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)對應(yīng)1-2篇筆記训裆。

筆記的方式钩骇，是引用一段個人覺得比較有亮點(diǎn)的英文原文，再給一段簡化的中文說明郑象，不采用中文版的翻譯贡这，不自行做直接翻譯，只說明要點(diǎn)厂榛。因?yàn)椴豢赡艽蠖未蠖蔚厝ヒ酶墙茫厝粫姓Z境的丟失，會做一些補(bǔ)充說明击奶，以“按：”開始辈双。對中文版翻譯進(jìn)行更正或調(diào)整的說明，以“注：”開始柜砾。偶爾也會插入自己的議論湃望，以“評：”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)的語言和語法
• 《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2來說明極限的真正含義证芭。

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

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)對象泛源，而且能定義任意兩個對象間的距離拔妥，那么我們就可以談?wù)撨@些對象的一個序列有沒有極限。如果兩個對象的距離（而不是差）小于δ达箍，那么它們可被稱為“δ-逼近的”没龙。（距離的概念將在度量空間[III.58]里進(jìn)一步討論）。

例如缎玫，空間中的點(diǎn)的序列可以擁有一個極限硬纤，函數(shù)的序列也可以。（函數(shù)間的距離的定義方式?jīng)]有那么顯然赃磨，不過存在很多自然的方式來對其進(jìn)行定義筝家。）一個更進(jìn)一步的例子來自分形的理論（見動力學(xué)[IV.15]）：在里面出現(xiàn)的復(fù)雜圖形，最好是定義為較簡單的圖形的極限邻辉。

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.)

我們說函數(shù)在a處連續(xù)溪王，如果

上面的公式其實(shí)就是說，f(x)作為對f(a)的估算值骇，無論我們希望這個估算多么精確莹菱，對應(yīng)的精度都是可以達(dá)到的，只要我們準(zhǔn)備好讓x是a的一個足夠好的近似吱瘩。如果一個函數(shù)在每個點(diǎn)a處都連續(xù)道伟，它就是連續(xù)函數(shù)。粗略地說使碾，就是它沒有突然的跳躍（這也就排除了會使得精確的估計(jì)變得困難的某一類急速震蕩）蜜徽。

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.

熱需要時間來在介質(zhì)中傳導(dǎo)票摇，所以雖然在很遠(yuǎn)處的點(diǎn) (x',y',z') 的溫度最終會影響到點(diǎn)(x,y,z)拘鞋，此刻（在時間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，那么我們就會預(yù)期點(diǎn)(x,y,z)的溫度會上升兄朋，如果平均更冷掐禁，那我們預(yù)期就會下降怜械。

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Δ颅和，稱為拉普拉斯算子傅事。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峡扩。

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.

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

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买置，一個函數(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强霎。結(jié)果是忿项，我們可以對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城舞。

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

的簡寫轩触。

算子

評：包含這兩個符號，不是因?yàn)檫@是新知識拉馋，只是在我少年時代初次接觸這個三角形榨为、正方形還有另外一個倒三角形時，對數(shù)學(xué)符號升起了某種神秘崇高的感覺煌茴，至今看到這幾個符號依然能喚起那時的感覺柠逞。我如此喜歡這種用幼兒時代就接觸的符號來濃縮中學(xué)時代才能理解的知識的方式。

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把積分和逆微分區(qū)分開來景馁，但是有一個稱為微積分基本定理的著名定理斷言這兩個程序事實(shí)上會給出相同的答案板壮，至少當(dāng)所考察的函數(shù)具有所有“合理”的函數(shù)一定會具有的某些連續(xù)性時是這樣的。因此合住，通常都認(rèn)為把積分看成微分的逆運(yùn)算是合法的绰精。確切些說，如果f是連續(xù)的透葛，而F(x)可以對于某個常數(shù)a定義為

評：這段話對于學(xué)過高等數(shù)學(xué)的同學(xué)可能感覺平淡無奇蹄胰，不過雙語對照閱讀起來，還是有一些新的感覺奕翔，所以摘錄出來裕寨。

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)可微是一個遠(yuǎn)比實(shí)可微要強(qiáng)得多的條件派继，我們也可以期待全純函數(shù)會具備許多有趣的屬性宾袜。

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)定義在整個復(fù)數(shù)平面C上驾窟，如果限制函數(shù)定義在整個復(fù)數(shù)平面的一個單連通區(qū)域庆猫，即沒有洞的開集合上，則一切依然成立绅络。如果區(qū)域里有洞月培，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)繞洞昨稼。因此节视，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)觀察假栓，在整個現(xiàn)代幾何學(xué)里非常多的引申與影響寻行。關(guān)于拓?fù)鋵W(xué)，可以進(jìn)一步參看代數(shù)拓?fù)?/a> 這一條目匾荆。

在第五節(jié)的最后牙丽，中文版討論了Liouville's theorem简卧，而英文電子版缺失。這個定理是說：如果函數(shù)f是定義在整個復(fù)平面上的全純函數(shù)烤芦，而且函數(shù)f是有界的（即存在一個常數(shù)C举娩，使得對于每一個復(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.

如果你還沒有見過一些進(jìn)階的概念铜涉，并且對于現(xiàn)代幾何學(xué)是什么樣的一無所知，那么你只要理解兩個基本的概念（幾何學(xué)與對稱之間的關(guān)系遂唧，以及流形的概念）芙代，就能從這本書收獲更多。

評：點(diǎn)出了現(xiàn)代幾何學(xué)的這兩個最為核心的概念的關(guān)鍵性盖彭。這段話在中文版中纹烹，錯誤地將“進(jìn)階”翻譯為“高深”页滚，并且將進(jìn)階概念與一無所知的并列關(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.”

一般來說铺呵，幾何學(xué)就是數(shù)學(xué)里涉及我們通常會按照慣例視為幾何語言的部分裹驰，如“點(diǎn)”、“直線”陪蜻、“平面”邦马、“空間”贱鼻、“曲線”宴卖、“球”、“立方體”邻悬、“距離”症昏，還有“角度”這樣的詞匯扮演了突出的角色。然而父丰，還存在一種更為深刻的觀點(diǎn)肝谭，最初為克萊因所主張，認(rèn)為變換才是這門科學(xué)的真正的主題蛾扇。所以除了上面列舉的這些詞以外攘烛，還要加上“反射”、“旋轉(zhuǎn)”镀首、“平移”坟漱、“拉伸”、“剪切”更哄、“投影”芋齿，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”成翩。

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.

可以有兩種方式來看待剛性變換觅捆，其一是將它們看作對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麻敌。

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.

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

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赢赊，它們既不能保持距離，也不能保持角度聂示，所以距離和角度都不是仿射幾何學(xué)的概念域携。

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被它們保持鱼喉。

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].

通過等價關(guān)系的概念秀鞭，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”這個思想表達(dá)得更確切趋观。

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概念時自然涌現(xiàn)的幾何學(xué)，其中我們說兩個圖形是等價的锋边，或者用技術(shù)的術(shù)語來說是同胚的皱坛，只要它們均可連續(xù)變形為另外一個。

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)豆巨，它是所有以經(jīng)過原點(diǎn)的直線為軸的旋轉(zhuǎn)剩辟。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往扔，叫做PSL(2,R)贩猎。

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.

為了從這個群中得出對應(yīng)的幾何學(xué)，我們必須先把它理解為萍膛，某個2維點(diǎn)集合的變換群吭服，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蝗罗。

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)的三個最常用的模型是半平面模型艇棕、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.)

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

對射影平面的第二種觀點(diǎn)桩匪，是把它看作R^3中過原點(diǎn)的直線的集合打瘪。因?yàn)橐粭l這樣的直線可由它與單位球面的兩個交點(diǎn)決定，所以也可以把這個集合看成就是單位球面吸祟，但是與普通的球面有一個值得注意的區(qū)別瑟慈，就是（單位球面上）相對的點(diǎn)可以視作同一點(diǎn)，因?yàn)樗鼈儗?yīng)于同一條直線屋匕。這一點(diǎn)很難想象葛碧，但并非不可能。假設(shè)有這樣一個世界过吻，在它的一邊發(fā)生的任何事进泼，該事件一個完全一致的副本，都會在另外一邊完全對應(yīng)的地方發(fā)生纤虽。（按：接下來作者舉了真實(shí)的“你”去副本的“巴黎”的例子）在這種情況下乳绕，說只有一個“巴黎”，只有一個“你”逼纸，那就更加自然了洋措，不過這時世界已經(jīng)不再是球面，而是一個射影平面了杰刽。

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保持其上“廣義距離”不變的從R^4到R4的線性映射菠发。

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.

讓我們想象一個為靜止水體所覆蓋的行星王滤，如果丟一塊大石頭到在北極的水里，水波會以半徑越來越大的圈傳播開去（在任何時刻滓鸠，這個圈都是一個緯圈）然而雁乡，到了一定時候，這個圈到達(dá)了赤道糜俗，此后它會開始收縮踱稍，直到最后，整個波同一時間到達(dá)南極悠抹，發(fā)生能量的突然爆發(fā)珠月。

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⌒颗ィ現(xiàn)在波前（又稱為波陣面）不再是一個圈桥温，而是一個不斷擴(kuò)展的球面引矩。邏輯上梁丘，這個球面可以擴(kuò)展到非常大然后又開始收縮，但并不是收縮回到原點(diǎn)旺韭，而是從里翻到外地收縮到宇宙另外一端的某一點(diǎn)上氛谜。……更重要的是区端，這樣的解釋可以變成一種對3維球面的數(shù)學(xué)上連貫自洽的真正的三維描述值漫。

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.

處理這個問題的一個不同的，而且更加一般的途徑是使用圖冊或者圖集（atlas）织盼。日常生活中的一本世界地圖冊是由許多平面的地圖頁訂成的杨何，加上對于它們之間的重疊的說明，說明某些頁面的一部分如何對應(yīng)另外一些頁面的一部分沥邻。雖然這樣的圖冊是在地圖上標(biāo)出存在于三維宇宙中的外部對象危虱，地球表面的球面幾何（所包含的信息）可以僅從這些圖冊中就讀出。

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.

圖冊的概念很容易推廣到三維的情形情況唐全。這時每一頁都是三維空間的一部分埃跷。專業(yè)術(shù)語中，不說頁邮利，而是說“區(qū)圖”（chart）弥雹。一個三維圖冊，就是區(qū)圖的集合延届，同樣加上對于一個區(qū)圖的某一部分如何對應(yīng)于另一區(qū)圖的哪一部分的說明剪勿。

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（atlas）的概念，人們可以說圖冊就是一個流形方庭。

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：就獲得對流形的直觀理解的目的而言厕吉，）最好以外在的方式（按：與內(nèi)蘊(yùn)（intrinsic）的方式相反赦颇，內(nèi)蘊(yùn)的方式不要參照任何包含其的空間）來看待一個d-流形，就像我們最初考察3維球面時一樣赴涵，將其視為一個存在于更高維的空間中的d維超曲面媒怯。實(shí)際上，有一個著名的納什定理髓窜，指出所有的流形都是這樣產(chǎn)生的扇苞。

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)（即給出兩個區(qū)圖間的重疊部分的對應(yīng)關(guān)系的函數(shù)）自身是可微的，那么這個函數(shù)對于兩個區(qū)圖同為可微或不可微就得到了保證寄纵。具備以上屬性的流形可以稱為可微流形鳖敷。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.

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

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

1. 兩點(diǎn)可能屬于不同的區(qū)圖恤浪；
2. 對同一流形有很多種選擇區(qū)圖的方式從而無法得到距離的唯一定義（而就算給定一個區(qū)圖畅哑，在重疊的部分距離的定義也未必兼容）；
3. 圖冊里的區(qū)圖是平坦的水由，因此圖冊內(nèi)的距離將難以體現(xiàn)流形上最短路徑的長度荠呐。

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砂客。黎曼幾何學(xué)的一個重大主題就是從其中選擇在某些方面最好的黎曼度量泥张。……更通用的方法是鞠值，要找出附加在黎曼度量上的額外條件媚创，這些額外的條件要足夠地強(qiáng)，使得只有一個黎曼度量能夠滿足它們齿诉，或者至少要使得滿足這些條件的黎曼度量族很小筝野。