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Math in Moscow Program Differential Geometry Problem Sets Math in Moscow program Differential Geometry Problem Sets by Prof. Maxim Kazarian Spring 2020 Differential Geometry Problem Set I 14.02.2020 @ @ 1. Find the expression for the basic vector fields @x and @y , and the basic differential 1-forms dx, dy in the polar coordinates on the plane. 2. Draw the phase portrait and find the phase flow for the following vector fields on the plane @ (a) @x ; @ @ (b) x @x + y @y ; @ @ (c) x @y − y @y ; @ @ (d) x @y − y @x ; @ @ (e) x @y + y @x . 3. Let N and E be the unit length vector fields on the sphere directed to the North and to the East, respectively. Compare the commutator of their flows with the flow of their commutator. 4. Check that the distribution defined by the following 1-form α is integrable. Find a func- tion f such that df = g α, g 6= 0. (a) α = z dx + z dy + 2(1 − x − y) dz; (b) α = 2 y z dx + 2 x z dy + (1 − xy) dz. P 5. Find the coordinate expression for the commutator of the vector fields A = ai@xi and P n B = bi@xi in R . 6. The coordinateless definition of the differential of a 1-form is du(A; B) = @A(u(B)) − @B(u(A)) − u([A; B]): Check that the expression on the right-hand side is bilinear with respect to the multiplica- tion of the fields A and B by functions. Differential Geometry Problem Set II 21.02.2020 2 Parametrization of a plane curve τ 7! γ(τ) 2 R is called natural if jγ_ j ≡ 1. The curvature is computed as k = jγ¨j for a natural parametrization. 1. Compute the curvature of the following curves: (a) y = x2; x2 y2 (b) + = 1. a2 b2 2. Find a formula for the curvature in arbitrary parametrization. 3. Compute and draw the caustic (evolute, focal set) of the curves in Problem 1. 2 2 4. The distance-square function Sq is defined by Sq(t) = jq − γ(t)j where γ : R ! R is a 2 plane curve and q 2 R is considered as a parameter. Prove: 0 Sq = 0 , q is on the normal to the curve; 0 00 Sq = Sq = 0 , q is a center of curvature (belongs to the evolute); 0 00 000 Sq = Sq = Sq = 0 , it is a singular point of the evolute (corresponding to an extremal point of the curvature). By the previous problem the evolute is the discriminant (the locus of parameter values for which the function has a degenerate critical point) of the family S. 5. Compute (and draw) the discriminant of the following family of polynomials: 4 2 fa;b(x) = x + ax + b x: 2 The evolute can also be identified as the locus of critical values of the map C × R ! R sending (x; t) to x + t nx where x 2 C is a point of the curve and nx is the normal unit vector at x. 6. Compute (and draw) the critical points and the critical values of the pleat map: x x3 + xy 7! y y Differential Geometry Problem Set II' 28.02.2020 1. Find a formula for the curvature and the torsion of a space curve in arbitrary parametriza- tion. 2. Compute the curvature and the torsion for the following curves: (a) (t; t2; t3); (b) (a cos t; a sin t; b t); (d) 3x2 + 15y2 = 1; z = xy. Differential Geometry Problem Set III 28.02.2020 3 Consider a surface in the Euclidean space R given by its parametrization r(v1; v2) = x(v1; v2); y(v1; v2); z(v1; v2) . Its basic tangent vectors and the unit normal vector are @r @r r1 × r2 r1 = ; r2 = ; n = : @v1 @v2 jr1 × r2j 2 2 The coefficients of the Riemannian metric g = g11 dv1 + 2g12 dv1dv2 + g22 dv2 are the pairwise scalar products of the basic tangent vectors, gij = (ri; rj): Alternatively, the metric can be obtained by expanding the expression g = dx2 + dy2 + dz2; where dx, dy, and dz are replaced by the total differentials of the corresponding coordinate functions. The Riemannian metric is sometimes referred to as the first quadratic form. The area form is p σ = det(g) dv1 ^ dv2: Its integral computes the areas of domains on the surface. 2 2 The coefficients of the second quadratic form h = h11 dv1 + 2h12 dv1dv2 + h22 dv2 are 2 determined by differentiating the basic tangent vectors r = @ri = @rj = @ r , ij @vj @vi @vi@vj hij = (rij; n): Alternatively, they can be computed by differentiating the coordinates of the normal vector, @n hij = − ri; : @vj The principal curvatures λ1; λ2 are the roots of the characteristic equation det(h−λ g) = 0. The principal directions are the corresponding eigenvectors. They are orthogonal to one another. The Gaussian curvature is det h K = λ λ = : 1 2 det g According to the Gauss's Theorema Egregium, the Gaussian curvature can be deter- mined by the coefficients of the Riemannian metric only. 1. Compute the Euclidean metric dx2 + dy2 and the area form dx ^ dy on the plane in the polar coordinates. 2. Compute the first, the second quadratic forms and the area 2-form of the surface given as the graph z = f(x; y) of a smooth function. 3. Compute the first, the second quadratic forms, the principal curvatures, and the Gaussian curvature for the following surfaces (a) the sphere x2 + y2 + z2 = R2 in spherical coordinates; a 2 2 (b) the paraboloid y = 2 x + y ; (c) the torus obtained by rotation of a circle of radius r around the axis on the distance R from the center of the circle. 4. Consider the following family of surfaces with local coordinates u; v and depending on the parameter θ: 8 x = cos θ sinh v sin u + sin θ cosh v cos u; <> y = − cos θ sinh v cos u + sin θ cosh v sin u; :>z = u cos θ + v sin θ: π Here θ = 0 or π corresponds to a helicoid, and θ = ± 2 corresponds to a catenoid. Prove that this family of surfaces is isometric (that is, the metric is independent of θ). Compute the Gaussian curvature. Differential Geometry Problem Set IV 06.03.2020 Here is of the ways to compute the curvature of a metric on a surface. Orthonormalizing the metric, represent it in the form 2 2 g = u1 + u2; where u1 and u2 are 1-forms. Then the area form is given by σ = u1 ^ u2: Define the connection 1-form α = α1u1 +α2u2 whose coefficients α1 and α2 are determined by the relations du1 = α1 σ; du2 = α2 σ; Then the curvature K of the metric is determined from the relation dα = −K σ: Theorem. 1. The curvature of a metric is independent of a choice of the basis u1; u2 of 1-forms used in its computation. 2. If the metric is the first quadratic form of a surface in the Euclidean 3-space then its curvature coincides with the Gaussian curvature. 1. For the surfaces of Problems 3 and 4 of the previous Problem Set (a) compute the curvature of the metric and compare with the Gaussian curvature. (b) In the case of the sphere and the torus compute the total area and the total integral of the curvature form K σ. 2. Determine the curvatures for the metrics below. (a) dx2 + sin2 x dy2; (b) dx2 + sinh2 x dy2; (c) dx2 + x2 dy2; 4 (dx2 + dy2) (d) ; (1 + k(x2 + y2))2 (e) conformal metric of the form g(x; y)(dx2 + dy2), where g(x; y) > 0 is a function; (f) g = dx2 + 2 cos ! dx dy + dy2, where ! = !(x; y) is some function. dv A family of tangent vectors v(t) to a surface along a curve γ(t) is called parallel if dt is orthogonal to the surface for all t. The parallel transport defines a linear transformation from the tangent space to the surface at the initial point of the curve to the tangent plane at its final point. The parallel transport is independent of the parametrization of the curve; but it does depend on a choice of a curve connecting two points. The parallel transport preserves lengths and angles. In other words, it is an orthogonal transformation. The parallel transport is determined by the intrinsic geometry of the surface. In particular, for flat surfaces (whose metric is isomorphic to the Euclidean one) it coincides with the usual parallel transport on the plane. If two surfaces are tangent to one another along a curve then the parallel transport along this curve on both surfaces coincides. A flat surface tangent to a given one along a curve is called its developable. 3. Compute the parallel transport on the sphere along a parallel with the altitude φ (a) from the definition, solving the corresponding differential equation; (b) from the geometry of the developable. The condition on a parallel family is equivalent to the equality rγ_ v(t) = 0 where r is the covariant derivative defined as the composition of the componentwise deriva- 3 tive of the tangent vector field considered as the R -valued function and the orthogonal projection to the tangent plane. Theorem. The covariant derivative is uniquely determined by the metric: in an orthonor- mal basis e1; e2 of tangent vector fields it is given by the equations ( rξe1 = α(ξ) e2; rξe2 = −α(ξ) e1; where α is the connection form computed in the dual basis u1; u2 of 1-forms.
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