Differential Geometry: Curves, Surfaces, and Metrics

This English version was machine-translated from the Chinese original. Technical terms may require verification.

Introduction

What's geometry?

We can think of geometry as a sort of a sort of a point-of-sense study of 3-dimensional Euclid space, both of which were done before the 18th century, which we now call classical geometry, with roughly the following branch:

• Primary Geometry: Primary Geometry in Commons Education, Research in Justice Systems
• Primary geometry: We're inParsing GeometryResearch in the use of digital methods such as coordinates for learning in middle schools
• Classical micro-diverse geometry: using coordinates to combine calculus and linear algebras, to study more general nature

As mathematics develops, geometry transcends classical geometry, creating branches of geometry, imitation geometry, spherical geometry and hyperbolic geometry, and tectonics. One hopes to find a unified theoretical study geometry. Klein suggested using a variant to study geometry, which was considered a set of points. The nature of the change remains unchanged.

Further, we can consider that geometry studies are a collection of knowledge that remains constant in nature, under a particular variable cluster.

Geometric Classification

We assume that there is already a plane with a standard arc-angled system, and the constant nature of the study point set under a given variant is geometry.

& Change Range

For the change below $$00begin{aligned} (\xi, A):&\mathbb{R}^2\mapsto\mathbb{R}^2,\&{\cHFFE7C5}$$$US$US$$US$$US$$US$$US$$US$$US$$US$$US$$US$$US$$US$$$US$$$US$$US$$US$$$US$$$US$$$US$$US$$$$US$$$US$$US$$$US$$$$$US$$$$US$$$$$US$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$...$$ of which$A\in O(2),\xi\in R^2$And we can see that this is a linear shift and a horizontal shift, and he keeps the size of the graphic (the distance between points) and shape (the angle between any vectors is the same) and this is the geometry of the plane, which is the representation of the variant. Euro-Specific Including the entire classical geometry.

Simulation of the variants

For the change below $$\left(\xi,A\right):\mathbb{R}^{2}\to\mathbb{R}^{2},x\mapsto Ax+\xi.$$ of which$A\in GL(2),\xi\in R^2$; we can see that this is a non-degradable linear transformation and a horizontal shift, at which point the ratio of lengths of two segments of a straight line remains unchanged, and this shift represents a fusion of the luminous changes Simulation Geometry Especially, the nature of any simulation geometry study is summarized in European geometry.

Insight variant

Consider Matrix $M=\begin{matrix}0&1&0\0&0&1\1&0&0\end{pmatrix}.$$ 我们进行划分 $$\boldsymbol{M}=\left(\begin{array}{ll|l} \boldsymbol{A} & \boldsymbol{B} \ \hline c \quad d & e I'm sorry, I'm sorry. It's just one. $2\times 2$- The matrix. $2\times 1$Matrix, three numbers.

We're considering a change. $$\left.\left[\begin{array}{c}x^{1}\x^{2}\end{array}\right.\right]\rightarrow\left.\left[A\left(\begin{array}{c}x^{1}\x^{2}\end{array}\right.\right)+B\right]/\left(cx^{1}+dx^{2}+e\right)=\left.\left[\begin{array}{c}x^{2}/x^{1}\1/x^{1}\end{array}\right.\right]$$ This transformation is a few points of conglomerate, which means... Insight Geometry

Consequenceally transform the group

Considering all the continuously changed formations, maintaining connectivity, the same nature of tightness, and representing Taku Po.

Follow-up

After this section, we begin to discuss more general geometry, or to follow up on our previous geometry. Search for standard forms, seek classification Basic thinking. The main discussion was on geometry in three-dimensional European space and some simple imitation geometry. Mainly,

• Micro-specimetric Foundation and Curves
• Simulation geometry base and imitation curve theory
• Simulate Curve
• Micro-geometrically

Vector and curve theory in 3-dimensional space

Curves in 3-dimensional-European space

We're already in the basic nature of the three-dimensional European space.Parsing Geometry Math Analysis 4 is described in detail in the theory of multiple calculus and points, where we simply revisit some of the more important and previously not very skilled elements as the triggers for starting to learn the geometry of the calibration.

Definitions: Establishment$r$Yes.$(a,b)$Present.$R^3$..the continuous map of the $$r:(a,b)\mapsto\mathbb{R}^{3}$$ And the map is satisfied $r^{\prime}\left(t\right)\neq0$ You're the one who's gonna get you. $C=\left{r\left(t\right)\in\mathbb{R}^{3}|t\in\left(a,b\right)\right}$ And then, "Could"$C$is the regular curve, $r$It's a curve.$C$A regular parameterization

I'm not sure.A curve can be regulated with multiple regular parameters

For the length of the curve, we can naturally define the following: Set$u(t)$It's a regular parameterization, then from point to point.$a\to b$The arc longs $S=int t b}lft^\lt\right\{2}dt=\int{t_{a}}^{t_{b}}\sqrt{\left(u^{\prime}\left(t\right),u^{\prime}\left(t\right)\right)_{E}}dt$$

There is a question, the arc is of a curve nature, but is there an uncertain amount: is it linked to a regular parameterization, if so, is the arc long or is it certain?

We're giving the following additional definition and proposition without additional explanation.

• Definition: Use of Maps$\mu$ It acts as a regular parameter, keeps the curve unchanged, and gets a new regular parameter that we call map$\mu$is a regexp, new regexp
• Title:Normalize the arc length
• Thesis: The equidistance of the curve does not affect the arc Long

Shape of a curve

Curvature

Starting with this section, we focus on finding a geometrical figure that allows us to paint the shape of a curve that should remain constant at the equidistance variant and the normal variant. Since the positive shifts are not the focus of our research, we are primarily considering the equidistance shifts.

After having a long arc, the motion of the research point in the arc becomes important. It is clear that the arc was painted before without the question of a point speed, so we may consider that the speed is always 1 and we can consider the following definition:

Definitions: assumptions $r:(a,b)\mapsto\mathbb{R}^3$For a long arc parameter $C^3$ Curve, which means $\dot{r}(s)|2\equiv1, define the two vectors below $$\\alpha\left(s\right): =dot{r}\left(s\right), \\beta\left(s\right): =ddot{r}\left(s\right)/\left|ddot{r}\left(s\right)\right|{2}.$$

Because...$\parallel\boldsymbol{\alpha}(s)\parallel_2\equiv1,\boldsymbol{\alpha}(s)\perp\boldsymbol{\beta}(s).$Remember$\kappa(s):=\parallel\ddot{r}(s)\parallel_2$called the curve in$r(s)$Light it.Curvature

Apparently, the above-mentioned $\alpha(s)$It's the direction of the curve. $\beta(s)$It's called a curve.Lord rule...watch, like Go.$\ddot{r}\left(s\right)=0$, and at this point there is no definition. ... $$\gamma\left(s\right):=\alpha\left(s\right)\times\beta\left(s\right)$$ Called the curve at this point.Sub-legal

The blogger says that the government is not a party to the law.$\alpha$It's speed, so it's always size one,$\beta$It's unitization acceleration. His size.$\kappa(s)$Reflects the slowness of curve conversion. This is a difference from the acceleration of research in physics, and it distinguishes the acceleration of the direction from that of the method, which is deducted from the effect of the original speed and is described in the shape of the motion curve itself.

Theoretically:The difference between the curve of the curve and the straight line, with a curve of 0

Close plane

Definitions: Yes $r(s)$Point, point. $\alpha(s)$The plane is called the curve at that point.Normal;specified$r(s)$Point ${\beta}(s)$The square of the pattern is the curve from the pointTotes plane;specified$r(s)$Point$\gamma(s)$It's a curved plane.Close plane

Note that definitions here require a given vector to be the vector, so the tangent corresponds to the plane, the main to the plane, and the secondary to the close plane.

Theorem (the geometry of the close plane): $$r:\left(a,b\right)\mapsto R^{3},s\mapsto r\left(s\right)$$ Yes.$C^3$ Regular Curve$C$. Set the curve at point$s_0$Queuerate$\kappa(s_0)\ne0$ So there is. $$\gamma\left(s_{0}\right)//\lim_{\Delta s\to0^{+}}\frac{\left(r\left(s_{0}\right)-r\left(s_{0}-\Delta s\right)\right)\times\left(r\left(s_{0}+\Delta s\right)-r\left(s_{0}\right)\right)}{\left(\Delta s\right)^{3}}.$$ The blogger adds: Find two points around the point of zero, three points into a triangle and a plane, the latter two points are now approaching, and the limit of their flat is its closeness.

Accumulation

Now we can give some natural observations, if the motion curve is a plane curve, then both the speed direction and the main method are on the sub-motion, and the close plane is fixed, the motion plane; if the curve is not on a plane, then the curve is fixed.The method vector of the close plane$\gamma$The rate of variation of this vector reflects the degree of deviation of the curve from the plane curve. So we can give the following description.

Definitions: Establishment $$r:\left(a,b\right)\mapsto R^{3},s\mapsto r\left(s\right)$$ Yes.$C^3$ Regular Curve$C$. Arc long parameterisation is defined$\gamma\left(s\right)=r\left(s\right)\times\frac{\ddot{r}\left(s\right)}{\left|\ddot{r}\left(s\right)\right|_{2}}$ Remember $$\dot{\gamma}\left(s\right)=-\tau\left(s\right)\beta\left(s\right).$$ This.$\tau(s)$Called the curve in$r(s)$Light it.Accumulation

Theoretically:The difference between the scratching measure curve and the plane curve, with a scratch rate of 0

Frenet frame and the curve in it

Frenet frame and Frenet formulae

Definitions: Establishment $r:\left(a,b\right)\mapsto R^{3}$ To a rule.$C^3$Regular Curve$C$The arc longer parameterization,$$\alpha\left(s\right):=\dot{r}\left(s\right),\beta\left(s\right):=\ddot{r}\left(s\right)/\left|\ddot{r}\left(s\right)\right|_{2},\gamma\left(s\right):=\alpha\left(s\right)\times\beta\left(s\right)$$Call$r\left(s\right)$♪ Is where it is ♪$\alpha\left(s\right)$、 $\beta(s)、\gamma(s)$The coordinates of the base are curved $r(s)$;natural, when$\dot{\alpha}(s)=0$ Time,$\boldsymbol\beta(s)$No definition. So Frenet frames are defined only at points where curve curvatures are not zero.

The point of the Frenet frame is that the curve itself is in the direction of the three right-hands at one point, not the direction we are artificially choosing. So the Frenet frame is in some way the nature of the curve itself (in fact, after learning this section, you see Frenet frames that paint all the characteristics of the curve: two curves with the same Frenet frame actually overlap).

This section will examine how the Frenet frame changes along the direction of the increase of the curve long parameters. More precisely, we will study the following:$\dot{\alpha}\left(s\right),\dot{\beta}\left(s\right),\dot{\gamma}\left(s\right).$

The results are known to be:$\dot{a}\left(s\right)=\kappa\left(s\right)\beta\left(s\right),\dot{\gamma}\left(s\right)=-\tau\left(s\right)\beta\left(s\right)$, now only needs to calculate$\beta(s)$ ...the term “calculation” means that$\dot{\beta}(s)$As$\alpha、\beta、\gamma$ The linear combination of the beta is known to be a unit vector, so...$\dot{\beta}\left(s\right)\perp\beta\left(s\right).$And... $$\dot{\beta}\left(s\right)=b_{1}\left(s\right)\alpha\left(s\right)+b_{3}\left(s\right)\gamma\left(s\right)$$ Calculating coefficients $$00begin{aligned}b s\right&=\left(\alpha\left(s\right),\dot{\beta}\left(s\right)\right){E}=\frac{d}{ds}\left(\alpha,\beta\right){E}\left(s\right)-\left(\dot{\alpha}\left(s\right),\beta\left(s\right)\right){E}\&=-\left(\dot{\alpha}\left(s\right),\beta\left(s\right)\right){E}=-\kappa\left(s\right);\b_{3}\left(s\right)&=\left(\gamma\left(s\right),\dot{\beta}\left(s\right)\right){E}=-\left(\dot{\gamma}\left(s\right),\beta\left(s\right)\right){E}=\tau\left(s\right).\end{aligned}$$ 因此我们可以得到$\dot{\alpha}\left(s\right),\dot{\beta}\left(s\right),\dot{\gamma}\left(s\right).$满足 $$\frac{\mathrm{d}}{\mathrm{d}s}\begin{bmatrix}\alpha\left(s\right)\\beta\left(s\right)\\gamma\left(s\right)\end{bmatrix}=\begin{bmatrix}0&\kappa\left(s\right)&0\-\kappa\left(s\right)&0&\tau\left(s\right)\0&-\tau\left(s\right)&♪ I'm gonna be a big fan of the world ♪ This is called the Frenet formula or the Pyramid Motion formula, which he explained is fully determined by the point-by-point curvature and scratching of the Frenet frame on the curve.

The uniqueness of the curve

Theorem (the only scrutinizing of curves): set$r:\left(a,b\right)\to\mathbb{R}^{3}$and$\tilde{r}:\left(a,b\right)\to\mathbb{R}^{3}$The difference is... $C^{3}$Regular Curve$C$and$\tilde{C}$..and the arc longer.$\forall s\in\left(a,b\right),\kappa\left(s\right)\neq0$And... $$\kappa\left(s\right)=\kappa\left(s\right)\neq0,\tau\left(s\right)=\tau\left(s\right).$$ So there's a equidistance shift.$(A,\xi)$♪ Make $$Ar\left(s\right)+\xi=\widetilde{r}\left(s\right),\forall s\in\left(a,b\right).$$

The only structology of the curve indicates the only thing that is the only thing that is the same curve rate and the gravitational effect of the curve in the equidistance variant, that is, the only geometry of the European formula.

Existence of Curves

Now we're looking at the only counterproblem, giving the curve to scratch, looking for the curve.

Theorem (local existence of curves): set$(a,b)\subset \mathbb{R},\kappa$、$\tau$Yes (Pretty)$a,b)$Defined $C^1$real functions, and $kappa>0.$则存在着$\mathbb{R}^3$中的弧长参数化曲线 $r:(a,b)\mapsto\mathbb{R}^3$使得 $\kappa$、$\tau is the curve's curve curvature and scratch rate.

Local existence is only one point.

Simulation of spatial geometry

Because of the relative independence of the simulation of space geometry, we'reSimulation of spatial geometryHe presented the discussion as a supplement to the geometry theory, and he paved the way for discussion of important issues.

The geometry of the curve is local.

Starting with this chapter, we'll discuss a question of whether we can give a piece of the film and not proceed."Pull!"and"Squeeze"Under the conditions, turn him into a piece of paper.

In this chapter, we study which parts of the curve are considered to be flat in the geometric sense of the interior; in fact, we find that the column, the cone and the ball, have a huge difference between the double sides, and introduce the Riemann curvature to distinguish them.

Starting with this chapter, we resume discussions on European space, not on the analogue of space geometry studied in the previous chapter.

The pyres bound to the curve.

O'Neil Space.

We're studying an O'Hara space.$E^3$ He's a mimic space.$\mathscr{A}^3$Plus measures, that's what's in there. $$(\overrightarrow{AB},\overrightarrow{CD})$$ And it can be understood that a dual function is defined in space. $$(\bullet,\bullet):\mathbb{R}^3\times\mathbb{R}^3\mapsto\mathbb{R}$$ He meets the nature of the measure, such as non-negative and interchangeal; this is what the internal buildup of the imitation space calls Euclid space.

Now, we can build a special imitation system... Standard positive-coordinate system The base vector is zero or one, depending on the same number, i.e.$(e_{i},e_{j})=\delta_{ij}$

The curve of Oxygen.

We've defined the song in the simulation space. Noodles. Imitation of the mimic space curve in spatial geometryNow it's natural to extend the definition into the O'D space, after all, he's a model space that increases the scale and creates a special coordinates. Yes

Assumptions$A\in S$ Yes.$A$..of the border.$U$Yes.$S$Localisation of Parameters $$\varphi:\left(-\varepsilon,\varepsilon\right)\times\left(-\varepsilon,\varepsilon\right)\mapsto S\cap U.$$ So we can give the quality easily. Points$P$The sports equation is $$\overrightarrow{OP}\left(t\right)=x^{i}\left(u\left(t\right),v\left(t\right)\right)e_{i}.$$ It's the linear combination of parameters and foundations.

So you can give the mass speed satisfaction (guide to parameters) $$V\left(t\right)=\dot{u}\left(t\right)\partial_{u}x^{i}e_{i}+\dot{v}\left(t\right)\partial_{v}x^{i}e_{i}=\dot{u}\left(t\right)\partial_{u}+\dot{v}\left(t\right)\partial_{v}.$$

Further search for guidance to give the acceleration equations as $$a(t)=\ddot{u}\left(t\right)\partial_{u}+\ddot{v}\left(t\right)\partial_{v}+\left(\dot{u}\left(t\right)\dot{u}\left(t\right)\partial_{uu}^{2}x^{i}+2\dot{u}\left(t\right)\dot{v}\left(t\right)\partial_{uv}^{2}x^{i}+\dot{v}\left(t\right)\dot{v}\left(t\right)\partial_{vv}^{2}x^i\right)e_i$$ This is about traditional polytherapy, but the symbols are special.

To facilitate our discussion of acceleration, we'll take a second-stage lead.$b$ The first lead is$p$ The latter can be further divided into two parts that are consistent with the curve and vertically. $$a=b+p=b+p_{\perp}+p_{\parallel}$$

And then we'll agree that the symbols below are $$\begin{matrix} y^{1}=u,y^{2}=v,\partial_{1}=\partial_{{y}^{1}}=\partial_{u},\partial_{2}=\partial_{{y}^{2}}=\partial_{v}\ g_{ab}=\left(\partial_{a},\partial_{b}\right)\ \Gamma_{ab}^{l}=\frac{1}{2}g^{lc}\left(\partial_{a}g_{bc}+\partial_{b}g_{ac}-\partial_{c}g_{ab}\right) \end{matrix}$$ So there is. $$p_{\parallel}=\dot{y}^{a}\dot{y}^{b}\Gamma_{ab}^{c}\partial_{c}$$ and $$a_{\parallel}=b+p_{\parallel}=\left(\ddot{y}^{a}+\dot{y}^{b}\dot{y}^{c}\Gamma_{bc}^{a}\right)\partial_{a}$$

Free motion and geodesy on the curve.

We consider the mass to move freely on the curve, but only in the grip of the grip, so the acceleration must be in the curve, which is...$a_{\parallel}=0$ The special track at this time is called the curved geodesic line, satisfying the equation. $$\left(\ddot{y}^{a}+\dot{y}^{b}\dot{y}^{c}\Gamma_{bc}^{a}\right)\partial_{a}=0$$

Metric

We discussed it in the calculations above. $$g_{ab}=\left(\partial_{a},\partial_{b}\right)$$ of which$\partial_{a},\partial_{b}$ It's the curve, which means we're calculating the amount of the amount of the vector; the measure of this definition in the cut-off space at a point is called the curve-based Riemann measure, and we'll study it further later.

Measurement and First Basic Forms

Now we'll think about it. $S\subset \mathbb{E}^3\text{以及E}^3$The standard positive-coordinate system on top${O,e_i}.$Consider $A\in\mathcal{S}$ ♪ Start ♪$U$Localisation on $$\left(-\varepsilon,\varepsilon\right)\times\left(-\varepsilon,\varepsilon\right)\mapsto S\cap U\mapsto\mathbb{R}^{3},$$ $$\left(y^{1},y^{2}\right)\mapsto P\mapsto\left(x^{1}\left(y^{1},y^{2}\right),x^{2}\left(y^{1},y^{2}\right),x^{3}\left(y^{1},y^{2}\right)\right).$$

Recalling the definitions in the previous section $$g_{ab}=\left(\partial_{a},\partial_{b}\right)=\sum_{i=1}^{3}\partial_{a}x^{i}\cdot\partial_{b}x^{i}$$ It's the amount of vector that's in the local parameter frame. And we can calculate the in-trodition of the vectors for two vectors in any space.

If two vectors$(X,Y)$ Yes. $\left(\partial_{u},\partial_{v}\right)$This group of guys wrote... $$X=X^{a}\partial_{a},Y=Y^{a}\partial_{a}.$$ Then there's an internality. $$\left(X,Y\right)=\left(X^{a}\partial_{a},Y^{b}\partial_{b}\right)=X^{a}Y^{b}\left(\partial_{a},\partial_{b}\right)=X^{a}Y^{b}g_{ab}$$ Here.$\partial_{1}=\partial_{y^{1}},\partial_{2}=\partial_{y^{2}}.$

Based on the definition we give here in "Measure" here, we define aRiemann measures, for the warp, the form of the above measure is calledFirst basic form of the curveFirst basic form geometry is calledGeometrics of InnerThe geometry of geometry in the study is calledGeometry within

Contact

We have found in the calculation of the "Physical Particle of Curved Particle" section of this paper that the acceleration of the pyretic point is actually the guide to the pyresity of the pyretic point. We've seen that the acceleration is not usually zero.Second basic form of the wareThe next chapter will discuss) and the acceleration is only one of the speed guides we're considering. But the advantage of doing so is that it is still the vector.

And we're going to define it in the light of this idea.

Definition (Riemann Contact Concord on Curve): Set$S\subset \mathbb{E}^3\text{以及E}^3$It's a full smooth side.${O,e_{i}}$As a standard positive-coordinate system (from variable to $x^{i}$- I'm sorry. $A\in S,U 为 A 的开邻域$ .

If you're in$U\cap S$Defined a smooth vector Field$X=X^ie_i$(Yes.$U\cap S$ ♪ Every point of the day ♪ $B$,specify a vector $X(B)\in T_B(S)).$ Set$Y\in T_A\left(S\right),\gamma_{:}\left(-\varepsilon,\epsilon\right)\to S$ Yes$S$ Previous regular curve, satisfied$\gamma(0)=A,\dot{\gamma}(0)=Y$, define vector field$X$Yes.$A$Point-by-Stand$Y$Contact Wizard in Direction$\nabla_YX$As Vector

$$\frac{d\left(X^{i}\circ\gamma\right)}{dt}|{t=0}eI'm sorry. Yes.$T_{A}\left(S\right).$Positive projection on top

Geodesy

This section of the report defines the geodesic line and is a reference to the line on the plane.

Definitions: Establishment $$\gamma:\left(-\varepsilon,\varepsilon\right)\mapsto S$$ To the Curve $S$And the curve is a smooth and normal curve. If it's parallel, that is, it's parallel along the curve itself, then it's called the geodesy line.

It's a very natural idea to understand why we call him a line on the side.

Theorem: The pattern of the vector of the geodesy is constant

The speed of the free mass is still the same, the binding vertical track is not working.

Typical is the cone, all his terminals are geodesics, and another is the big one on Earth. Circle

Arc Long Variable

The geodesic line is a proliferation of straight lines on the curve, and one of the most important features of the straight segment is that he is the shortest line between the two points, and the geodesic line has similar results.

Theorem: Set $A,B\in S$ For two points on the smooth side. $\Gamma_\mathrm{AB}$For me.$A$ Here we go. $B$ It's... ♪ And the positive curves come together ♪$\gamma\in\Gamma_\mathrm{AB}$For the shorter of which is the arc,$\gamma$For a geodesic line

Prove that theorem requires knowledge of arc-long fractions.

Local flat curvature and Riemann curvature

As the end of this chapter, we answer the question at the beginning of the chapter, how to determine how a curve can be spread to plane near a point.

Local flat plane

Definition: A curve $S$At some point. $A$ It's called flat nearby, if there's an opening.$A\in U,S\cap U$There is a local parameterization (known as local standard cross-parametric) $$\varphi:\left(-\varepsilon,\varepsilon\right)\times\left(-\varepsilon,\varepsilon\right)\mapsto S\cap U,\(y^{1},y^{2})\mapsto P\in S\cap U,$$ Here. $\varphi(0,0)=A$And...$\left{\partial_{1},\partial_{2}\right}$It's a local unit that's turning over the frame, which means it's in...$S\cap U$Go, go, go! $$(\partial_{i},\partial_{j})=\delta_{ij}$$

The definition tells us if it's on the side of the curve ($A$The Queen of the West is a very small country, and it is a small country.$A$The next curve becomes flat.

Now, we want to judge whether a curve is flat, which requires us to judge whether or not a single coordinate can be created. Yes

Riemann curvature

We introduce the concept of Riemann curvature and prove that the curve is almost flat equal to zero in the vicinity of a point.

We define $$\begin{matrix} \Phi:\left(-\varepsilon,\varepsilon\right)\times\left(-\varepsilon,\varepsilon\right)\mapsto S\cap U\ \left(t,u\right)\mapsto\exp_{\gamma\left(t\right)}\left(uX\left(t\right)\right) \end{matrix}$$ of which$A$Yes.$S$Up a bit. $\gamma$It's through.$A$A geodesic line,$X$The Earth's landline is the vector field.$\exp_{\gamma\left(t\right)}\left(uX\left(t\right)\right)$It's an index map. It means...$\gamma(t)$As Start$X(t)$For the first speed line, check it out.$u$The value of the moment.

We're saying if Riemann's chorus is zero, then...$\Phi$Yeah.$A$A local standard in the vicinity is being condensed. It's a flat spot.

Definitions:Riemann curvatureDefine $$\left(\nabla_{\partial_{a}}\nabla_{\partial_{b}}-\nabla_{\partial_{b}}\nabla_{\partial_{a}}\right)\partial_{c}=R_{abc}^{d}\partial_{d}$$

Special, the Riemann curvature is given in local parameters.$\Phi$Calculating, in fact,The Riemann curve is the same result, calculated by any local parameter. So we just need to find any local parameter, and we can calculate the Riemann curvature to determine whether the area is flat, without having to study the more complex local standard cross-reference.

The geometry of the local exterior of the curve

Inheritance and Inheritance

The outer ones mean what we care about from now on.Geometrics are not just dependent on the first basic form.To describe the outer and inner differences, the example of the column mentioned in the previous chapter is still considered. The column is cut off along the parent line and can be flat, and we know that this is because its Riemann curve is zero. But the column is not flat, but it is still “twirled”. This requires geometry.

To illustrate the difference between two bends, we consider the distance between the two points on the side of the curve. Form. Set.$S$The blogger says:$A,B\in S.$One way is... $$D\left(A,B\right)=\inf\left{S\text{上从}A\text{ 到 }B\text{ 的正则曲线的弧长}\right}.$$ Another way is to define it directly as two points in three-dimensional Oscillation.$A、B$The distance of the O'Hara, as it is.$d\left(A,B\right).$Always. $d\left(A,B\right)\leqslant D\left(A,B\right).$

Attention.$,D\left(A,B\right)$ And so all the studies are the same. $D(A,B)$The geometry is determined by the first basic form alone.

If we're going to study the curvature,$d(A,B)$The nature of the problem cannot be determined by the first basic form alone.

Second basic form

The second basic form of physics

We continue to discuss the acceleration on the side of the curve, and in the first basic form we have studied the vector weight and made it clear that the weight of the weight must be in the last three of the acceleration expression. And the latter three are the secondary variant of speed, which means that the weight of acceleration is also the secondary variant of speed, which is the second basic form we want to introduce.

If$\gamma:(-\varepsilon,\varepsilon)\mapsto\mathcal{S}$The equation (and also a curve) of the mass,$\dot{\gamma}^ie_i$And for its speed, the acceleration is: $$a=\partial_{\dot{\gamma}}\dot{\gamma}^{i}e_{i}.$$ The definition of the second basic form above can be written in the form of $$$\left (\dot{gamma},\dot{gamma}\right){i}=\left(\partial\dot{\gamma e right , $$ Vertical symbol means the amount of the sum that is actually in the right line of the curve.

Definition of second basic form

Definitions: Establishment$\mathbb{E}^3$There's a curve. $S$ And a standard positive-coordinate system.${O,e_{i}}$(Record coordinates as$x^{i}).$ Set$A\in S$For a point on the curve,$U$ Yes$A$ A neighborhood. $X,Y$ Defined in two$U\cap S$The vector field. Definitions $$\Pi\left(X,Y\right)=\left(\partial_{X}Y^{i}e_{i}\right)_\perp$$ For the second basic form of the warp. Here.$\partial_x$ Yes$X$ # The directional guides,$\bot$It means taking the weight of this vector in the curved direction.

We can easily prove the nature of the following.

Theorem (essential nature of the second basic form): Establishment$\Pi$To the Curve$S$Second basic form of the Convention$X,Y$For the Curve$A$Two vectors of the dot,$f$Defines the smooth function on S.

• $\Pi\left(X,Y\right)=\Pi\left(Y,X\right)$,
• $\Pi\left(fX,Y\right)=\Pi\left(X,fY\right)=f\Pi\left(X,Y\right).$

Gauss is the perfect theorem.

We use the great Gauss theorem as the end of this chapter and the entire microgeometrical inference; it is in a way one of the most important local theorems in the early history of microgeometry; it is precisely because this theorem begins to recognize the difference between internal geometry and external condensation that one finds that the first basic form also contains crooked information.

Let's start with Gauss curvature.$S.$Yes$E^3$A smooth side of the middle $A\in S$Yes.$A$The point is to set up a standard and direct coordinates. Yes$\left{A,e_i\right}$♪ Always make ♪ $S$ Yes.$A$ Near as a binary function$f$The image is... $$P\in S\cap U\Leftrightarrow x^{3}\left(P\right)=f\left(x^{1}\left(P\right),x^{2}\left(P\right)\right),x^{1}\left(A\right)=x^{2}\left(A\right)=x^{3}\left(A\right)=0.$$ Here. $x^i(P)$Finger. $P$ No, no, no.$i$ The coordinates. And... $\partial_af(0,0)=0,a=1,2.f$ Yes.$(0,0)$Light it. The one-way value of the Hessian matrix, known as the curved face $S$ Yes.$A$Point of Gauss curvature.

This odd definition is based on the omission of many necessary descriptions and now adds.

• The reason why this weird system was created is not to use the original point of origin.
• The Hessian matrix corresponds to a quintessentially extended curvature in the second level of the original, and his shape directly influences the nature of the curvature at the original point.
• The Hessian Matrix is now...$2\times2$, so it must be able to be polarized. The value of the row is the value of the diagonal diagonal. Jack.
• And then the last one, the average of these two characteristic values also means, called the mean curvature.

Theorem (Gauss excellent theorem): Gauss curvature is fully determined (i.e. first basic form)

The great thing about this is that the process of defining the Gauss curvature is quite external, and we rely on an external coordinate system to calculate the Hessian matrix; yet he is an inherent geometry that is not directly related to how the curve is present in the external coordinate system.

Gauss’s brilliant theorem reveals that reliance on the first basic form alone carries a considerable amount of distorted information.