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Math 3968 - Differential Geometry MATH 3968 - DIFFERENTIAL GEOMETRY ANDREW TULLOCH Contents 1. Curves in RN 2 2. General Analysis 2 3. Surfaces in R3 3 3.1. The Gauss Bonnet Theorem 8 4. Abstract Manifolds 9 1 MATH 3968 - DIFFERENTIAL GEOMETRY 2 1. Curves in RN Definition 1.1. A curve is said to be regular if α0(t) =6 0 for all t Definition 1.2. Let α be a curve parametrized by arc length. The number jα00(s)j = k(s) is called the curvature of α at s. Definition 1.3 (Frenet equations). t0 = kn n0 = −kt + τb b0 = −τn Definition 1.4. Total curvature of a unit speed curve α(s) from a to b is defined as Z b kds = θ(b) − θ(a) a Definition 1.5. The winding number k is defined as the integer such that, for a closed curve, the total curvature Z b kds = 2πk a Theorem 1.6 (Rotation Theorem). For a piecewise smooth simple closed curve, traversed in an anti-clockwise direction around a bounded region of the plane, the winding number is 1 2. General Analysis Definition 2.1 (Homeomorphism). A map is a homeomorphism is it is continuous and has a continuous inverse. Definition 2.2 (Diffeomorphism). A map is a diffeomorphism is it is smooth and has a smooth inverse Definition 2.3 (Differential of a smooth map). The differential of a smooth map ' : RN ! RM is defined as follows. Let w 2 RM , and let α be a differentiable curve such that α(0) = p, α0(0) = w. Then the composition β = ' ◦ α is differentiable, and we have 0 0 d'p(w) = β (0) = (' ◦ α) (0) Theorem 2.4 (Inverse function theorem). Let ϕ : U ⊂ RN ! RM be a differentiable mapping and suppose that the differential dϕ is an isomorphism at p 2 U. Then there exists a neighbourhood V ⊂ U and a neighbourhood W of F (p) such that ϕ is a diffeomorphism. Theorem 2.5 (Implicit function theorem). MATH 3968 - DIFFERENTIAL GEOMETRY 3 Definition 2.6. A map ϕ is regular at a point p if the matrix of dϕ has full rank at p. A critical point has that the matrix of dϕ does not have a full rank at p. Definition 2.7 (Implicitly defined surfaces). Let f : R3 ! R be a smooth function. Let p be a point in R3 with f(p) = 0 and f 0(p) =6 0. Then there exists a neighbourhood of the point a such that there exists a smooth parameterisation ϕ : U ! R3 of the set of solutions to f(x) = 0. Remark. This allows us to define surfaces implicitly as the set of solutions to some function. E.g. the sphere in R3, the sphere S2 is defined as the set of solutions to f(x; y; z) = x2 + y2 + z2 − 1 = 0 3. Surfaces in R3 Definition 3.1. A regular surface is defined intuitively that for every point p 2 S, there exists a diffeomorphism between some open set U 2 R2 and a neighbourhood V of that point p 2 S 3 Definition 3.2. Let ϕ : U ! R be a parametrization of a surface S. The tangent space TpS to the surface at a point p 2 U is the image of the mapping 2 3 dϕp : R ! R Theorem 3.3 (Surface as a graph). Let S ⊂ R3 be a regular surface and p 2 S. Then there exists a neighbourhood V of p in S such that V is the graph of a differentiable function which has one of the following forms: z = f(x; y); y = g(z; x); x = h(z; y). Definition 3.4 (Surface of revolution). Let C(v) = (α(v); 0; β(v)) be a curve in the xz plane. Then we can parameterise the surface formed by revolving the curve about the z axis by ϕ(u; v) = (α(v) cos u; α(v) sin u; βv) Definition 3.5 (First fundamental form). Let S be a surface parameterised by ϕ :(u1; u2) ⊂ R2 ! R3 @ϕ . Write Ei = @ui . Then we can define the matrix g of the second fundamental form I as gij = hEi;Eji Definition 3.6. The arc length of a parameterised curve α(t) = ϕ(u1(t); u2(t)) is given by Z Z t t qX 0 0i 0j s(t) = jα (t)jdt = giju u 0 0 Definition 3.7 (Area of a regular surface). Let R ⊂ S be a bounded region of a regular surface, contained in the coordinate neighbourhood of the parameterization ϕ : U ⊂ R2 ! R3. The positive number ZZ 1 2 −1 jE1 × E2jdu du = A(R);Q = ϕ (R) Q MATH 3968 - DIFFERENTIAL GEOMETRY 4 Definition 3.8 (Orientation). A regular surface is orientable if it is possible to cover it with a family of coordinate neighbourhoods in such a way that if a point p 2 S belongs to two neighbourhoods of this family, then the change of coordinate matrix has a positive determinant at p. If such a choice is not possible, then the surface is called non-orientable. Definition 3.9 (Normal vector). We define the normal vector N to a regular surface S as E × E N(p) = 1 2 (p) jE1 × E2j Definition 3.10 (Gauss Map). Let S ⊂ R3 be a regular surface with an orientation N. The map N : S ! R3 takes its values in the unit sphere S2, and this map N : S ! S2 is called the Gauss map. Proposition 3.11. The differential dNp : Tp(S) ! Tp(S) is a self adjoint linear map - i.e. hdNp(w); vi = hw; dNp(v)i Definition 3.12. The matrix of the second fundamental form is given by −⟨ i −⟨ i h i IIij = hij = Nui ; Ej = dN(Ei); Ej = N; Eij We have IIp(X) = −⟨dNp(X);Xi Definition 3.13 (Normal curvature). The normal curvature of a space curve kn is defined as kn = kcosθ, where θ is the angle between N and n. Definition 3.14. The principle curvatures are the eigenvalues of −dNp. Theorem 3.15. If the normal curvatures IIp(X) with jXj = 1 are not all equal, then there is an orthonormal basis e1; e2 of TpS such that 2 2 IIp(cos θe1 + sin θe2 = k1 cos θ + k2 sin θ where k1; k2 are the principle curvatures as defined above. Definition 3.16. A point p 2 S where k1 = k2 is called an umbilical point Definition 3.17. A curve α(t) is asymptotic if its normal curvature is zero for all t. 0 Definition 3.18. A curve is a line of curvature if α (t) is an eigenvector of −dNα(t) for all t Definition 3.19. The Gauss curvature of an oriented surface at a point p is K(p) = det(−dNp) = k1(p)k2(p) The mean curvature of S at p is 1 H(p) = trace(−dN ) 2 p MATH 3968 - DIFFERENTIAL GEOMETRY 5 Theorem 3.20. T −1 −(dNp) = (II)(I) h h − h2 K = 11 22 12 − 2 g11g22 g12 h g − 2h g + h g H = 11 22 12 12 22 11 − 2 2(g11g22 g12) Definition 3.21 (Minimal surface). A surface is a minimal surface if its mean curvature is iden- tically zero. H ≡ 0 2 3 Definition 3.22. A coordinate chart ϕ : U ⊂ R ! Σ ⊂ R is called isothermal if g11 = g22; g12 = g22 = 0. This is equivalent to saying (I) is a multiple of the identity. Theorem 3.23. If ϕ is isothermal, then ϕ(U) = Σ is a minimal surface if and only if r2ϕ = 0 Definition 3.24 (Isometry). A diffeomorphism : x1 ! S2 is an isometry if it preserves the inner product, i.e. for all p 2 S1 and X; Y 2 TP S1 hX; Y ip = hd p(X); d p(Y )i (p) This is equivalent to Ip(X) = I (p)(d p(X) If for each p 2 S1, there is an open neighbourhood U1 of p and a map : U1 ! S2, then we say that S1 is locally isometric to S2. Theorem 3.25. If two surfaces : U ! S1 and ' : U ! S2 have the same coefficients for gij, then S1 and S2 are locally isometric. i Definition 3.26 (Christoffel symbols). Define the Christoffel symbols Γjk by 1 2 ϕu1u1 = Γ11ϕu1 + Γ11ϕu2 + L1N 1 2 ϕu1u2 = Γ12ϕu1 + Γ12ϕu2 + L2N 1 2 ^ ϕu2u1 = Γ21ϕu1 + Γ21ϕu2 + L2N 1 2 ϕu2u2 = Γ22ϕu1 + Γ22ϕu2 + L3N Taking inner products with Ei gives MATH 3968 - DIFFERENTIAL GEOMETRY 6 1 Γ1 g + Γ2 g = (g ) 11 11 11 12 2 11 u1 1 Γ1 g + Γ2 g = (g ) − (g ) 11 12 11 22 12 u1 2 11 u2 1 Γ1 g + Γ2 g = (g ) 12 11 12 12 2 11 u2 1 Γ1 g + Γ2 g = (g ) 12 12 12 22 2 22 u1 1 Γ1 g + Γ2 g = (g ) − (g ) 22 11 22 12 12 u2 2 22 u1 1 Γ1 g + Γ2 g = (g ) 22 12 22 22 2 22 u2 Theorem 3.27 (Gauss’ Theorem). 2 − 2 2 2 1 2 − 1 2 − 2 2 g11K = (Γ11)u2 (Γ12)u1 + Γ11Γ22 + Γ11Γ12 Γ12Γ11 Γ12Γ12 Theorem 3.28 (Mainardi-Codazzi equations). − 1 2 − 1 − 2 (h11)u2 (h12)u1 = h11Γ12 + h12(Γ12 Γ11) h22Γ11 − 1 2 − 1 − 2 (h12)u2 (h22)u1 = h11Γ22 + h12(Γ22 Γ12) h22Γ12 Theorem 3.29 (Christoffel symbols in terms of the metric). 1 Γl = gjl (g + g − g ) ik 2 ij;k jk;i ki;j Definition 3.30.
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