tài liệu khác Geodesics

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY ——————–o0o——————— NGUYEN THI NHU QUYNH GEODESICS BACHELOR THESIS HA NOI, 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY ——————–o0o——————— NGUYEN THI NHU QUYNH GEODESICS BACHELOR THESIS Major: Geometry Supervisor: NGUYEN THAC DUNG HA NOI, 2019 Thesis Assurance The thesis is completed after self-study and synthetic process with the guidance of Assoc Prof Dr Nguyen Thac Dung It is written by following the textbook entitled ”Elementary Differential Geometry” by Andrew Pressley Most of the contents of this thesis are taken from the chapter 9: ”Geodesics” in the mentioned textbook While the thesis is completed, I consulted some documents which have been mentioned in the bibliography section I assure that this thesis is not copied from any other thesis I certify that these statements are true And I will be responsible for their correctness Ha Noi, May 5, 2019 Student Nguyen Thi Nhu Quynh Bachelor thesis NGUYEN THI NHU QUYNH Thesis Acknowledgement This thesis is conducted at the Department of Mathematics, HANOI PEDAGOGICAL UNIVERSITY The lecturers have imparted valuable knowledge a`n facilitated for me to complete the course and the thesis I would like to express my deep respect and gratitude to Dr Nguyen Thac Dung, who has direct guidance, help me to complete this thesis I also want to thank to MS Nguyen Thac Dung for his valuable advice and assistance in the course of my degree Due to time, capacity and conditions are limited, so the thesis can not avoid errors So I am looking forward to receiving valuable comments from teachers and friends Ha Noi, May 5, 2019 Student Nguyen Thi Nhu Quynh i Contents Thesis Assurance Notation Preface 1 PRELIMINARIES 1.1 Curves in the plane and in the space 1.2 Curvatures on a smooth surfaces 1.3 Surfaces and Isometries of surface GEODESICS 2.1 Definition and basic properties 2.2 Geodesic equations 13 2.3 Geodesics on surfaces of revolution 18 2.4 Geodesics as shortest paths 27 2.5 Geodesic coordinates 34 CONCLUSION 38 38 ii Bachelor thesis NGUYEN THI NHU QUYNH Notation S, S1 , S2 : surface p,q : point γ : curve γ : parametrized curve : vector (t) : speed ă : acceleration kg : geodesic curvature N: Standard unit normal θ : the angle between q and p ϕ :the angle between q and positive x-axis Tp S : Tangent plane to S at p Π : plane is perpendicular to the tangent plane of the surface at point of γ σ(u, v): surface patch Γ: Christoffel symbols E, G, F : coefficient of the first fundamental form Bachelor thesis NGUYEN THI NHU QUYNH Preface Geodesics are the curves in a surface that make turns just to stay on the surface and never move sideways A bug living in the surface and following such a curve would perceive it to be straight We will begin with a definition of geodesics, then present various method for finding geodesics on surfaces, and later reveal their relationships to shortest paths.The term geodesic comes from the science of geodesy, which is concerned with measurements of the earth’s surface Chapter PRELIMINARIES 1.1 Curves in the plane and in the space Definition 1.1.1 A parametrized curve in Rn is a smooth map γ : (α, β) −→ Rn , for some α, β with −∞ ≤ α ≤ β ≤ ∞ From now on, all parametrized curves will be assumed to be smooth ˙ Definition 1.1.2 If γ is a parametrized curve, its first derivative γ(t) is called the tangent vector of γ at the point γ(t) Definition 1.1.3 If γ : (α, β) −→ Rn is a parametrized curve, its speed ˙ ˙ at the point γ(t) is γ(t) and γ is said to be a unit-speed curve if γ(t) is a unit vector for all t ∈ (α, β) Definition 1.1.4 A parametrized curve γ : (α, β) −→ Rn is a reparametrization of a parametrized curve γ : (α, β) −→ Rn if there is a smooth bijection map φ : (α, β) −→ (α, β) ( the reparametrization map) such that the inverse map φ−1 : (α, β) −→ (α, β) is also smooth and γ(t) = γ(φ(t)) f or all t ∈ (α, β) Bachelor thesis 1.2 NGUYEN THI NHU QUYNH Curvatures on a smooth surfaces Definition 1.2.1 If γ is a unit-speed curve with parameter t, its curvature k(t) at the point γ(t) is defined to be ă (t) (t)2 Definition 1.2.2 If is a unit-speed curve on an oriented surface S, then γ˙ is a unit vector and is a tangent vector to S Hence, γ˙ is perpendicular to the unit normal N of S, so γ, ˙ N and N ×γ˙ are perpendicular unit vectors Since is unit-speed, ă is perpendicular to γ, ˙ and hence is a linear combination of N and Nì: ă = kn N + kg N × γ˙ The scalars kn and kg are called the normal curvature and the geodesic curvature of γ, respectively Definition 1.2.3 γ is a normal section of the surface if γ is the intersection of the surface with the a plane Π that is perpendicular to the tangent plane of the surface at every point of γ Corollary 1.2.4 The curvature k, normal curvature kn and geodesic curvature kg of a normal section of a surface are related by kn = ±k, kg = Proposition 1.2.5 (Gauss Equations) Let σ(u, v) be a surface patch with first and second fundamental forms Edu2 + 2F dudv + Gdv and Ldu2 + 2M dudv + N dv Bachelor thesis NGUYEN THI NHU QUYNH where Then, σuu = Γ111 σu + Γ211 σv + LN, σuv = Γ112 σu + Γ212 σv + M N, σvv = Γ122 σu + Γ222 σv + N N, where GEu − 2F Fu + F Ev , 2(EG − F ) GEv − F Gu , Γ112 = 2(EG − F ) 2GFv − GGu + F Gv Γ122 = , 2(EG − F ) 2EFu − EEv + F Eu , 2(EG − F ) EGu − 2F Ev , Γ212 = 2(EG − F ) EGv − 2F Fv + F Gu Γ222 = 2(EG − F ) Γ111 = Γ211 = The six Γ coefficients in these formulas are called Christoffel symbols Note that they depend only on the first fundamental form of σ Proposition 1.2.6 Let γ(t) = σ(u(t), v(t)) be a curve on a surface patch σ, and let v(t) = α(t)σu + β(t)σv be a tangent vector field along γ, where α and β are smooth function of t Then, v is parallel along γ if and only if the following equations are satisfied: α˙ + (Γ111 u˙ + Γ112 v)α ˙ + (Γ112 u˙ + Γ122 v)β ˙ =0 β˙ + (Γ211 u˙ + Γ212 v)α ˙ + (Γ212 u˙ + Γ222 v)β ˙ =0 These equations involve only the first fundamental form of σ Bachelor thesis NGUYEN THI NHU QUYNH So, (v − v0 )2 + w2 = Ω2 where v0 is a constant So the geodesics are the images under σ of the parts of the circles in the vw-plane given by equation 2.10 and lying in the region w > Note that these circles all have center on the v-axis, and so intersect the v-axis perpendicularly The meridians correspond to straight lines perpendicular to the v-axis Figure 2.7: The corresponding geodesics on the pseudosphere itself are shown below Note that the geodesics cannot be extended indefinitely, in one direction in the case of the meridians and in both directions for the others This is because the geodesics ’run into’ the circular edge of the pseudosphere in the xy-plane A bug walking at constant speed along such a geodesic would reach he edge in the finite time, and thus would suffer the fate feared by ancient mariners of falling off the edge of the world In terms of the vw-plane, the reason for this is that the line w = is a boundary of the region that corresponds to the pseudosphere and the straight lines and semicircles that correspond to the geodesics cross 25 Bachelor thesis NGUYEN THI NHU QUYNH this line Figure 2.8: Clairaut’s theorem can often be used to determine the qualitative behaviour of the geodesics on a surface S, when solving the geodesic differential equations explicitly may be difficult or impossible Note first that, in general, there are two geodesics passing through any given point p ∈ S with a given angular momentum Ω, for v˙ is determined by equation 2.8 and u˙ up to sign by equation 2.9 In fact, one geodesic is obtained from the other by reflecting in the plane through containing the of rotation ( which changes Ω to −Ω) followed by changing the parameter t of the geodesic to −t ( which changes the angular momentum back to Ω again) The discussion in the preceding paragraph shows that we may as well assume that Ω > 0, which we from now on Then,equation 2.9 shows that the geodesic is confined to the part of S which is at a distance ≥ Ω from the axis If all of S is a distance > Ω from the axis, the geodesic will cross every parallel of S For otherwise, u would be bounded above or below on S, say the former Let u0 be the least upper bound of u on the geodesic, 26 Bachelor thesis NGUYEN THI NHU QUYNH and let Ω + , where > 0, be the radius of the parallel u = u0 If u is sufficiently close to u0 , the radius os the corresponding parallel will be Ω + , and on the part of the geodesic lying in this region we shall have |u| ˙ 1− Ω Ω+ >0 by equation 2.9 But this clearly implies that the geodesic will cross u = u0 contradiction our assumption Thus, the interesting case is that in which part of S is within a distance Ω of the axis The discussion of this case will be clearer if we consider a concrete example whose geodesics nevertheless exhibit essentially all possible forms of behaviour 2.4 Geodesics as shortest paths To see the connection between geodesics and shortest paths on an arbitrary surface S, we consider a unit-speed curve γ on S passing through two fixed point p, q ∈ S If γ is a shortest path on S from p to q then the part of γ contained in any surface patch of S must be the shortest path between any two of its points.For if p’ and q’ are any two points of γ in σ, and if there were a shortest path in σ from p’ to q’ than γ we could replace the part of γ between p’ and q’ by this shortest path, thus giving a shortest path from p to q in S We may therefore consider a path γ entirely contained in a surface patch σ To test whether γ has smaller length than any other path in σ passing through two fixed points p, q on σ; we embed γ in a smooth family of curves on σ passing through p and q By such a family, we 27 Bachelor thesis NGUYEN THI NHU QUYNH mean a curve γ τ on σ, for each τ in an open interval (−δ, δ), such that (i) there is an > such that γ τ (t) is defined for all t ∈ (− , ) and all τ ∈ (−δ, δ); (ii) for some a,b with − < a < b < we have γ τ (a) = p and γ τ (a) = q for all τ ∈ (− , ); (iii) the map from the rectangle (−δ, δ) × (− , ) into R3 given by (τ, t) −→ γ τ (t) is smooth; (iv) γ = γ Figure 2.9: The length of the part of γ τ between p and q is b ˙γ τ dt, L(τ ) = a where a dot denotes d/dt Theorem 2.4.1 The unit-speed curve γ is a geodesic if and only if d L(τ ) = when τ = dτ 28 Bachelor thesis NGUYEN THI NHU QUYNH for all families of curves γ τ with γ = γ We assumed that γ = γ is unit-speed, we cannot assume that γ τ is unit-speed if τ = 0, as this would imply that the length of the segment of γ τ corresponding to a ≤ t ≤ b is independent of τ Proof We use the formula for ’ differentiating under the integral sign’ : if f (τ, t) is smooth, d dτ f (τ, t)dt = ∂f dt ∂τ Thus, d d L(τ ) = dτ dτ d = dτ b γ˙τ dt a b (E u˙ + 2F u˙ v˙ + Gv˙ )1/2 dt a b ∂ (g(τ, t)1/2 )dt a ∂τ b ∂g = g(τ, t)−1/2 dt, a ∂τ = where g(τ, t) = E u˙ + 2F u˙ v˙ + Gv˙ and a dot denotes d/dt Now, 29 (2.14) Bachelor thesis NGUYEN THI NHU QUYNH ∂g ∂E ∂F ∂G ∂ u˙ ∂ v˙ ∂ v˙ ∂ u˙ = u˙ + u˙ v˙ + v˙ + 2E u˙ + 2F v˙ + u˙ + 2Gv˙ ∂τ ∂τ ∂τ ∂τ ∂τ ∂τ ∂τ ∂τ ∂u ∂v ∂u ∂v ∂u ∂v = Eu + Ev u˙ + Fu + Fv u˙ v˙ + Gu + Gv v˙ ∂τ ∂τ ∂τ ∂τ ∂τ ∂τ ∂ 2u ∂ 2v ∂ 2v ∂ 2u + 2E u˙ + 2F v˙ + u˙ + 2Gv˙ ∂τ ∂t ∂τ ∂t ∂τ ∂t ∂τ ∂t ∂v ∂u + Ev u˙ + 2Fv u˙ v˙ + Gv v˙ = Eu u˙ + 2Fu u˙ v˙ + Gu v˙ ∂τ ∂τ 2 ∂ u ∂ v + (E u˙ + F v) ˙ + (F u˙ + Gv) ˙ ∂τ ∂t ∂τ ∂t The contribution to the integral in the above equation coming from the terms involving the second partial derivatives is b g −1/2 a ∂ 2v ∂ 2u + (F u˙ + Gv) ˙ dt (E u˙ + F v) ˙ ∂τ ∂t ∂τ ∂t ∂v ∂u + (F u˙ + Gv) ˙ (E u˙ + F v) ˙ ∂τ ∂τ =g −1/2 b − a t=b (2.15) t=a ∂ ∂u ∂ ∂v g −1/2 (E u˙ + F v) ˙ + g −1/2 (F u˙ + Gv) ˙ ∂t ∂τ ∂t ∂τ using integration by parts Now, since γ τ (a) and γ τ (b) are independent of τ ( being equal to p and q, respectively), we have ∂γ τ = when t = a or b ∂τ Since ∂u ∂v ∂γ τ = σu + σv , ∂τ ∂τ ∂τ We see that ∂u ∂v = = when t = a or b ∂τ ∂τ 30 dt, Bachelor thesis NGUYEN THI NHU QUYNH Hence, the first term on the right-hand side of equation 2.15 is zero Inserting the remaining terms in equation 2.15 back into 2.14, we get d L(τ ) = dτ b U a ∂u ∂v +V ∂τ ∂τ dt, (2.16) where d U (τ, t) = g −1/2 (Eu u˙ + 2Fu u˙ v˙ + Gu v˙ ) − g −1/2 (E u˙ + F v) ˙ , dt d V (τ, t) = g −1/2 (Ev u˙ + 2Fv u˙ v˙ + Gv v˙ ) − g −1/2 (F u˙ + Gv) ˙ dt (2.17) Now γ = γ is unit-speed, so since γ˙τ = g(τ, t), we have g(τ, t) = for all t when τ = Comparing the equation 2.17 with the geodesic equation, we see that, if γ is a geodesic, the U = V = when τ = 0, and hence by equation 2.16, d L(τ ) = when τ = dτ For the converse, we have to show that, if b U a ∂u ∂v +V ∂τ ∂τ dt = when τ = (2.18) for all families of curves γ τ , the U = V = when τ = ( since this will prove that γ satisfies the geodesic equation) Assume, then, that condition (2.18) holds, and suppose, for example, that U = when τ = We will show that this leads to a contradiction Since U = when τ = , there is some t0 ∈ (a, b) such that U (0, t0 ) = 0, say U (0, t0 ) > Since U is a continuous function, there exists η > 31 Bachelor thesis NGUYEN THI NHU QUYNH such that U (0, t) > if t ∈ (t0 − η, t0 + η) Let φ be smooth function such that φ(t) > if t ∈ (t0 − η, t0 + η) and φ(t) = if t ∈ / (t0 − η, t0 + η) (2.19) Suppose that γ(t) = σ(u(t), v(t)), and consider the family of curves γ τ (t) = σ(u(τ, t), v(τ, t)) where u(τ, t) = u(t) + τ φ(t), v(τ, t) = v(t) Then, ∂u/∂τ = φ and , ∂v/∂τ = for all τ and t, so equation 2.18 gives b 0= a ∂v ∂u +V U ∂τ ∂τ t0 +η dt = U (0, t)φ(t)dt (2.20) t0 −η τ =0 But U (0, t) and φ(t) are both > for all t ∈ (t0 −η, t0 +η), so the integral on the right-hand side of equation 2.20 is > This contradiction proves that we must have U (0, t) = for all t ∈ (a, b) One proves similarly that V (0, t) = for all t ∈ (a, b) Together, these results prove that γ satisfies the geodesic equations The proof is complete Theorem 2.4.2 Let γ be a shortest path on a surface S connecting two points p and q Then the part of γ contained in any surface patch σ of S must be a geodesic The converse of the statement in the theorem is not necessarily true, however If γ is a geodesic on σ connecting p and q Then γ need not be a shortest path between the two points The great circle connecting two 32 Bachelor thesis NGUYEN THI NHU QUYNH points p and q on a sphere is split into two circular arcs by the points Both arcs are geodesics Only the shorter one of the two is the shortest path joining p and q Figure 2.10: In general, a shortest path joining two points on the surface may not exist For example, consider the surface P which is the xy-plane with its origin removed There is no shortest path from the point p=(-1,0) to the point q(1,0) Such a path would be the line segment, except it passes through the origin and does not lie entirely on the surface Any short path in P from p to q would walk in a straight line as long as possible, and then move around the origin, and continue in a straight line It can always be improved on by moving a little closed to the origin before circling around it Therefore, there is no shortest path connecting the two points If a surface S is a closed subset of R3 , and if there is some path in S joining any two points p and q, then there always exists a shortest path joining the two points For instance, a sphere is a closed subset of R3 , and the short great circle arc joining two points on the sphere is their shortest path The surface P , the xy-plane with its origin removed, is not a closed subset of R3 because any open ball containing the origin must contain of P ( so the set of points not in P is not open) 33 Bachelor thesis 2.5 NGUYEN THI NHU QUYNH Geodesic coordinates The existence of geodesics on a surface S allows us to construct a very useful atlas for S For this, let p ∈ S and let γ, with parameter v say, be a unit-speed geodesic on S with γ(0) = p For any value of v, let γ v , with parameter u, say, be a unit-speed geodesic such that γ v = γ(v) and which is perpendicular to γ at γ(v) ( γ v is unique up to the reparametrization u −→ −u) We define σ(u, v) = γ v (u) Proposition 2.5.1 There is an open subset U of R2 containing (0,0) such that σ : U −→ R3 is an allowable surface patch of S Moreover, the first fundamental form of σ is du2 + G(u, v)dv , where G is a smooth function on U such that G(0, v) = 1, Gu (0, v) = 0, whenever (0,v) ∈ U Figure 2.11: 34 Bachelor thesis NGUYEN THI NHU QUYNH Proof The proof that σ is ( for a suitable open set U ) an allowable surface patch makes use of the inverse function theorem Note first that, for any value of v, σu (0, v) = d v d d γ (u)|u=0 , σv (0, v) = γ v (u) = (γ(v)), du dv dv and that these are perpendicular unit vectors by construction.If σ(u, v) = (f (u, v), g(u, v), h(u, v)), it follows that the Jacobian matrix   f f  u v    gu gv    hu hv has rank when u = v = Hence, at least one of its three × submatrices is invertible at (0,0), say   fu fv gu gv   (2.21) By the Inverse Function Theorem , there is an open subset U of R2 such that the map given by F (u, v) = (f (u, v), g(u, v)) is a bijection from U to an open subset F (U ) of R2 , and such that its inverse map F (U ) −→ U is also smooth The matrix (2.21) is then 35 Bachelor thesis NGUYEN THI NHU QUYNH invertible for all (u, v) ∈ U , and so σu and σv are linearly independent for (u, v) ∈ U It follows that σ : U −→ R3 is a surface patch As to the first fundamental form of σ, note first that E = σu d v = γ (u) du =1 because γ v is unit-speed curve Next, we apply the second of the geodesic equations to γ v The unit-speed parameter is u and v is constant, so we get Fu = 0, But when u = 0, we have already seen that σu and σv are perpendicular, so F = It follows that F = everywhere Hence, the first fundamental form of σ is du2 + G(u, v)dv We have dγ G(0, v) = σv (0, v) = dv 2 =1 because γ is unit-speed Finally, from the first geodesic equation in (2.2) applied to the geodesic γ, for which u = and v is unit-speed parameter, we get Gu (0, v) = The proof is complete A surface patch σ constructed as above is called a geodesic patch, and u and v are called geodesic coordinates Example 2.5.2 Consider a point p on the equator of the unit sphere S Let the equator be parametrized with the longitude φ as γ(φ) Let βφ be the meridian parametrized by the latitude φ and passing through the point on the equator with longitude φ The corresponding geodesic patch is the usual one in latitude and longitude, for which the first 36 Bachelor thesis NGUYEN THI NHU QUYNH fundamental form is dθ2 + cos2 θ dφ2 37 Bachelor thesis NGUYEN THI NHU QUYNH CONCLUSION In this thesis we have presented systematically the following results: Recall the definitions about surface tangent vector, unit-speed curve, curvature, surface of revolution, parallels, meridians,normal section, Moreover, we recall the first fundamental form, the second fundamental form We have shown the detail definition and basic properties of geodesics, geodesic equations We consider geodesics on the surfaces of the revolution,shown and proof Clairaut’s theorem, take some examples related to them Proving geodesics is a shortest path and shown geodesic coordinates 38 Bibliography [1] Andrew Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series [2] B O’Neill, Elementary Differential Geometry, Acedamic Press, Inc., 1966 39 ... GEODESICS 2.1 Definition and basic properties 2.2 Geodesic equations 13 2.3 Geodesics on surfaces of revolution 18 2.4 Geodesics as... Proposition 2.2.4 that there are no other geodesics Corollary 2.2.6 Any local isometry between two surfaces takes the geodesics of one surface to the geodesics of the other Proof Let S1 and S2... consider a concrete example whose geodesics nevertheless exhibit essentially all possible forms of behaviour 2.4 Geodesics as shortest paths To see the connection between geodesics and shortest paths