<Differential Geometry: A Geometric Introduction> (by David W. Henderson ) Notes

<Differential Geometry: A Geometric Introduction> (by David W. Henderson ) Note

 

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CH1
直线意味着存在一种对称变换
Lines which are intrinsically straight on a surface are often called 【geodesics】.
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Two geometric spaces, G and H, are said to be 【locally isometric】 at points G in G and H in H if the local intrinsic experience at G is the same as the experience at H.
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A surface that is locally isometric to the plane is traditionally called 【developable】.
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CH2
We call a figure in our f.o.v. a 【point】 if it does not have two parts which are distinguishable from each other.
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We can define limits in a f.o.v. by asserting that the sequence {x n } converges to y if eventually x n is indistinguishable from y.
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If γ is a smooth curve with nonzero curvature κ p at the point p, then the 【osculating circle】 at p is the circle C p through p which has the same curvature vector and unit tangent vector as γ.
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A unit vector perpendicular to the osculating plane at p is called the【binormal】, Bp。
We pick the direction of B p by specifying that {Tp , Np , Bp } be right handed,
The three unit vectors, Tp , Np , Bp , are called the 【Frenét frame】 at the point p。
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Most books define the 【torsion(扭力/扭矩)】 (vector) of a curve to be the rate of change (with respect to arc length) of the binormal, in symbols τ p = B'p
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a curve with well-defined (nonzero) curvature is planar if and only if the binormal is constant (or, its torsion is everywhere zero).
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【Frenét-Serret Equations】:
κ ≡ T′(s) = κ(s)N(s),
N′(s) = −κ(s)T(s) + τ(s)B(s),
τ ≡ B′(s) = −τ(s)N(s),
where κ(s) = |T′(s)| is the 【scalar curvature】 and
τ(s) = −[B′(s)·N(s)]  is the 【scalar torsion】.
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Chapter 3 Extrinsic Descriptions of Intrinsic Curvature
intrinsic curvature=geodesic curvature
The 【normal space】 N p M at a point p in the smooth surface M is the union of all lines which are perpendicular to T p M at p. In 3-space the normal space is a line called the 【normal line】.
We are, in effect, defining intrinsic curvature as the curvature observed by the bug.
For now all that we can do is to find a formal extrinsic definition of the intrinsic curvature by defining the 【intrinsic curvature】 to be the projection of the 【extrinsic curvature】 onto the tangent plane.
【intrinsic curvature (or geodesic curvature)】：
κg = [projection of κ onto the tangent plane Tp S ]
【normal curvature】：
κn = [projection of κ onto the normal space Np S ].
The directions of the three different curvatures, κ , κn , κg , give rise to three different normals:(Figure 3.5)
k:   the extrinsic normal to the curve, （extrinsic curvature）
kn: the normal to the surface,  
kg: the intrinsic normal to the curve in the surface.(把圆锥摊平)
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The curvature κ is the extrinsic curvature of the curve as a curve in space without reference to any surface containing it. （extrinsic curvature κ与曲线所在的面无关！）
However, the normal and intrinsic curvatures depend on the surface that one is considering.（normal kn，intrinsic curvatures kg与曲线所在的面有关）
κ = κn + κg
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The curve γ is called a geodesic if and only if κ g = 0 at every point.
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A smooth surface M is called a 【regular ruled surface】 if on M there is a smooth curve α(t) (parametrized by arclength t) and at each point α(t) of the curve a unit vector r(t) such that
1. r(t) is a differentiable function of t,
2. each point α(t) is in the interior of an (extrinsically) straight segment in M that is parallel to r(t),
3. there is a (global) coordinate patch for M which can be expressed in the form:
x(t,s) = α(t) + sr(t), and
4. the vectors, x1(t,s) = α′(t) + sr′(t), x2(t,s) = r(t) form a basis for the tangent space.
The curve α is called the 【directrix准线】 of the surface, and the extrinsically straight segments are called the 【rulings】 of the surface.
（x1表示x对第一个变量求偏导）
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Chapter.4.Tangent.Space.Metric(度量,度规?).and.Directional.Derivative
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In Chapter 5 we developed geometrically intrinsic descriptions of holonomy, parallel transport, and curvature of surfaces.
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 In Chapter 6 we developed extrinsic descriptions of Gaussian curvature and showed that it was the same as the intrinsic curvature for all C$^2$ surfaces.
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 In Chapter 7, we found intrinsic local descriptions of Gaussian (intrinsic) curvature with respect to extrinsically defined local coordinates, using (extrinsic) directional derivatives. 
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in  chapter 8，we will develop an intrinsic directional derivative that will allow intrinsic local descriptions of parallel transport.
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intrinsic directional derivative:
∇$_X$f =Xf − <Xf, n(p)> n
where X is a tangent vector at p
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CH6
PROBLEM 6.2. Second Fundamental Form
if Xp , Yp∈TpM for a smooth surface M in R3 , then we define the second fundamental form to be: II(Xp, Yp) = <Xp, −Yp n> ,
c. Show that 
II(x1 ,x2 ) = 〈x21 ,n〉 = 〈x12 ,n〉 = II(x2 ,x1 )  
and that
II(Xp, Yp ) = II(Yp, Xp)
解答：
Since x 1 and x 2 are tangent vectors, 〈x 1 ,n〉 = 0 = 〈x 2 ,n〉. Thus, 
0 = x 2 〈x 1 ,n〉 = 〈x 2 x 1 ,n〉 + 〈x 1 ,x 2 n〉 and 
0 = x 1 〈x 2 ,n〉 = 〈x 1 x 2 ,n〉 + 〈x 2 ,x 1 n〉. 
Therefore, II(x1, x2) ≡ 〈x 1 ,− x 2 n〉 = −〈x1 ,x2 n〉 = 〈x21 ,n〉 = 〈x 12 ,n〉 = −〈x 2 ,x 1 n〉 = 〈x 2 ,− x 1 n〉 = II(x 2 ,x 1 )
−〈x1, x2 n〉= 〈x21, n〉？为什么？
因为法线n和切向量x1相互垂直，所以x1*n=0,两边对第二个坐标参数求偏导得：
x21*n + x1 *x2 n = 0，即〈x21, n〉+ 〈x1, x2 n〉=0
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º¹²³⁴ⁿ₁₂₃₄·∶αβγδεζηθικλμνξοπρστυφχψω∽≌⊥∠⊙∈∩∪∑∫∞≡≠±≈＄㏒㎡㎥㎎㎏㎜
⊂∈∂
http://www.math.sinica.edu.tw/www/tex/online_latex.jsp得，N$_u$*x$_u$+N, x$_{uu}$=0，得证。