)集合的變換群，一旦我們做到了這一點(diǎn)，我們就有了二維雙曲幾何的模型。

The three most commonly used models of hyperbolic geometry are called the half-plane model, the disk model, and the hyperboloid model.

雙曲幾何學(xué)的三個(gè)最常用的模型是半平面模型、圓盤模型和雙曲面模型。

Here are two ways of regarding the projective plane. The first is that the set of points is the ordinary plane, together with a “point at infinity.” The group of transformations consists of functions known as projections.

A second view of the projective plane is that it is the set of all lines in R^3 that go through the origin. Since a line is determined by the two points where it intersects the unit sphere, one can regard this set as a sphere, but with the significant difference that opposite points are regarded as the same—because they correspond to the same line. (This is quite hard to imagine, but not impossible. Suppose that, whatever happened on one side of the world, an identical copy of that event happened at the exactly corresponding place on the opposite side. ...... It might under such circumstances be more natural to say that there was only one Paris and only one you and that the world was not a sphere but a projective plane.)

對(duì)射影平面有兩種觀點(diǎn)：第一種觀點(diǎn)認(rèn)為，這個(gè)點(diǎn)集合其實(shí)就是普通的平面加上無(wú)窮遠(yuǎn)點(diǎn)。組成射影變換群的函數(shù)我們稱為投影。

對(duì)射影平面的第二種觀點(diǎn)，是把它看作R^3中過(guò)原點(diǎn)的直線的集合。因?yàn)橐粭l這樣的直線可由它與單位球面的兩個(gè)交點(diǎn)決定，所以也可以把這個(gè)集合看成就是單位球面，但是與普通的球面有一個(gè)值得注意的區(qū)別，就是（單位球面上）相對(duì)的點(diǎn)可以視作同一點(diǎn)，因?yàn)樗鼈儗?duì)應(yīng)于同一條直線。這一點(diǎn)很難想象，但并非不可能。假設(shè)有這樣一個(gè)世界，在它的一邊發(fā)生的任何事，該事件一個(gè)完全一致的副本，都會(huì)在另外一邊完全對(duì)應(yīng)的地方發(fā)生。（按：接下來(lái)作者舉了真實(shí)的“你”去副本的“巴黎”的例子）在這種情況下，說(shuō)只有一個(gè)“巴黎”，只有一個(gè)“你”，那就更加自然了，不過(guò)這時(shí)世界已經(jīng)不再是球面，而是一個(gè)射影平面了。

A Lorentz transformation is a linear map from R^4 to R^4 that preserves these “generalized distances.”

洛倫茲變換就是一個(gè)保持其上“廣義距離”不變的從R^4到R4的線性映射。

Let us therefore imagine a planet covered with calm water. If you drop a large rock into the water at the North Pole, a wave will propagate out in a circle of everincreasing radius. (At any one moment, it will be a circle of constant latitude.) In due course, however, this circle will reach the equator, after which it will start to shrink, until eventually the whole wave reaches the South Pole at once, in a sudden burst of energy.

讓我們想象一個(gè)為靜止水體所覆蓋的行星，如果丟一塊大石頭到在北極的水里，水波會(huì)以半徑越來(lái)越大的圈傳播開去（在任何時(shí)刻，這個(gè)圈都是一個(gè)緯圈）然而，到了一定時(shí)候，這個(gè)圈到達(dá)了赤道，此后它會(huì)開始收縮，直到最后，整個(gè)波同一時(shí)間到達(dá)南極，發(fā)生能量的突然爆發(fā)。

Now imagine setting off a three-dimensional wave in space—it could, for example, be a light wave caused by the switching on of a bright light. The front of this wave would now be not a circle but an ever-expanding spherical surface. It is logically possible that this surface could expand until it became very large and then contract again, not by shrinking back to where it started, but by turning itself inside out, so to speak, and shrinking to another point on the opposite side of the universe. ......More to the point, this account can be turned into a mathematically coherent and genuinely three-dimensional description of the 3-sphere.

現(xiàn)在來(lái)想象三維的空間里突然發(fā)出的波——例如，它可以是打開一盞明亮的燈所產(chǎn)生的光波?，F(xiàn)在波前（又稱為波陣面）不再是一個(gè)圈，而是一個(gè)不斷擴(kuò)展的球面。邏輯上，這個(gè)球面可以擴(kuò)展到非常大然后又開始收縮，但并不是收縮回到原點(diǎn)，而是從里翻到外地收縮到宇宙另外一端的某一點(diǎn)上?！匾氖?，這樣的解釋可以變成一種對(duì)3維球面的數(shù)學(xué)上連貫自洽的真正的三維描述。

A different and more general approach is to use what is called an atlas. An atlas of the world (in the normal, everyday sense) consists of a number of flat pages, together with an indication of their overlaps: that is, of how parts of some pages correspond to parts of others. Now, although such an atlas is mapping out an external object that lives in a three-dimensional universe, the spherical geometry of Earth’s surface can be read off from the atlas alone.

處理這個(gè)問(wèn)題的一個(gè)不同的，而且更加一般的途徑是使用圖冊(cè)或者圖集（atlas）。日常生活中的一本世界地圖冊(cè)是由許多平面的地圖頁(yè)訂成的，加上對(duì)于它們之間的重疊的說(shuō)明，說(shuō)明某些頁(yè)面的一部分如何對(duì)應(yīng)另外一些頁(yè)面的一部分。雖然這樣的圖冊(cè)是在地圖上標(biāo)出存在于三維宇宙中的外部對(duì)象，地球表面的球面幾何（所包含的信息）可以僅從這些圖冊(cè)中就讀出。

The idea of an atlas can easily be generalized to three dimensions. A “page” now becomes a portion of threedimensional space. The technical term is not “page” but “chart,” and a three-dimensional atlas is a collection of charts, again with specifications of which parts of one chart correspond to which parts of another.

圖冊(cè)的概念很容易推廣到三維的情形情況。這時(shí)每一頁(yè)都是三維空間的一部分。專業(yè)術(shù)語(yǔ)中，不說(shuō)頁(yè)，而是說(shuō)“區(qū)圖”（chart）。一個(gè)三維圖冊(cè)，就是區(qū)圖的集合，同樣加上對(duì)于一個(gè)區(qū)圖的某一部分如何對(duì)應(yīng)于另一區(qū)圖的哪一部分的說(shuō)明。

The formal definition of a manifold uses the idea of atlases: indeed, one says that the atlas is a manifold.

流形（manifold）的正式定義中使用了圖冊(cè)（atlas）的概念，人們可以說(shuō)圖冊(cè)就是一個(gè)流形。

It may be better to think of a d-manifold in the “extrinsic” way that we first thought about the 3-sphere: as a d-dimensional “hypersurface” living in some higher-dimensional space. Indeed, there is a famous theorem of Nash that states that all manifolds arise in this way.

（按：就獲得對(duì)流形的直觀理解的目的而言，）最好以外在的方式（按：與內(nèi)蘊(yùn)（intrinsic）的方式相反，內(nèi)蘊(yùn)的方式不要參照任何包含其的空間）來(lái)看待一個(gè)d-流形，就像我們最初考察3維球面時(shí)一樣，將其視為一個(gè)存在于更高維的空間中的d維超曲面。實(shí)際上，有一個(gè)著名的納什定理，指出所有的流形都是這樣產(chǎn)生的。

This is guaranteed if the function that gives the correspondence between the overlapping overlapping parts (known as a transition function) is itself differentiable. Manifolds with this property are called differentiable manifolds: manifolds for which the transition functions are continuous but not necessarily differentiable are called topological manifolds. The availability of calculus makes the theory of differentiable manifolds very different from that of topological manifolds.

如果轉(zhuǎn)移函數(shù)（即給出兩個(gè)區(qū)圖間的重疊部分的對(duì)應(yīng)關(guān)系的函數(shù)）自身是可微的，那么這個(gè)函數(shù)對(duì)于兩個(gè)區(qū)圖同為可微或不可微就得到了保證。具備以上屬性的流形可以稱為可微流形。而具備僅連續(xù)但不一定可微的轉(zhuǎn)移函數(shù)的流形被稱為拓?fù)淞餍巍Ｎ⒎值目捎檬沟梦⒎挚晌⒘餍闻c拓?fù)淞餍蔚睦碚撳漠悺?/p>

The single most important moral to draw from the above problems is that if we wish to define a notion of distance for a given manifold, we have a great deal of choice about how to do so. Very roughly, a Riemannian metric is a way of making such a choice.

從以上問(wèn)題所能得到最重要的教訓(xùn)就是，如果想在一個(gè)給定的流形上定義距離的概念，有很多種方式可以可供選擇。而粗略地說(shuō)，黎曼度量就是進(jìn)行選擇的方法。

按：這之前討論了采用區(qū)圖中的相應(yīng)點(diǎn)間的距離來(lái)定義流形中的兩點(diǎn)距離的三個(gè)問(wèn)題，分別是：

1. 兩點(diǎn)可能屬于不同的區(qū)圖；
2. 對(duì)同一流形有很多種選擇區(qū)圖的方式從而無(wú)法得到距離的唯一定義（而就算給定一個(gè)區(qū)圖，在重疊的部分距離的定義也未必兼容）；
3. 圖冊(cè)里的區(qū)圖是平坦的，因此圖冊(cè)內(nèi)的距離將難以體現(xiàn)流形上最短路徑的長(zhǎng)度。

As should be clear by now from the above discussion, on any given manifold there is a multitude of possible Riemannian metrics. A major theme in Riemannian geometry is to choose one that is “best” in some way. ......More generally, one searches for extra conditions to impose on Riemannian metrics. Ideally, these conditions should
be strong enough that there is just one Riemannian metric that satisfies them, or at least that the family of such metrics should be very small.

從以上的討論中應(yīng)該可以清晰看出，在任意給定的流形上總有許多可能的黎曼度量。黎曼幾何學(xué)的一個(gè)重大主題就是從其中選擇在某些方面最好的黎曼度量?！ㄓ玫姆椒ㄊ?，要找出附加在黎曼度量上的額外條件，這些額外的條件要足夠地強(qiáng)，使得只有一個(gè)黎曼度量能夠滿足它們，或者至少要使得滿足這些條件的黎曼度量族很小。