<<Differential Geometry of Curves and Surfaces>>笔记

<Differential Geometry of Curves and Surfaces> by Manfredo P. do Carmo

real line R
interval I
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CH1 Curves
DEFINITION. A parametrized differentiable curve is a differentiable map a: I --> R3 of 【an open interval I = (a, b)】 of the 【real line R】 into R3
。
Therefore, we call any point t where a'(t) = 0 a 【singular point】 of a.
。
DEFINITION. A parametrized differentiable curve a : I --> R3 is said to be【 regular】 if a'(t) != 0 for all t ∈ I.
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THE THEOREM OF TURNING TANGENTS .
The rotation index of a simple closed curve is & 1, where the sign depends on the orientation of the curve.
。
A 【vertex】 of a regular plane curve a : [a, b]-->R2 is a point t∈[a, b] where k'(t) = 0
。
THEOREM 2 (The Four-Vertex Theorem).
A simple closed convex curve has at Ienst four vertices.

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CH2
. P53
The vector e$_1$ is tangent to the curve u --> (u, v0)
dx$_q$(e$_1$)  = (∂x/∂u, ∂y/∂u, ∂z/∂u) =  ∂x/∂u

Similarly, using the coordinate curve u = u0,  v --> (u0, v)
dx$_q$(e$_2$)  = (∂x/∂v, ∂y/∂v, ∂z/∂v) =  ∂x/∂v

Thus, the matrix of the linear map dx, in the referred basis is
dx$_q$ = (∂x/∂u,  ∂x/∂v)
。
P55
Regurlar surface 之所以是regurlar的条件(确保存在切平面)可以表述为(下面3个等价)：
1）∂x/∂u与∂x/∂v线性无关
2）∂x/∂u /\ ∂x/∂v  != 0
3）that one of the minors of order 2 of the matrix of dx, that is, one of the Jacobian determinants be different from zero a t q.
 Jacobian determinants:
∂(x,y)/∂(u,v) ≡
|∂x/∂u   ∂x/∂v|
|∂y/∂u   ∂y/∂v|
∂(y,z)/∂(u,v) ≡ ...
∂(x,z)/∂(u,v) ≡ ...
.

DEFINITION 2. Given a diflerentiable map F : U ⊂ Rn  -->  Rm defined in an open set U of Rn we say that p ∈ U is a 【critical point】 of F if the differential dF$_p$: Rn --> Rm is not a 【surjective(满射) (or onto) mapping】. The image F(p) ∈ Rm of a critical point is called a 【critical value】 of F. A point of Rm which is not a critical value is called a 【regular value】 of F.
.
判断是否是regular surface：
PROPOSITION 2.
 If f: U ⊂ R3 ---> R is a diferentiable function and a ∈ f(U) is a regular value of f, then f$^-1$(a) is a reghlar surface in R3
.
A surface S ⊂ R3 is said to be 【connected】 if any two of its points can be joined by a continuous curve in S。
判断S是否是连通曲面的方法：
If f : S ⊂ R3 R is a nonzero continuous function defined on a connected surface S, then f does not change sign on S。
。
Proposition 1 says that the graph of a differentiable function is a regular surface. The proposition 3 provides a local converse of this; that is, any regular surface is locally the graph of a differentiable function.
。
Proposition 4 says that if we already know that S is a regular surface and we have a candidate x for a parametrization, we do not have to check that x$^-1$ is continuous, provided that the other conditions hold.
。
We shaIl frequently make the notational abuse of indicating 【f 】and 【f o x】 by the same symbol 【f(u, v)】, and say that f(u, v) is the expression of f in the system of coordinates x.
。
Two regular surfaces S1 and S2 are diffeomorphic if there exists a differentiable map φ : S1--> S2 with a differentiable inverse φ$^-I$: S1-->S2, Such a φ is called a 【diffeomorphism】from S1 to S2
。
diffeomorphism（regular surfaces） ～ isomorphism（vector spaces）～congruence（Euclidean geometry）.
.
P75
A 【regular curve】 in R3 is a subset C⊂R3 with the following property: For each point p∈C there is a neighborhood V of p in R3 and a differentiable homeomorphism α:I ⊂R-->V∩C such that the differentiable dαt is one-to-one for each t∈I
.
extended surfaces of revolution.
.
DEFINITION 2. A 【parametrized surface】 x:. U ⊂ R2 --> R3 is a differentiable map x from an open set U ⊂ R2 into R3 . The set x(U) ⊂ R3 is called the trace of x. x is regular if the diferential dx,: R2 --> R3 is one-to-one for all q E U (i.e., the vectors dx/du, dx/dv are linearly independent for all q ∈ U). A point p ∈ U where dx, is not one-to-one is called a 【singular point】 of x.
。
Let a:I-->R3 be a regular parametrized curve. Define
x(t, v) = a(t) + v*a‘(t),        (t, v)∈I x R
x is a parametrized surface called the 【tangent surface】 of a.
.
.
【tangent plane】 to S at p and will be denoted by T$_p$(S). The choice of the parametrization x determines a basis {∂x/∂u(q), (∂x/∂v)(q)} of T$_p$(S), called the basis associated to x. Sometimes it is convenient to write ∂x/∂u = x$_u$ and ∂x/∂v = x$_v$
.
The linear map 【dφ$_p$】 defined by Prop2 is called the 【differeniial of φ at p∈S1】 . In a similar way we define the differential of a (differentiable) function f:U⊂S-->R at p∈U as a linear map 【df$_p$】
。
A 【critical point】 of a differentiable functionf: S --, R defined on a regular sur-
face S is a point p∈S such that df$_p$=0
.
(Chain Rule.) Show that if φ : S1-->S2 and ψ : S2-->S3 are differentiable maps
and p∈S1 , then
d(ψ o φ)$_p$ = dψ$_{φ(p)}$ o dφ$_p$
.
The first fundamental form( I$_p$(w)=<w, w>$_p$ , w∈Tp(S))的意义：
the first fundamental form is merely the expression of how the surface S inherits the natural inner product of R3. Geometrically, as we shall see in a while, the first fundamental form allows us to make measurements on the surface (lengths of curves, angles of tangent vectors, areas of regions) without referring back to the ambient space R3 where the surface lies.
the importance of the first fundamental form I comes from the fact that by knowing I we can treat metric questions on a regular surface without further references to the ambient space R3
。
I$_p$(α'(p)) = E*u'^2 + 2F*u'v' +G*v'^2
E(u0, v0) = <x$_u$, x$_u$>$_p$
F(u0, v0) = <x$_u$, x$_v$>$_p$
G(u0, v0) = <x$_v$, x$_v$>$_p$
。
the coordinate curves of a parametrization are orthogonal if and only if F(u, v) = 0 for all (u, v). Such a parametrization is called an 【orthogonal parametrization】
.
 ds ：the "element" of arc length  of S
 if a(t) = x(u(t),v(t)) is a curve on S and s = s(t) is its arc length, then
(ds/dt)² = E(du/dt)² +2F(du/dt)(dv/dt) + G(dv/dt)²
简记为ds² = E*du² + 2F*dudv + G*dv²
。
DEFINITION 2. Let R⊂S be a bounded region of a regular surface contained in the coordinate neighborhood of the parametrization x : U⊂R2 S.The positive number is called the 【area of R】.
∫∫$_Q$ |x$_u$ /\ x$_v$| dudv ≡ A(R) ≡Area 【面积】
因为|x$_u$ /\ x$_v$|² + <x$_u$, x$_v$>²=  |x$_u$|²|x$_v$|²
所以|x$_u$ /\ x$_v$| = sqrt(EG-F²)
所以∫∫$_Q$ sqrt(EG-F²) dudv ≡ A(R)
。
Gradient on Surfaces
df$_p$(v)≡<grad f(p), v>$_p$     for all v∈T$_p$(S)
.
Orthogonal Families of Curves
.
2- 6. Orientation o f Surfaces
DEFINITION 1. A regular surface S is called 【orientable】 if it is possible to cover it with a family of coordinate neighborhoods in such a way that if a point p∈S belongs to two neighborhoods of this family, then the change of coordinates has positive Jacobian at p. The choice of such a family is called an 【orientation】 of S, and S, in this case, is called 【oriented】. v s u c h a choice is not possible, the surface is called nonorientable.
.
Orientation is a global property
orientability is preserved by difleomorphisms
orientable and oriented
.
Let A be a subset of R3, We say that p∈R is 【a limit point】 of A if every neighborhood of p in R3 contains a point of A distinct from p.
A is said to be 【closed 】if it contains all its limit points.
A is 【bounded】 if it is contained in some ball of R3.
If A is closed and bounded, it is called a 【compact set】。
。
Properties of compact subsets of R3
（   d(p, q)：The distance between two points p,q∈R3   ）
。
B. Differentiability in Rⁿ
f is diferentiable at x0, if it has continuous derivatives of all orders at x0,
f is diferentiable in U if it is differentiable at all points in U
。
Remark. We use the word differentiable for what is sometimes called infinitely differentiable (or of class C∞). Our usage should not be confused with the usage of elementary calculus, where a function is called differentiable if its first derivative exists
。
It is an important fact that when f is differentiable the partial derivatives
off are independent of the order in which they are performed
。
the matrix (∂fi/∂xj), i = 1, . . . , m, j = I , . . . , n, is called the 【Jucobian matrix】 of f at p.
When n = m, this is a square matrix and its determinant is called the 【Jacobian determinant】; it is usual to denote it by   det(∂fi/∂xj) ≡ ∂(f1,...,fn)/∂(x1,...xn)

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CH3 The Geometry of the Gauss Map
if V⊂S is an open set in S and N: V-->R3 is a differentiable map which associates to each q∈V a unit normal vector at q, we say that N is a【 diferentiable field of unit normal vectors】 on V
。
We shall say that a regular surface is 【orientable】 if it admits a differentiable field of unit normal vectors defined on the whole surface
。
Define a basis {v, w}∈ Tp(S) to be 【positive】 if (v /\ w, N) is positive.
.
We say that a linear map A: V-->V is 【self-adjoint】 if (Av, w) = (v, Aw) for all v, w∈V
。
dNp: The differential value of N at p
The tangent vector N'(0) ≡ dNp(a'(0)) is a vector in Tp(S) (Fig. 3-3)
.
The fact that dNp: Tp(S)-->Tp(S) is a self-adjoint linear map allows us to associate to dNp a quadratic form Q in Tp(S), given by Q(v) = <dNp(v), v>,   v∈Tp(S)
几何解释：
IIp(α'(0)) = k$_n$(p), In other words, the value of the second fundamental form IIp for a unit
vector v∈Tp(S) is equal to the normal curvature of a regular curve passing through p and tangent to v.
这也意味着：
PROPOSITION 2 (Meusnier). All curves lying on a surface S and having at a given point p∈S the same tangent line have at this point the same normal curvatures.【C and Cn have the same normal curvature at p along v】
.
S上每个点p处的切面上都有正交基{e1, e2}， 有dNp(e1) = -k1e1, dNp(e2) = -k2e2， （k1>=k2）. k1, k2是切面上单位圆的IIp的极值
.
DEFINITION 4. The maximum normal curvature k1 and the minimum normal curvature k2 are called the 【principal curvatures】 at p; the corresponding directions, that is, the directions given by the eigenvectors e1, e2, are called 【principal directions】 at p.
。

IIp(v)  ≡ -<dNp(v), v>     v∈Tp(S)  （IIp由N的微分来描述， 对比Ip使用切向量来描述的）
.
DEFINITION 3.（由曲面的曲率得到曲线法线曲率）
 Let C be a regular curve in S passing through p∈S, k the curvature of C at p, and cosθ=<n, N>, where n is the normal vector to C and N is the normal vector to S at p. The number k$_n$ = k*cosθ is then called the 【normal curvature of C⊂S at p】.
。
DEFINITION 5. If a regular connected curve C on S is such that for all p∈C the tangent line of C is a principal direction at p, then C is said to be 【a line of curvature of S】
。
PROPOSITION 3 (Olinde Rodrigues). A necessary and sufficient condition for a connected regular curve C on S to be a line of curvature of S is that
N'(t) = λ(t) a'(t)
for any parametrization a(t) of C , where N(t) = N o a(t) and R(t) is a differentiable function of t . In this case, -λ(t) is the (principal) curvature along a'(t)
.
Euler formula:
v∈Tp(S), θ is the angle from e1 to v in the orientation of Tp(S)
沿v方向的曲线法线曲率: kn = k1*cos²θ+k2*sin²θ
。
DEFINITION 6. Let p∈S and let dNp: Tp(S) -->Tp(S) be the differential of the Gauss map. The det(dNp) is the 【Gaussian curvature】 K of S at p. The negative of half of the trace of dN, is called the 【mean curvature】 H of S at p. 记为：
K = k1*k2
H = (k1+k2)/2
.
DEFINITION 7. A point of a surface S is called
1. Elliptic if det(dNp) > 0.
2. Hyperbolic if det(dNp) < 0.
3. Parabolic if det(dNp) = 0, with dNp != 0.
4. Planar if dNp = 0.
.
DEFINITION 8. If at p∈S, k1=k2 then p is called an 【umbilical point脐点】 of S; in particular, the planar points (k1=k2=0) are umbilical points.
。
PROPOSITION 4, I f all points of a connected surface S are umbilical points, then S is either contained in a sphere or in aplane.
。
DEFINITION 9. Let p be a point in S. An 【asymptotic(渐进线) direction of S at p】 is a direction of Tp(S) for which the normal curvature is 0. An asymptotic curve of S is a regular connected curve C⊂S such that for each p∈C the tangent line of C at p is an asymptotic direction.
。
Let p be a point in S. The 【Dupin indicatrix at p】 is the set of vectors w of Tp(S) such that IIp(w) = ±1.（即 ±1 = k1* x^2 + k2* y^2）
。
Dupin indicatrix 用来求conjugate directions（P150）
.
DEFINITION 10, Let p be a point on a surface S.
- Two nonzero vectors w1, w2∈Tp(S) are 【conjugate】 if <dNp(w1), w2> = <w1, dNp(w2)> = 0.
- Two directions r1, r2 at p are 【conjugate】 if a pair of nonzero vectors w1, w2 parallel to r1 and r2 respectively, are conjugate.
。
the principal directions are conjugate, and that an asymptotic(渐进线) direction is conjugate to itself

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3-3, The Gauss Map in Loca/ Coordinates
The tangent vector to a(t) at p is α' = x$_u$u' + x$_v$v' and
dN(α') = N'(u(t), v(t)) = N$_u$u' + N$_v$v'
Since N$_u$ and N$_v$ belong to Tp(S), we may write
N$_u$ = a11 x$_u$ + a21 x$_v$,   (Eq. (I))
N$_v$ = a12 x$_u$ + a22 x$_v$    (Eq. (2))
and therefore,
dN(α') = (a11 u' + a12 v') x$_u$ + (a21u' + a22 v') x$_v$
hence

On the other hand,
IIp(α') ≡ -<dN(α'), α'> = -<N$_u$u' + N$_v$v', x$_u$u' + x$_v$v'> = e*(u')² + 2f*u'v' + g*(v')²
where, since <N, x$_u$>= <N, x$_v$> = 0,
e = -<N$_u$, x$_u$> = <N, x$_{uu}$>   (因为法线N和切向量x$_u$相互垂直，所以N*x$_u$=0, 等式两边对第二个坐标参数求偏导得，N$_u$*x$_u$+N, x$_{uu}$=0，得证。)
f =  -<N$_v$, x$_u$> = <N, x$_{uv}$> = <N, x$_{vu}$> = -<N$_u$, x$_v$>
g = -<N$_v$, x$_v$> = <N, x$_{vv}$>
。
-f = <N$_u$, x$_v$> = a11 F + a21 G   (在Eq. (I)两端乘x$_v$*cosθ)
-f = <N$_v$, x$_u$> = a12 F + a22 G   (在Eq. (2)两端乘x$_u$*cosθ)
-e = <N$_u$, x$_u$> = a11 E + a21 F   (在Eq. (I)两端乘x$_u$*cosθ)
-g = <N$_v$, x$_v$> = a12 F + a22 G   (在Eq. (2)两端乘x$_v$*cosθ)
where E, F, and G are the coefficients of the first fundamental form in the basis {x$_u$, x$_v$}

。
(u, v, w)≡<u /\ v, w>
N =  (x$_u$ /\  x$_v$) / |x$_u$ /\  x$_v$|
.
II(a'(t)) = e*(u')² + 2f*u'v' + g*(v')² = 0 is called 【the diferential equation of the asymptotic curves】.
。

消去λ，再次得the diferential equation of the asymptotic curves，
写成行列式形式：
| (v')²    -u'v'    (u')² |
|  E        F        G    | = 0   （Eq. 8）
|  e         f         g    |
。
the principal directions are orthogonal to each other.
.
a necessary and suficient condition for     [the coordinate curves of a parametrization] to be [lines of curvature] in a neighborhood of a nonumbilicalpoint    is that    F = f = 0
.
a convenient expression for the Gaussian curvature of a surface of revolution: K = -φ''/φ
.
If a parametrization of a regular surface is such that F = f = 0, then the principal curvatures are given by e/E and g/G.
In this case, the Gaussian curvature K and the mean curvature H are given by:
K = (eg)/(EG),   H = 1/2 * (e/E - g/G)
.
Each point of S has a neighborhood that can be written as z = h(x, y).
the Hessian of h at (0,O) is the second fundamental form of S at p.
.
用Gauss map来解释Gauss curvature的几何意义(P167)：
K(p) = lim{A->0} A'/A
A‘ ≡ ∫∫$_R$ | N$_u$ /\ N$_v$ | dudv =  ∫∫$_R$ K* | x$_u$ /\ x$_v$ | dudv
.
k ≡ lim{s->0} θ/s = lim{s->0} σ/s (indicatrix of tangents)
Gaussian curvature K is the analogue of the curvature k of plane curves.
.
The notion of 【contact of order】>= n is invariant by diffeomorphisms.
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3-4. Vector Fields
A 【vector field】 in an open set U c R2 is a map which assigns to each q∈U a vector w(q)∈R2.  The vector field w is said to be differentiable if writing q = (x, y) and w(q) = (a(x, y), b(x, y)), the functions a and b are differentiable functions in U.
。
Given a vector field w, it is natural to ask whether there exists a【 trajectory】of this field, that is, whether there exists a differentiable parametrized curve α(t) = (x(t), y(t)), t∈I,
such that α‘(t) = w(α(t))  ： α是w的trajectory
。
The map a is calIed the (local)【flow】 of w at p.
A 【field of directions】 r in an open set U⊂R2 is a correspondence which assigns to each p∈U a line r(p) in R2 passing through p. 【r is said to be diferentiable】 at p∈U if there exists a nonzero differentiable vector field w, defined in a neighborhood V⊂U of p, such that for each q∈V, w(q) ！= 0 is a 【basis of r(q)】; r is diferentiable in U if it is differentiable for every p∈U.
。
A regular connected curve C c U is 【an integral curre of a field of directions r 】defined in U⊂R2 if r(q) is the tangent line to C at q for all q∈C.  给定 r ，则存在C。
。
two nonzero vectors w1 and w2  are 【equivalent】 if w1 = λw2 for some λ∈∂R, λ！=0.
In the language of differential equations, a field of directions r is usually given by
a*dx + b*dy = 0
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3-5. Ruled Surfaces and Minimal Surfaces
.
Given a one-parameter family of Iines {a(t), w(t)}, the parametrized surface:
x(t, v) = α(t)+ v*w(t)
is called the 【ruled surface】 generated by the family {α(t), w(t)}. The lines Lt are called the 【rulings】, and the curve α(t) is called a 【directrix】 of the surface x.
.>>>>>>>>>>>>>>>>>>>>>>>>>>>
摘自其他地方的一些资料：http://encyclopedia2.thefreedictionary.com/ruled+surface
a surface that can be thought of as a one-parameter family of lines; it may be generated by moving a line 【the generatrix(rulings)】) along some curve (the 【directrix】).
Ruled surfaces are divided into developable surfaces and nondevelopable surfaces (skew ruled surfaces).
a developable ruled surface is the envelope of a one-parameter family of planes.
For a nondevelopable ruled surface the tangent planes are different at different points of a given generatrix. When the point of tangency moves along the generatrix, the tangent plane rotates about the generatrix. A full rotation of the tangent plane, corresponding to a traversal of the entire generatrix by the point of tangency, is equal to 180°. Every generatrix has a point such that for each of the two parts into which it divides the generatrix a full rotation of the tangent plane is equal to 90°. This point (point 0 in Figure 2) is called 【the center of the generatrix】.

Figure 2

The tangent of the angle between the planes tangent to the surface at the center O and at some other point O’; of the same generatrix is proportional to the distance OO’. The proportionality factor is called the 【distribution parameter of the ruled surface】. The absolute value of the total curvature of a ruled surface reaches its maximum value on a given generatrix at its center and decreases as we move away from the center along the generatrix. The locus of the centers of generatrices is called 【the curve of striction】. For example, for a helicoid—a ruled surface described by the uniform spiral motion of a line about some axis (which the moving line intersects at a right angle)—the axis (AB in Figure 2) is the curve of striction. Quadric ruled surfaces—the hyperbolic paraboloid, the hyperboloid of one sheet—have two different systems of rectilinear generatrices (the radio tower of V. G. Shukhov’s system, which is located at Shabolovka in Moscow, was constructed from hyperboloids of one sheet). Quadric ruled surfaces are the only doubly ruled surfaces.

If two ruled surfaces can be rolled out on each other, then it is possible to roll one along the other in such a way that they will have a common generatrix. The application of ruled surfaces in the theory of mechanisms is based on this fact.
<<<<<<<<<<<<<<<<<<<<<<<<<<<<
.
Sometimes we use the expression ruled surface to mean the trace of x.
The assumption w‘(t) ！= 0, t∈I, is usually expressed by saying that the ruled surface x is 【noncylindrical】。
β(t) =  α(t)+ u(t)*w(t)， u = -<α', w'>/<w', w'>
β is called the【 line of striction】,and its points are called the 【central points of the ruled surface】
β不依赖于α
用β表示x：x(t, v) = β(t)+ u*w(t),
计算gaussian曲率K = -λ²/(λ²+u²)² <=0    (Eq.6)
central point的几何解释：the points of a ruling, except perhaps the central point, are regular points of the surface. If A!=0, the function |K(u)I is a continuous function on the ruling and, by Eq. (6), the central point is characterized by the fact that IK(u)I has a maximum there.
。
α(t) is called the 【distribution parameter】 of x(t, v)
。
λ的解释：tanθ = u/λ
.
The surface (8) is said to be 【developable】 if (w, w', α')≡0  ==> Gaussian curvature K=0
.
B, Minimal Surfaces
A regular parametrized surface is caIled minimal if its mean curvature vanishes everywhere.
（wiki：a minimal surface is a surface that locally minimizes its area. This is equivalent to  having a mean curvature of zero.）
A regular parametrized surface x = x(u, u) is said to be 【isothermal等温线的】 if (x$_u$, x$_u$) = (x$_v$, x$_v$) and (x$_u$, x$_v$) = 0
.
PROPOSITION 2.
Let x = x(u, v) be a regular parametrized surface and assume that x is isothermal. Then   x$_{uu}$ + x$_{vv}$ = 2λ² H,   where λ² = <x$_u$, x$_u$> = <x$_v$, x$_v$>
。
Δf = ∂²f/∂u² + ∂²f/∂v², We say that f is 【harmonic】 in U if Δf = 0
。
Let x(u, v) = (x(u, v), y(u, v), z(u, v)) be a parametrized surface and assume that x is isothermal. Then x is minimal if and only if its coordinate functions x, y, z are harmonic
。
Let C denote the complex plane, which is, as usual, identified with R2 by setting ζ = u+iv, ζ∈C, (u, v)∈R2. We recall that a function f: U∈C-->C is 【analytic】 when, by writing
f(ζ) = f1(u, v) + i f2(u, v)
the real functions f1 and f2 have continuous partial derivatives of first order which satisfy the so-called 【Cauchy-Riemann equations】:   ∂f1/∂u = ∂f2/∂v,   ∂f1/∂v = - ∂f2/∂u
f1 and f2 are said to be 【harmonic conjugate】
.
Now let x: U⊂R2-->R3 be a regular parametrized surface and define complex functions φ1， φ2， φ3, by： φ1(ζ)= ∂x/∂u - i ∂x/∂v,   φ2(ζ)= ∂y/∂u - i ∂y/∂v,   φ3(ζ)= ∂z/∂u - i ∂z/∂v,
where x, y, and z are the component functions of x.

LEMMA, x is isothermal if and only if φ1²+ φ2² + φ3² ≡ 0. If this last condition is satisfied, x is minimal if and only if φ1 , φ2, and φ3 are analytic functions.
.
=============================================================
CH4 The Intrinsic Geometry of Surfaces
DEFINITION 1. A diffeomorphism φ : S1-->S2 is an isometry i f for all p∈S and all pairs w1, w2∈Tp(S) we have
<w1, w2>$_p$ = <dφ$_p$(w1), dφ$_p$(w2)>$_{φ(p)}$
The surfaces S1 and S2 are then said to be 【isometric】.
In other words, a diffeomorphism φ is an isometry if the differential dφ preserves the inner product. It follows that, dφ being an isometry, for all w∈T$_p$(S):
I$_p$(w)≡<w, w>$_p$ = <dφ$_p$(w1), dφ$_p$(w2)>$_{φ(p)}$ ≡ I$_{φ(p)}$(dφ$_p$(w))
.
DEFINITION 2. A map φ: V->S~ of a neighborhood V of p∈S is a local isometry at p if there exists a neighborhood V~ of φ(p)∈S~ such that φ: V->V~  is an isometry. If there exists a local isometry into S at every p∈S, the surface S is said to be locally isometric to S~.
S and S~ are locally isometric if S is locally isometric to S and S is 【locally isometric】 to S
。
>>>>>>>>>>>>>>>>>>>>>>>>>>
homeomorphism同胚  v.s.  isometry等距
 (http://mathworld.wolfram.com/Homeomorphism.html)
A 【homeomorphism】, also called a 【continuous transformation】, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions.
A homeomorphism which also preserves distances is called an 【isometry】.
Affine transformations are another type of common geometric homeomorphism.
.
homeomorphism  v.s.  Homomorphism同态/同形
 (http://mathworld.wolfram.com/Homomorphism.html)
Homomorphism is a term used in category theory to mean a [general morphism]. The term derives from the Greek omicronmuomicron (omo) "alike" and muomicronrhophiomegasigmaiotasigma (morphosis), "to form" or "to shape."
The similarity in meaning and form of the words "homomorphism" and "homeomorphism" is unfortunate and a common source of confusion.
A morphism is a map between two objects in an abstract category.
An isomorphism between an object and itself is called an 【automorphism】.
<<<<<<<<<<<<<<<<<<<<<<<<<<
The notion of isometry is the natural concept of equivalence for the metric properties of regular surfaces. In the same way as diffeomorphic surfaces are equivalent from the differentiability viewpoint, so isometric surfaces are equivalent from the metric viewpoint。
。
It is possible to define further types of equivalence in the study of surfaces. From our point of view, diffeomorphisms and isometries are the most important. However, when dealing with problems associated with 【analytic functions of complex variables】, it is important to introduce the 【conformal equivalence】。
。
DEFINITION 3. A diffeomorphism φ : S->S~ is called a 【conformal map】if for all p∈S and all v1, v2∈Tp(S) we have
<dφ$_p$(v1), dφ$_p$(v2)> = λ²(p) *<v1, v2>$_p$.
where λ² is a nowhere-zero diflerentiable function on S. the surfaces S and S~are then said to be conformal.
.
PROPOSITION 2. Let x:U-->S and x~:U-->S~ be parametrizations such that E = λ²E, F = λ²F, G = λ²G in U, where λ² is a nowhere-zero differentiable function in U. Then the map φ = x~ o x$^{-1}$: x(U)-->S~ is a 【local conformal map】.
.
THEOREM. Any two regular surfaces are locally conformal.
.
The proof that 【there exist isothermal coordinate systems on any regular surface】 is delicate and will not be taken up here
。
EXERCISE 7  【linear isometry】
When W = V, a linear isometry is often called an 【orthogonal transformation】
EXERCISE 9  【group of isometries】
EXERCISE 13 【linear conformal map】
if φ is area-preserving and conformal, then φ is an isometry。
---------------------------------------------
4-3. The Gauss Theorem and the Equations of Compatibility
The coefficients Γijk,, i, j, k = 1, 2, are called the 【Christoffel symbols】 of S in the parametrization x.
。
All geometric concepts and properties expressed in terms of the Christoffel symbols are invariant under isometries.
。
THEOREMA EGREGIUM (Gauss). The Gaussian curvature K of a surface is invariant by local isometries。
The Gauss formuIa and the Mainardi-Codazzi(Eq 6,6a) equations are known under the name of compatibility equations of the theory of surfaces
.
A natural question is whether there exist further relations of compatibility
between the first and the second fundamentaI forms besides those already
obtained. The Gauss-Bonnet theorem says, No.
In other words, by successive derivations or any other process we would obtain
no further relations among the coefficients E, F, G , e, f, g and their deriva-
tives. Actually, the theorem is more explicit and asserts that the knowledge
of the first and second fundamental forms determines a surface locally.
。
THEOREM (Bonnet)
x~ = T ο ρ ο x
where T: translation , ρ: linear orthogonal transformation.
(An orthogonal transformation is a linear transformation T:V->V which preserves a symmetric inner product. In particular, an orthogonal transformation (technically, an orthonormal transformation) preserves lengths of vectors and angles between vectors
(source http://mathworld.wolfram.com/OrthogonalTransformation.html) )
-------------------------------------------------
4-4. Parallel Transport. Geodesics
DEFINITION 1. Let w be a diferentiable vector field in an open set U⊂S and p∈U . Let y∈Tp(S). Consider a parametrized curve
α:(-ε, ε)-->U,
with α(0) = p and α'(0) = y, and let w(t), t∈(-ε, ε) , be the restriction of the vector field w to the curve α . The vector obtained by the normal projection of (dw/dt)(0) onto the plane Tp(S) is called the 【covariant derivative共变导数】at p of the vector field w relative to the vector y.
This covariant derivative is denoted by (Dw/dt)(0) or (D$_y$w)(p)
（dw/dt在切平面上的投影是Dw/dt ）
 
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
http://blog.csdn.net/wangxiaojun911/article/details/17198771
共变导数则是在流体上定义导数的方法。
在基于欧几里得空间的笛卡尔坐标系里，对向量场求导数的方法与上文类似，即取两个空间坐标相近的点，然后考察其向量差与位置改变的比值。如果位置改变量是无穷小量，那么可以得到该点的导数。
但是，在流形的球面上，位置改变量的计算则根本不切实际，因为当移动一个向量的时候，随着路径的不同，结果根本就不一样。一个向量沿着球面转动一圈，因为曲率不为零，可能根本就不是原来那个向量了。换句话说，在曲面上的每个点上没有统一的坐标系，所以要把坐标系的变化考虑在内。或者说是共变导数是不依赖坐标系的求导方法。

联络(Connection)
联络描述了空间中某一点，对应于另外一点的空间转换。此表述隐含了一些假设。
首先曲面上每一点定义一个相互独立的空间，称为切空间(Tangent Space)。切空间是由该点的所有切向量(Tangent Vector)组成的空间, 这些切向量都是垂直于该点法线方向的向量。其次，定义在不同切空间中的切向量是不能相互运算，比如相加和相减的。因为曲率不加以考虑的话，这些运算都没有意义。
但是，这些不同点的切空间之间是有联系的，这些联系就叫联络。联络可以把无穷接近的两个切空间中的向量，转换到同一个切空间中。联络实际上是反映了切空间的弯曲程度。
有很多种实现联络的方法。但前提是，这些不同切空间中相应的向量的分量是需要可以相互对应的。
如何使用联络定义共变导数? 使用倒三角加上两个位于同一点的向量（比如v，u）来表示。可以写作Dvu，读作向量u沿着向量v的共变导数。定义参考向量v的意义在于，移动后的u向量要额外考虑它原本参考系中的变化（联络，即跟空间的结构变化有关），这是它与普通导数的最大区别。

wiki: https://en.wikipedia.org/wiki/Covariant_derivative
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
.
The covariant derivative is, therefore, a generalization of the usual derivative of vectors in the plane
.
covariant derivative的定义可以扩展到vector field上，为此先定义如下几个定义
。
DEFINITION 2. A parametrized curve a: [0, I] -->S is the 【restriction】 to [0, l] of a differentiable mapping of (0-ε, l+ε), ε>0, into S. If α(0) = p and a(l) = q, we say that a 【joins】 p to q. α is regular if α'(t)!=0 for t ∈[0, I].
(use I denote interval [0, I])
。
DEFINITION 3. Let α: I-->S be aparametrized curve in S . A 【vector field】 w along a is a correspondence that assigns to each t∈I a vector w(t)∈Tp(S)
。
把covariant derivative扩展到vector field上：
DEFINITION 4. Let w be a diferentiable vector field along α:I-->S.
The expression (1) of (Dw/dt)(t), t∈I, is well defined and is called the 【covariant derivative】 of w at t.
。
DEFINITION 5. A vector field w along a parametrized curve α: I -->S is said to be 【paralIel】 if Dw/dt = 0 for every t ∈ I .
。
PROPOSITION 2. Let a: I--> S be a parametrized curve in S and let w0∈Tα(t0)(S), t0∈I. Then there exists a unique parallel vector field w(t) along α(t), with w(t0) = w0.
Proposition 2 allows us to talk about parallel transport of a vector along a parametrized curve.
.
用 tangent vector field is parallel来定义geodesic：
DEFINITION 8, A nonconstant, parametrized curve r : I-->S is said to be 【geodesic】 at t∈I if the field of its tangent vectors r'(t) is parallel along r at t ; that is,
Dr'(t)/dt = 0
r is a parametrized geodesic if it is geodesic for all t∈I.
.
DEFINITION 9. Let w be a diflerentiable field of unit vectors along a parametrized curve α: I-->S on an oriented surface S. Since w(t), t∈I, is a unit vector field, (dw/dt)(t) is normal to w(t), and therefore
Dw/dt  = λ(N /\ w(t))   （dw/dt在切平面上的投影是Dw/dt）
The real number λ=λ(t) , denoted by [Dw/dt], is called the 【algebraic value of the covariant derivative】 of w at t.
the sign of [Dw/dt] depends on the orientation of S and that [Dw/dt] = <dw/dt, N/\w>.
.
geodesic与面的朝向无关，但geodesic的曲率与面的朝向有关。
。
[Dα‘(s)/ds]≡k$_g$   is called the 【geodesic curvature】（当α被投影到切平面上得到曲线β，geodesic curvature等于β的曲率）
。
α‘’(s)=k*n在切平面上的分量是k$_g$ （k is the curvature of C at p and n is the normal vector of C at p）， 所以，k² = k$_n$² + k$_g$²

Figure 4-17

。
为什么要引入vector field？原因是为了测量geodesic curvature.
如何测量geodesic curvature？仅凭α‘(s)(切线)的角度的变化率是无法衡量geodesic curvature的。所以，又加了一个parallel field v(s) 做为参考----用切线与v(s)的夹角的变化率来衡量曲率.
。

differential equations of the geodesics of S

.
PROPOSITION 3. Let x(u, v) be an orthogonal parametrization (that is, F = 0) of a neighborhood of an oriented surface S, and w(t) be a differentiablefield of unit vectors along the curve x(u(t), v(t)). Then
[Dw/dt] = 1/(2*sqrt(EG)) * (G$_u$ dv/dt - E$_v$ du/dt) + dφ/dt
where φ(t) is the angle from x$_u$ to w(t) in the given orientation.
.
PROPOSITION 5. Given a point p ∈S and a vector w∈ Tp(S), w!= 0, there exist an ε > 0 and a unique parametrized geodesic r : (-ε, ε) --> S such that r(0) = p, r'(0)= w. (给定点p和方向w，则唯一确定geodesic)
。
---------------------------------------------
4-5. The Gauss-Bonnet Theorem and Its App/ications
one of the most important features of the Gauss-Bonnet theorem is that it provides a remarkable relation between the topology of a compact surface and the integral of its curvature。
。
Let S be an oriented surface. A region R⊂S (union of a connected open set with its boundary) is called a【 simple region】 if R is homeomorphic to a disk and the boundary ∂R of R is the trace of a simple, closed, piecewise regular, parametrized curve α: I -->S
We say then that a is 【positively oriented】if 沿α 的正向的话，α延逆时针方向把R围起来。
。
∫∫$_{x^-1(R)}$ f(u,v) sqrt(EG-F²) dudv, 这个式子不依赖于x，所以被称为:
the integral of f over the region R, 记为：∫∫$_R$ f dσ
.
lim$_{R->p}$ Δφ/A(R) = K$_p$ （高斯曲率的几何解释）
（Δφ与w0无关，与α(0)无关）
-----------------------------------------

GAUSS-BONNET THEOREM (Local).

。
Euler-Poincare' characteristic of the triangulation： F - E + V = χ

GLOBAL GAUSS-BONNET THEOREM

COROLLARY 2. Let S he an orientable compact surface; then   ∫∫$_S$ K dσ = 2π χ(S)
.
applications:
1. A compact surface of positive curvature is homeomorphic to a sphere.
2. Let S be an orientable surface of negative or zero curvature. Then two geodesics r1 and r2, which start from a point p∈S cannot meet again at a point q∈S in such a way that the traces of r1 and r2, constitute the boundary of a simple region R of S.
3.Lek S be a surface homeomorphic to a cylinder with Gaussian curvature K < 0. Then S has at most one simple closed geodesic
4. If there exist two simple closed geodesics Γ1 and Γ2 on a compact surface S of positive curvature, then Γ1 and Γ2 intersect.
6.  ∫∫$_T$ K dσ = ∑φ$_i$ - π
If K != 0 on T, this is the area of the image N (T) of T by the Gauss map N: S-->S2. This was the form in which Gauss himself stated his theorem: The excess of a geodesic triangle T is equal to the area of its spherical image N(T).
.
The above fact is related to a historical controversy about the possibility
of proving Euclid's fifth axiom (the axiom of the parallels), from which it
follows that the sum of the interior angles of any triangle is equal to n. By
considering the geodesics as straight lines, it is possible to show that the
surfaces of constant negative curvature constitute a (local) model of a geome-
try where Euclid's axioms hold, except for the fifth and the axiom which
guarantees the possibility of extending straight lines indefinitely. Actually,
Hilbert showed that there does not exist in R3 a surface of constant negative
curvature, the geodesics of which can be extended indefinitely (the pseudo-
sphere of Exercise 6 , Sec. 3-3, has an edge of singular points). Therefore, the
surfaces of R3 with constant negative Gaussian curvature do not yield a
model to test the independence of the fifth axiom alone. However, by using
the notion of 【abstract surface】, it is possible to bypass this inconvenience and
to buiId a model of geometry where all of Euclid's axioms but the fifth are
valid. This axiom is, therefore, independent of the others.
。
POINCARE'S THEOREM. The sum of the indices of (a diferentiable vector field v with) isolated singular points on a compact surface S is equal to the Euler-Poincare characteristic of S:
∑I$_i$ = 1/(2PI) * ∫∫$_S$ K dσ = χ(S)
It implies that ∑I$_i$ does not depend on v but only on the topology of S. For instance, in any surface homeomorphic to a sphere, all vector fields with isolated singularities must have the sum of their indices equal to 2. In particular, no such surface can have a differentiable vector field without singular points.
--------------------------------
4-6. The Exponential Map. Geodesic Polar Coordinates
LEMMA1. If the geodesic r(t, v) is defined for t∈(-ε, ε), then the geodesic r(t, λv), λ∈R, λ!=0, is defined for t∈(-ε/λ, ε/λ), and r(t, λv) = r(λt, v).
We shall now introduce the following notation. If v=Tp(S), v!=0, is such that r(|v|, v/|vI) = r(1, v) is defined, we set:
  exp$_p$(v) ≡ r(1, v) and exp$_p$(0) = p
.
【The important point is that exp$_p$ is always defined and differentiable in some neighborhood of p.】
.
PROPOSITION 2. exp$_p$: Bε⊂Tp(S) --> S is a diffeomorphism in a neighborhood U⊂Bε of the origin 0 of Tp(S).
d(exp$_p$)表示exp$_p$对t的导数, 也可记为 d exp$_p$
d(exp$_p$(tv))/dt |t=0 = v, 记为(d exp$_p$)$_0$(v) = v
.
Since the exponential map at p E S is a diffeomorphism on U, it may be
used to introduce coordinates in V
1. The 【normal coordinates】 which correspond to a system of rectangular coordinates in the tangent plane Tp(S).
2. The 【geodesic polar coordinates】 which correspond to polar coordinates in the tangent plane Tp(S)

map(ρ, θ) in Tp(S)  to  exp$_p$(ρ, θ) in S

。
The images by exp$_p$: U-->V of circles in U centered in 0 will be called 【geodesic circles】 of V, and the images of exp$_p$ of the lines through 0 will be called 【radial geodesics】 of V.
。
Remark 1. The geometric meaning of the fact that F = 0 is that in a normal neighborhood [the family of geodesic circles] is orthogonal to [the family of radial geodesics]. This fact is known as the Gauss lemma
.
Since in a polar system E = 1 and F = 0, the Gaussian curvature K can be written
K(ρ, θ) = - sqrt(G)$_{ρρ}$ / sqrt(G)  （F$_{ρ}$表示F对ρ求2次导数）
==> sqrt(G)$_{ρρ}$ + K * sqrt(G) = 0   (2)    
.
THEOREM (Minding). Any two regular surfaces with the same constant Gaussian curvature are locally isometric. More precisely, let S1, S2 be two regular surfaces with the same constant curvature K . Choose points p1∈ S1, p2∈S2, and orthonormal basis {el, e2}∈Tp1(S1), {f1, f2} ∈Tp2(S2). Then there exist neighborhoods V1 of p1 , V2 of p2 and an isometry ψ: V1-->V2, such that dψ(e1) = f1, dψ(e2) = f2.（ψ = exp$_{p2}$ o  ψ  o exp$_{p2}$$^{-1}$）
。
a geometrical interpretation of the Gaussian curvature K.
K(p) = lim$_{r->0}$ 3/pi * (2pi*r - L)/r^3
.
【A fundamental property of a geodesic is the fact that, locally, it minimizes arc length.】
-----------------------------------------------------
4-7. Further Properties of G e o d e s i c s， Convex Neighborhoodst
Br(p):在S上以p为中心，弧长r为半径的区域
=============================================================
CH5 Global Differen tial Geometry
We shall prove that the sphere is 【rigid】 in the following sense. Let φ:∑-->S be an isometry of a sphere ∑⊂R3 onto a regular surface S =φ(C)⊂R3. Then S is a sphere. Intuitively, this means that it is not possible to deform a sphere made of a flexible but inelastic material.
.
THEOREM I. Let S be a compact, connected, regular surface with constant Gaussian ctrrvature K. Then S is a sphere.
。
Let S be a regular, compact, and connected surface with Gaussian curvature K > 0 and mean curvature H constant. Then S is a sphere.
。
An ovaloid（椭圆形的;卵形的） of constant mean curvature is a sphere.
。
A regular surface of constant mean curvature that is homemorphic to a sphere is a sphere
。
A regular, compact, and connected surface of constant mean curvature is a sphere.
。
Two isometric ovaloids difler by an orthogonal linear transformation of R3
。
Theorem 1 is a typical result of global differential geometry, that is, information on local entities (in this case, the curvature) together with weak global hypotheses (in this case, compactness and connectedness) imply strong restrictions on the entire surface (in this case, being a sphere).
。
---------------------------------------------
5-3. Complete Surfaces.Theorem of Hop f-Rinow
DEFINITION 1. A regular (connected) surface S is said to be 【extendable】 if there exists a regular (connected) surface S～ such that S⊂S~ as a proper subset. If there exists no such S～, S said to be nonextendable.
DEFINITION 2, A regular surface S is said to be 【complete】 when for every point p∈S, any parametrized geodesic r : [0, ε)-->S of S, starting from p = r(0), may be extended into a parametrized geodesic r~: R --> S, defined on the entire line R.
In other words, S is 【complete】 when for every p∈S the mapping exp$_P$:Tp(S) --> S is defined for every v∈Tp(S).
（Complete Surface：  A surface which has no edges. http://mathworld.wolfram.com/CompleteSurface.html）
.
nonextendability   ⊂   completeness   ⊂    compactness (条件限制从weak到strong)
。
given two points p, q∈S of a complete surface S there exists a geodesic joining p to q which is minimal。
This theorem is the main reason the complete surfaces are more adequate for differential geometry than the nonextendable ones.
。
a surface S - {p} obtained by removing a point p from a complete surface S is not complete. In fact, a geodesic r of S should pass through p. By taking a point q, nearby p on r (Fig. 5-3), there exists a parametrized geodesic of S - {p} that starts from q and cannot be extended through p.
.
PROPOSITION 1. A complete surface S is nonextendable.
。
PROPOSITION 2. Given two points p, q∈S of a regular (connected) surface S, there exists a parametried piecewise diferentiable curve joining p to q.
。
DEFINITION 3. The (intrinsic) distance d(p, q) from the point p∈S to the point q∈S is the number： d(p, q) = inf l(α$_{p,q}$)
where the inf is taken over all piecewise diferentiable curves joining p to q.
.
PROPOSITION 3. The distance d dejned above has the following properties,
- d(p, q) = d(q, p)
- d(p, q) + d(q, r) >= d(p, r)
- d(p, q) >=0
- d(p, q) = 0 if and only if p=q
where p, q, r are arbitrary points of S.
.
PROPOSITION 4. If we let p0∈S be a point of S, then the function f: S-->R given by f(p) = d(p0, p), p∈S, is 【continuous】 on S. （d(p, q)是连续函数）
。
PROPOSITION 5. A closed surface S⊂R3 is complete.
。
COROLLARY. A compact surface is complete.
。
THEOREM (Hopf-Rinow). Let S be a complete surface. Given two points p, q∈S, there exists a minimal geodesic joiniflg p to q.
。
COROLLARY 1. Let S be complete. Then for every point p∈S the map exp$_p$: Tp(S)-->S is 【onto】 S。
COROLLARY 2. Let S be complete and bounded in the metric d (that is, there exists r > 0 such that d(p, q) < r for every pair p, q∈S). Then S is compact。
。
the diameter(直径) ρ(S) of a surface S is, by definition： ρ(S) = sup{p,q∈S} d(p,q)
-------------------------------------
5-4. First and Second Variations of Arc Length; Bonnet's Theorem
a complete surface S with Gaussian curvature K >= δ > 0 is compact (Bonnet's theorem)
。
。变分法：
DEFINITION I. Let α : [0,l]-->S be a regular parametrized curve, where the parameter s∈[0, I] is the arc length. A 【variation of α】 is a differentiable map h: [0, I]x(-ε, ε)⊂R2-->S such that
h(s, 0)= α(s),    s∈[0, 1]
For each t∈(-ε, ε), the curve h$_t$: [0, l] -->S, given by h$_t$(s) = h(s, t), is called 【a curve of the variation h】.
A variation h is said to be 【proper】 if    h(0, t) = α(0),  h(l, t) = α(l),  t∈(-ε, ε)
.

。
Intuitively, a variation of a is a family h, of curves depending differentiably on a parameter t∈(-ε, ε) and such that ho agrees with a (Fig. 5-13).The condition of being proper means that all curves h, have the same initial point a(0) and the same end point a(l).
。
It follows that a variation h of a determines a differentiable vector field V(s) along a by
V(s)=∂h/∂s (s, 0) , s∈[0,l]
V is called the variational vectorfield of h
.事实上，h(s,t) = exp$_{α(s)}$ tV(s)
the arc length of h$_t$ : L(t) = ∫{0, l} |∂h/∂s (s, t)| ds, t∈(-ε, ε)     (Eq.1)
LEMMA 1. L'(t) =  ∫{0, l} ∂/∂t|∂h/∂s (s, t)| ds
。
LEMMA 2.   D/dt (f(t)*w(t)) = f(t)*Dw/dt + df/dt *w(t)
covariant derivative
.
Let h:[0, 1]x(-ε, ε) -->S be a differentiable map. A 【differentiable vector field】 along a differentiable map h is:     V:[0, l]x(-ε,ε) --> S⊂R3
such that V(s, t)∈T$_{h(s,t)}$(S) for each (s, t)∈[0, I] x (-ε,ε).
This generalizes the definition of a differentiable vector field along a parametrized curve.
For instance, the vector fields (∂h/∂s)(s, t) and, (∂h/∂t)(s, t) are vector fields along h.
。
LEMMA 4.   D/∂s ∂h/∂t(s,t) = D/∂t ∂h/∂s(s,t) ,交换求导顺序，值不变。
PROPOSITION 2.   L'(0) =  ∫{0, l} <A(s), V(s)>ds,     (2)  where A(s)=(D/∂s)(∂h/∂s)(s,0)
Remark I. The vector A(s) is called the 【acceleration vector of the curve α】, and its norm is nothing but the absolute value of the geodesic curvature of α. Observe that L'(0) depends only on the variational field V(s) and not on the variation h itself. (Eq.2) is usually called【the formula for the first variation】 of the arc length of the curve α.
。
the geodesics as solutions of a "variational problem"：
PROPOSITION 3. A regular parametrized curve α:[0, 1]-->S, where the parameter s∈[0, I] is the arc length of α, is a geodesic if and only if, for every proper variation h:[0,l]x(-ε,ε)-->S of α, L'(0) = 0.
。
orthogonal variations: variational field V(s) satisfies the condition (V(s), r'(s)) = 0,
.
L''$_V$(0) ≡L''(0)
.
The second variation L"(0) of the arc length is the tool that we need to prove the crucial step in Bonnet's theorem
.
THEOREM (Bonnet) Let  the Gaussian curvature K of a complete surface S satisfy the condition
K>=δ>0.
Then S is compact and the diameter ρ of S satisfies the inequality: ρ <= π/sqrt(δ)
（因为δ是K的极值，所以前面说了那么多变分法的内容。）
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5-5. Jacobi Fields and Conjugate Points
DEFINITION I. Let γ: [0, I]-->S be a parametrized geodesic on S and let h:[0,l]x[-ε, ε]-->S be a variation of γ such that for every t∈(-ε, ε) the curve h$_t$(s) ≡ h(s, t), s∈[0, I], is aparametrized geodesic (not necessarily parametrized by arc length). The variational field (∂h/∂t)(s, 0) ≡ J(s) is called a 【Jacobi field】 along γ.
.
PROPOSITION 1. Let J(s) be a Jacobi field along γ : [0, 1] --> S, s∈[0,l]. Then J(s) satisjies the so-called 【Jacobi equation】: D²(J(s))/ds² + K(s)*( γ'(s)/\J(s) ) /\ γ'(s) = 0     (Eq.1)
where K(s) is the Gaussian curvature of S at γ(s).（Jacobi field满足Jacobi equation）
.
LEMMA 1(如何构造Jacobi field). Let p∈S and choose v, w∈Tp(S), with |v|=l. Let γ : [0, I]-->S be the geodesic on S given by:
    γ(s) ≡ exp$_p$(sv),  s∈[0, l].  
Then, the vector field J(s) along γ given by :
    J(s) ≡ s(d exp$_p$)$_{sv}$(w), s∈[0, l]
 is a Jacobi field. Furthermore, J(0)= 0, (DJ/ds)(0)=w.
 
.
PROPOSITION 2. If we let J(s) be a diflerentiable vector field along γ: [0, I] -->S, s∈[0, I], satisfying the Jacobi equation, with J(0) = 0, then J(s) is a Jacobi field along γ.（满足jacobi equation的那些differentialbe vector field是jacobi field）
。
DEFINITION 2. Let y : [0, I]-->S be a geodesic of S with γ(0) = p. We say that the point q=γ(s0), s0∈[0, I], is 【conjugate】 to p relative to the geodesic γ if there exists a Jacobi field J(s) which is not identically zero along γ with J(0) = J(s0) = 0.
（用jacobi field来描述两个点是“conjugate共轭的“）
。
conjugate locus。。。 P363
。
PROPOSITION 3. Let J1(s) and J2(s) be Jacobi fields along γ: [0, I]-->S, s∈[0, I], Then
<DJ1/ds, J2(s)>  -  <J1(s), DJ2/ds> = const.
.
PROPOSITION 4. Let J(s) be a Jacobi field along γ:[0, l] -->S, with
<J(s1), γ'(s1)> = <J(s2), γ'(s2)> = 0, s1,s2∈[0,l], s1!=s2
then   <J(s), γ'(s)> = 0,    s∈[0, l]
.
the conjugate points may be characterized by the behavior of the exponential map.
.
THEOREM.  a surface of curvature K<= 0 does not have conjugate points.
.
COROLLARY. Assume the Gaussian curvature K <=0. Then for every p∈S, the mapping exp$_p$: Tp(S)-->S is a local diffeomorphisrn.  (when this local diffeornorphisrn is a global diffeomorphism? See 5-6)
.
LEMMA 2 (Gauss). Let p∈ S be a point of a (complete) surface S and let u∈Tp(S) and w∈(Tp(S))$_u$. Then     <u, w> = <(d exp$_p$)$_u$(u),  (d exp$_p$)$_u$(w)>
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5-6. Covering Spaces;The Theorems of Hadamard
PROPOSITION 1. Let π : B~-->B be a local homeomorphism, B~ compact and B connected. Then π is a covering map.
.
Let π:B~-->B be a covering map. when B is arcwise connected there exists a one-to-one correspondence between the sets π$^{-1}$(p) and π$^{-1}$(q), where p and q are two arbitrary points of B.
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THEOREM 1 (Hadamard). Let S be a simply connected, complete surface, with Gaussian curvature K<=0. Then exp$_p$: Tp(S) --> S, p∈S, is a diffeomorphism; that is, S is diffeornorphic to a plane.
THEOREM 2 (Hadamard). Let S be an ovaloid. Then the Gauss map N: S-->S2 is a diffeomorphism. In particular, S is diffeomorphic to a sphere.。
.
PROPOSITION 1. A plane, regular, closed curve is convex if and only if it is simple and its curvature k does not change sign.
.

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5-7- Global Theorems for Curves; The Fary-Milnor Theorem
.The most important property of degree is its invariance under homotopy.
.
.THEOREM 3 (Fenchel's Theorem). The total curvature of a simple closed curve is >= 2π, and equality holds if and only if the curve is a plane convex curve.
.
5-8. Surfaces of Zero Gaussian Curvature
THEOREM. Let S c R3 be a complete surface with zero Gaussian curvature. Then S is a cylinder or a plane.
。
int P： the interior of P， the set of points which have a neighborhood entirely
contained in P. int P is an open set in S which contains only planar points.
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5-10 Abstract Surfaces; Further Generalizations
Historically, it[Abstract Surface] took a long time to appear, probably due to the fact that the
fundamental role of the change of parameters in the definition of a surface in R3 was not clearly understood.
.
DEFINITION 1. An 【abstract surface]】 (diflerentiable manifold of dimension 2) is a set S together with a family of one-to-one maps x$_α$ : Uα-->S of open sets Uα⊂R2 info S such that:
1. ∪$_α$ x$_α$(U$_α$) = S
2. For each pair α, β with x$_α$(U$_α$) ∩ x$_β$(U$_β$) = W != Φ, we have that x$_α$^{-1}$(W), x$_β$^{-1}$(W) are open sets in R2, and x$_β$$^{-1}$ ο x$_α$, x$_α$$^{-1}$ ο x$_β$ are differentiable maps
.
DEFINITION 2. Let S1 and S2 be abstract surfaces. A map φ : S1-->S2 is 【differentiable】 at p ∈S, if given a parametrization y: V⊂R2-->S2 around φ(p) there exists a parametrization x: U⊂R2-->S1 around p such that φ(x(U))⊂y(V) and the map y$^{-1}$ ο φ o x: x$^{-1}$(U)⊂R2-->R2。。。。。 (1)
is diflerentiable at x$^{-1}$(p).
φ is differentiable on S1 i f i t is differentiable at every p∈S1.
。
The map (1) is called the 【expression】 of φ in the parametrizations x, y.
。
DEFINITION 4. Let S1 and S2 be abstract surfaces and let φ: S1-->S2 be a differentiable map. For each p∈S1 and each w ∈Tp(S1), consider a differentiable curve α : (-ε, ε)-->S1, with α(0)= p, α '(0) = w. Set β= φ ο α. The map dφ$_p$:Tp(S1)-->Tp(S2) given by dφ$_p$(w) = β'(0) is a well-defned linear map, called the 【differential】 of φ at p。
。
DEFINITION 5. A 【geometric surface】 (Riemannian mangold of dimension 2) is an abstract surface S together with the choice of an inner product < , >p at each Tp(S), p∈S, which varies diflerentiably with p in the following sense. For some (and hence all) parametrization x: U -->S around p, the functions: E(u, v) = <∂/∂u, ∂/∂u>,   F(u, v) = <∂/∂u, ∂/∂v>,   G(u, v) = <∂/∂v, ∂/∂v>,
are differentiable functions in U . The inner product < , > is often called a 【(Riemannian) metric】 on S.
。
DEFINITION 6. A differentiable map φ : S-->R3 of an abstract surface S into R3 is an 【immersion陷入】 if the differential dφ$_p$: Tp(S)--> Tp(R3) is injective单射.
If, in addition, S has a metric <,> and <dφ$_p$(v), dφ$_p$(w)>$_{φ(p)}$ = <v, w>$_p$,    v, w∈Tp(S), then φ is said to be an 【isometric immersion】.
。
wiki：In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.[1] Explicitly, f : M → N is an immersion if
    D_pf : T_p M \to T_{f(p)}N\,
.
DEFINITION 7. Let S be an abstract surface. A differentiable map p: S-->Rn is an 【embedding 】if φ is an immersion and a homeomorphism onto its image.
。
DEFINITION 1a. A 【differentiable manifold】 of dimension n is a set M together with a family of one-to-one maps x$_α$: Uα-->M of open sets Uα⊂Rn into M such that
1. ∪  x$_α$(U$_α$) = M.
2. For each pair α, β with x$_α$(Uα) ∩ x$_β$(Uβ) = W !=Φ, we have that x$_α$$^{-1}$(W), x$_β$$^{-1}$(W) are open sets in Rn and that x$_β$$^{-1}$ ο x$_α$, x$_α$$^{-1}$ ο x$_β$ are differentiable maps.
3.The family (Uα, x$_α$) is maximal relative to conditions 1 and 2.
A family (Uα, x$_α$ satisfying conditions 1 and 2 is called a 【drfferentiable structure】 on M
。
Let S be an abstract surface and let T(S) = {(p, w), p∈S, w∈Tp(S)}. We shall show that the set T(S) can be given a differentiable structure (of dimension 4) to be called the 【tangent bundle】 of S
。
DEFINITION 5a. A【 Riemannian manifold】 is an n-dimensional differentiable manfold M together with a choice, for each p∈M , of an inner product <, >$_p$ in Tp(M) that  varies differentiably with p in the following sense. For some (hence, all) parametrization xα : Uα-->M  with p∈x$_α$(U$_α$), the functions:
        gij(u1, u2,. . . , un) = <∂/∂ui,∂/∂uj>,     i,j = 1, 2, ..., n,
are differentiable at x$_α$$^{-1}$ (p); here (u1, u2,. . . , un) are the coordinates of Uα⊂Rn.
The differentiable family {<, >$_p$, p∈M} is called a 【Riemannian structure
(or Riemannian metric)】 for M.
.
Thus, given a Riemannian structure for M , there exists a unique covariant derivative on M (also called the 【Levi-Civita connection】 of the given Riemannian structure) satisfying Eqs. (5)-(8).
D$_{fu +gw}$(v) = fD$_u$v + gD$_w$v                               (5)
D$_u$(fv +gw) = fD$_u$v + gD$_w$v + ∂f/∂u v + ∂g/∂u w   (6)
D$_{xi}$ xj = D$_{xj}$ xi     for all i,j                                               (7)
∂/∂uk <xi, xj> = <D$_{xk}$xi, xj> + <xi, D$_{xk}$xj>               (8)
.
The following concept, due to Riemann, is probably the best analogue in Riemannian geometry of the Gaussian curvature.
Let p ∈ M and let σ ⊂Tp(M) be a two-dimensional subspace of the tangent space Tp(M). Consider all those geodesics of M that start from p and are tangent to σ. From the fact that the exponential map is a local diffeomorphism at the origin of Tp(M), it can be shown that small segments of such geodesics make up an abstract surface S containing p . S has a natural geometric structure induced by the Riemannian structure of M. The Gaussian curvature of S at p is called the 【sectional curvatrrre K(p, σ)】 of M at p along σ.
。
5-1 1. Hilbert's Theorem
THEOREM. A complete geometric surface S with constant negative curvature cannot be isometrically immersed in R3
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Appendix point-set Topology of Euclidean Spaces
PROPOSITION 1. A map F: U⊂Rn Rm is covltivluous a t p0∈U if and only if for each converging sequence {pi} -->p0 in U, the sequence {F(pi)} converges to F(p0).
.
DEFINITION 2. A poivlt p∈Rn is a【 limit point】 of a set A⊂Rn if every neighborhood of p in Rn contains one point of A distinct from p.
.
To avoid some confusion with the notion of limit of a sequence, a limit point is sometimes called a 【cluster point】 or an 【accumulation point】.
。
DEFINITION 3. A set F⊂Rn is 【closed】 if every limit poivlt of F belongs to F. The 【closure】 of A⊂Rn denoted by A～, is the union of A with its limit points.
.
C. Compact Sets
DEFINITION 13. A set A⊂ Rn is bounded if it is contained in some ball of Rn. A set K⊂Rn is compact if it is closed and bounded.
.
PROPOSITION 12. Let F: K⊂Rn-->Rm be continuous and let K be compact. Then F(K) is compact
.
D. Connected Components

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