Μηδείς ἀγεωμέτρητος εἰσίτω μου τὴν στέγην.
This course is being taught at the following universities:
Lectures take place Mondays 14:15-15:45 EET in person at Physicum A101.
Lectures take place Fridays 14:15-15:45 EET in person at Physicum A101, with two exceptions:
Lectures take place Wednesdays 12:15-13:45 CET online at BigBlueButton (via StudIP).
Lectures take place Wednesdays 12:15-13:45 CET online at BigBlueButton (via StudIP).
Tartu | Oldenburg | Topic |
---|---|---|
11.09.2023 | 18.10.2023 | Introduction. Manifolds. Coordinates. Maps. |
18.09.2023 | 25.10.2023 | Fiber bundles. Sections. |
25.09.2023 | 01.11.2023 | Vector bundles. Affine bundles. |
02.10.2023 | 08.11.2023 | Operations on vector bundles. Tensor product. Whitney sum. |
09.10.2023 | 15.11.2023 | Tangent bundle. Vector fields. |
16.10.2023 | 22.11.2023 | Cotangent bundle. Tensor bundles. |
23.10.2023 | 29.11.2023 | Differential forms. Exterior derivative. |
30.10.2023 | 06.12.2023 | Pushforward. Pullback. Coordinate transformations. |
06.11.2023 | 13.12.2023 | Lie groups. Lie group actions. |
13.11.2023 | 20.12.2023 | Diffeomorphism groups. Lie derivative. Flow. |
20.11.2023 | 10.01.2024 | Principal fiber bundles. Associated fiber bundles. |
27.11.2023 | 17.01.2024 | Jets. Jet manifolds. Jet bundles. |
04.12.2023 | 24.01.2024 | Connections on fiber bundles. Connection forms. Covariant derivative. |
11.12.2023 | 31.01.2024 | Properties of connections. Curvature. Torsion. |
18.12.2023 | 03.04.2024 | Tensor densities. Covariant derivative of tensors. |
16.02.2024 | 10.04.2024 | Integration. Line integrals. Integrals over chain complexes. |
23.02.2024 | 17.04.2024 | Homogeneous spaces. Klein geometry. Cartan geometry. |
01.03.2024 | 24.04.2024 | Geometry on the tangent bundle. Non-linear connections. Sprays. |
08.03.2024 | 08.05.2024 | Finsler geometry. Motion of point particles. |
15.03.2024 | 15.05.2024 | Affine connections. Torsion. Autoparallels. |
22.03.2024 | 22.05.2024 | Riemannian geometry. Metric. Levi-Civita connection. Hodge dual. |
29.03.2024 | 29.05.2024 | Complex geometry. Spin. |
05.04.2024 | 05.06.2024 | Variational problem. Lagrange formalism. |
12.04.2024 | 12.06.2024 | Action principle. Euler-Lagrange equations. |
19.04.2024 | - | Lepage forms. |
26.04.2024 | 19.06.2024 | 1. Noether theorem. |
03.05.2024 | 26.06.2024 | Gauge theories I. Gauge transformations and gauge bundles. |
10.05.2024 | - | Gauge theories II. Matter fields. |
17.05.2024 | - | Gauge theories III. Gauge fields. |
24.05.2024 | - | 2. Noether theorem. |
- | - | Diffeomorphism invariance and energy-momentum conservation. |
31.05.2024 | 03.07.2024 | Symplectic geometry. Hamilton formalism. |
For each lecture there will be set of homework exercises, which are weighted to give up to 5 points. Course grade are given based on the results of the homework:
% | [0, 50) | [50, 55) | [55, 60) | [60, 65) | [65, 70) | [70, 75) | [75, 80) | [80, 85) | [85, 90) | [90, 95) | [95, 100] |
---|---|---|---|---|---|---|---|---|---|---|---|
Grade Tartu | F | E | D | C | B | A | |||||
Grade Oldenburg | - | 4,0 | 3,7 | 3,3 | 3,0 | 2,7 | 2,3 | 2,0 | 1,7 | 1,3 | 1,0 |
Please upload your answers on NextCloud. The password is given in the first lecture. Please upload your solutions as PDF files named according to the scheme LastName_Ex_N.pdf, where \(N \in \{01, \ldots, 32\}\) is the exercise number, and replace in your last name the letters ä → a, ö → o, ü → u, õ → o, ß → ss.
Map composition (2 p.)
Let \(L, M, N\) be manifolds and \(f: L \to M\) and \(g: M \to N\) smooth maps.
Charts on the sphere (3 p.)
Consider the sphere as the set \(M = (X^1, X^2, X^3) \in \mathbb{R}^3 : (X^1)^2 + (X^2)^2 + (X^3)^2 = 1\), as well as the function \(\tilde{\psi}: \tilde{U} \to \mathbb{R}^3\) with \(\tilde{U} = (0, \pi) \times (−\pi, \pi)\) defned by \[\tilde{\psi}(\theta, \phi) = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\,.\]
Deadline Tartu: 17. 09. 2023 23:59 EET
Deadline Oldenburg: 24. 10. 2023 23:59 CET
Product manifold (1 p.)
Let \(M\) and \(N\) be manifolds and \(M \times N\) their product manifold, together with the projections \(\mathrm{pr}_1\) and \(\mathrm{pr}_2\) onto the factors. Show that \((M \times N, M, \mathrm{pr}_1 , N)\) and \((M \times N, N, \mathrm{pr}_2 , M)\) are fiber bundles.
Möbius strip (4 p.)
Let \((M, S^1, \pi, (−1, 1))\) be the Möbius strip.
Deadline Tartu: 24. 09. 2023 23:59 EET
Deadline Oldenburg: 31. 10. 2023 23:59 CET
Zero section (1 p.)
Show that the zero section \(0 : B \to E\) of a vector bundle is indeed a section.
Trivial line bundle (3 p.)
Let \(B\) be a smooth manifold and \(E = B \times \mathbb{R}\).
Vector bundle as affine bundle (1 p.)
Show that every vector bundle \((E, B, \pi, F)\) is an affine bundle modeled over itself.
Deadline Tartu: 01. 10. 2023 23:59 EET
Deadline Oldenburg: 07. 11. 2023 23:59 CET
Möbius ⊕ Möbius = ? (5 p.)
Consider the circle \(B = S^1\), the product manifold \(E = S^1 \times \mathbb{R}^2\) together with projection \(\pi = \mathrm{pr}_1 : E \to B\) onto the first factor and the two functions \(f_1, f_2\) defined by
where coordinates have the ranges \((\phi, x, y) \in (0, 2\pi) \times \mathbb{R}^2\) and cover \(E\) except for \(\{o\} \times \mathbb{R}^2\) for some point \(o \in B\).
Deadline Tartu: 08. 10. 2023 23:59 EET
Deadline Oldenburg: 14. 11. 2023 23:59 CET
Vector fields on the sphere (2 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\theta \in (0, \pi)\) and \(\phi \in (0, 2\pi)\). Within this chart we define the function and vector fields
which can be uniquely smoothly extended to the complete sphere.
Point on a rolling wheel (3 p.)
Consider the motion of a rolling wheel of radius \(R\) in \(\mathbb{R}^2\). The wheel is rotating with angular velocity \(\omega\), such that its center follows the curve given by \((x, y) = (R\omega t, 0)\). At \(t = 0\), when the center of the wheel crosses the \(y\)-axis, mark a point along the \(y\)-axis with distance \(d\) over the center of the wheel, i.e., the point \((x, y) = (0, d)\). Let this point be fixed to the wheel, so that it keeps its distance to the centre of the wheel and follows its rotation.
Deadline Tartu: 15. 10. 2023 23:59 EET
Deadline Oldenburg: 21. 11. 2023 23:59 CET
Tensor fields on the sphere (3 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\theta \in (0, \pi)\) and \(\phi \in (0, 2\pi)\). Within this chart we define the function and vector fields
which can be uniquely smoothly extended to the complete sphere.
Inverse metric (2 p.)
Consider the tensor field \[g = a\mathrm{d}x^1 \otimes \mathrm{d}x^1 + b\mathrm{d}x^2 \otimes \mathrm{d}x^2 + c(\mathrm{d}x^1 \otimes \mathrm{d}x^2 + \mathrm{d}x^2 \otimes \mathrm{d}x^1) \in \Gamma(T^0_2M)\] with constants \(a, b, c\) satisfying \(ab − c^2 > 0\) in Cartesian coordinates \((x^1, x^2)\) on \(M = \mathbb{R}^2\). Let us further define \[G = A\partial_1 \otimes \partial_1 + B\partial_2 \otimes \partial_2 + C(\partial_1 \otimes \partial_2 + \partial_2 \otimes \partial_1) \in \Gamma(T^2_0M)\] with constants \(A, B, C\).
Deadline Tartu: 22. 10. 2023 23:59 EET
Deadline Oldenburg: 28. 11. 2023 23:59 CET
Differential forms on the sphere (3 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\theta \in (0, \pi)\) and \(\phi \in (0, 2\pi)\). Within this chart we define the functions
which can be uniquely smoothly extended to the complete sphere.
Exterior derivative and interior product (2 p.)
Let \((x^a)\) be coordinates on a manifold \(M\). Consider a 1-form \(\omega = \omega_a\mathrm{d}x^a\) and vector fields \(X = X^a\partial_a\) and \(Y = Y^a\partial_a\).
Deadline Tartu: 29. 10. 2023 23:59 EET
Deadline Oldenburg: 05. 12. 2023 23:59 CET
Pushforward and pullback via the embedding of the sphere (3 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\theta \in (0, \pi)\) and \(\phi \in (0, 2\pi)\). Use this chart to define the map \[\psi: S^2 \to \mathbb{R}^3, \psi(\theta, \phi) = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\,.\]
Pullback of functions and 1-forms (1 p.)
Let \(M\) and \(N\) be manifolds and \(\psi: M \to N\) a smooth map. Show (without using coordinates) that the pullback of a real function \(f \in C^{\infty}(N, \mathbb{R})\) and its total derivative \(\mathrm{d}f \in \Omega^1(N)\) satisfy \(\psi^*(\mathrm{d}f) = d\psi^*(f)\). Use the fact that a function \(g \in C^{\infty}(M, \mathbb{R})\) and a vector \(v \in T_pM\) satisfy \(v(g) = \langle v, \mathrm{d}g(p) \rangle\).
Pullback of a volume form along a diffeomorphism (1 p.)
Let \(M\) and \(N\) be manifolds of dimension \(n\) and \(\psi: M \to N\) a diffeomorphism. Use coordinates \((x^a)\) on \(M\) and \((y^a)\) on \(N\) and let \[\omega = w\,\mathrm{d}y^1 \wedge \cdots \wedge \mathrm{d}y^n \in \Omega^n(N)\] be a volume form on \(N\). Show that \[\psi^*(\omega) = w\,\det\left(\frac{\partial y^a}{\partial x^b}\right)\,\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n \in \Omega^n(M)\,.\]
Deadline Tartu: 05. 11. 2023 23:59 EET
Deadline Oldenburg: 12. 12. 2023 23:59 CET
Double cover of the rotation group (5 p.)
Consider the special unitary group \(\mathrm{SU}(2)\) of 2 × 2 matrices with determinant 1 such that \(AA^{\dagger} = \unicode{x1D7D9}_2\).
Deadline Tartu: 12. 11. 2023 23:59 EET
Deadline Oldenburg: 19. 12. 2023 23:59 CET
Vector field on the sphere (2 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\theta \in (0, \pi)\) and \(\phi \in (0, 2\pi)\), as well as the vector field \(X = \partial_{\phi}\).
Commutator of Lie derivatives (3 p.)
Demonstrate the formula \(\mathcal{L}_{[X,Y]}T = \mathcal{L}_X\mathcal{L}_YT - \mathcal{L}_Y\mathcal{L}_XT\) for the commutator of Lie derivative for
This can be shown without using coordinates, using only formulas that appear in the lecture notes.
Deadline Tartu: 19. 11. 2023 23:59 EET
Deadline Oldenburg: 09. 01. 2024 23:59 CET
Cross product (1 p.)
Consider the Lie group \(G = \mathrm{SO}(3)\) as given in the example in the lecture notes. Let \(M = \mathbb{R}^3 \times \mathbb{R}^3\) with left action \(\rho_M(g, (x,y)) = (gx,gy)\) and \(N = \mathbb{R}^3\) with left action \(\rho_N(g,x) = gx\), where \(gx\) denotes the multiplication of a matrix and a vector. Prove that the vector cross product \(\times: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3\) is an equivariant map.
Hopf fibration (2 p.)
Consider the Lie groups \(G = \mathrm{SU}(2)\) and \(H = \mathrm{SO}(2)\), and recall that as manifolds these are isomorphic to the spheres \[\mathrm{SU}(2) = \left\{\left.\begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix} \right| a,b \in \mathbb{C}, a\bar{a} + b\bar{b} = 1\right\} \cong S^3\,,\]\[\mathrm{SO}(2) = \left\{\left.\begin{pmatrix} c & d \\ -d & c \end{pmatrix} \right| c,d \in \mathbb{R}, c^2 + d^2 = 1\right\} \cong S^1\,.\]
Tangent bundle (2 p.)
Let \(M\) be a manifold of dimension \(\dim M = n\) and \(\mathrm{GL}(M)\) its general linear frame bundle. Consider the left action of \(G = GL(n)\) on \(F = \mathbb{R}^n\).
Deadline Tartu: 26. 11. 2023 23:59 EET
Deadline Oldenburg: 16. 01. 2024 23:59 CET
Dimension of jet bundles (1 p.)
Let \(\pi: T^r_sM \to M\) be the tensor bundle of type \((r,s)\) over a manifold \(M\) of dimension \(\dim M = n\). What is the dimension of the \(k\)'th order jet bundle \(J^k(T^r_sM)\)?
Coordinate transformation on jet manifolds (4 p.)
Let \(M = \mathbb{R}\) with coordinate \(x\) and \(N = \mathbb{R}\) with coordinate \(y\), and \(p \in M\) the point with coordinate \(x = 0\).
Hint: You should find the derivatives \(\mathrm{d}y/\mathrm{d}x(0), \mathrm{d}^2y/\mathrm{d}x^2(0)\) of the coordinate expression \(y(x)\) of \(h\) somewhere, and the same for its coordinate expression \(\tilde{y}(\tilde{x})\). How are they related?
Deadline Tartu: 03. 12. 2023 23:59 EET
Deadline Oldenburg: 23. 01. 2024 23:59 CET
Connection on a principal bundle (3 p.)
Let \(M = \mathbb{R}\) with coordinate \(t\), \(S^1\) the circle with coordinate \(\phi\) and \(P = M \times S^1\) the trivial bundle (the cylinder). This bundle is a principal \(\mathrm{U}(1)\) -bundle. Recall that we can represent elements of \(S^1\) by unit complex numbers \(e^{i\phi}\), and that \(G = \mathrm{U}(1)\) can be expressed also by unit complex numbers \(g = e^{i\psi}\), which act on the circle by multiplication.
Connection on an associated vector bundle (2 p.)
Let \(\pi: P \to M\) be the fiber bundle from above, \(F = \mathbb{C}\) with coordinates \(z = x + iy\) and \(\rho: \mathrm{U}(1) \times \mathbb{C} \to \mathbb{C}\) defined by \(\rho(g, z) = gz\). Use coordinates \((t, z = x + iy)\) on the associated bundle \(P \times_{\rho} F\).
Deadline Tartu: 10. 12. 2023 23:59 EET
Deadline Oldenburg: 30. 01. 2024 23:59 CET
Connection on a bundle over the sphere (5 p.)
Consider the following geometric objects:
Use coordinates \((\vartheta, \varphi)\) on \(M\) and \((\vartheta, \varphi, \psi)\) on \(P\), such that
Consider further:
Consider a connection on \(P\) defined by the horizontal lift map \[\begin{array}{rcccc} \eta & : & \pi^*TM & \to & TP \\ && (\vec{x}, \vec{y}, \vec{u}) & \mapsto & (\vec{x}, \vec{y}, \vec{u}, \vec{v}) \end{array}\] with \(\vec{v} = -\vec{y} \times (\vec{x} \times \vec{u})\).
This calculation has a physical background: The manifold \(M\) describes Earth's surface, and \(P\) the directions of motion of a pendulum (unit vectors tangent to Earth's surface), parametrized by an angle \(\psi\). This angle changes if the pendulum is transported parallelly around a circle with constant latitude \(\pi/2 - \Theta\). The rate of this change describes the motion of Foucault's pendulum.
Deadline Tartu: 17. 12. 2023 23:59 EET
Deadline Oldenburg: 06. 02. 2024 23:59 CET
Leibniz rule (2 p.)
Consider a vector density \(\mathfrak{Z} \in \Gamma(D^+_w(TM) \otimes TM)\) of weight \(w\) and a one-form density \(\mathfrak{A} \in \Gamma(D^+_{w'}(TM) \otimes T^*M)\) of weight \(w'\). Using their components with respect to a coordinate basis of \(TM\), verify the Leibniz rule for \(\mathfrak{Z} \otimes \mathfrak{A}\) by calculating
Hint: What is the weight of \(\mathfrak{Z} \otimes \mathfrak{A}\)?
Levi-Civita densities in the tangent bundle (3 p.)
Consider the canonical Levi-Civita densities \(\mathfrak{E}\) and \(\mathfrak{e}\) in the tangent bundle \(\tau: TM \to M\).
Hint: You might need the total antisymmetry of the Levi-Civita symbol and its relation to the Kronecker symbol.
Deadline Tartu: 24. 12. 2023 23:59 EET
Deadline Oldenburg: 09. 04. 2024 23:59 CET
Integral over the sphere (2 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\theta \in (0, \pi)\) and \(\phi \in (0, 2\pi)\). Use this chart to define the 2-form \(\omega = \sin\theta\,\mathrm{d}\theta \wedge \mathrm{d}\phi\).
Stokes in \(\mathbb{R}^3\) (3 p.)
Let \(M = \mathbb{R}^3\) and \(\omega = x\,\mathrm{d}y \wedge \mathrm{d}z \in \Omega^2(M)\). Consider the 3-cube \[\begin{array}{rcccc} c & : & [0, 1]^3 & \to & M \\ && (x, y, z) & \mapsto & (x, y, z) \end{array}\,.\]
Deadline Tartu: 22. 02. 2024 23:59 EET
Deadline Oldenburg: 16. 04. 2024 23:59 CET
Hyperbolic Klein geometry (3 p.)
Consider the Klein geometry \((G,H)\) with \(G = \mathrm{SL}(2,\mathbb{R})\) and \[H = \left\{\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, t \in \mathbb{R}\right\}\,.\]
Cartan connection (2 p.)
Consider the same Klein geometry as above and the matrix-valued 1-form \[A = \begin{pmatrix} \frac{\mathrm{d}r}{r} - xr^2\mathrm{d}\theta & \mathrm{d}x + 2\frac{x}{r}\mathrm{d}r - \frac{1 + x^2r^4}{r^2}\mathrm{d}\theta \\ r^2\mathrm{d}\theta & xr^2\mathrm{d}\theta - \frac{\mathrm{d}r}{r} \end{pmatrix}\] in coordinates \(\theta \in (-\pi,\pi), r \in \mathbb{R}^+, x \in \mathbb{R}\) on \(\mathrm{SL}(2,\mathbb{R})\), defined such that \[g = \begin{pmatrix} r\cos\theta & xr\cos\theta - \frac{\sin\theta}{r} \\ r\sin\theta & xr\sin\theta + \frac{\cos\theta}{r} \end{pmatrix}\,.\]
Deadline Tartu: 29. 02. 2024 23:59 EET
Deadline Oldenburg: 23. 04. 2024 23:59 CET
Connection coefficients (2 p.)
Consider a manifold \(M\) equipped with a non-linear connection. Show that the connection is…
Hint: You might find some helpful conditions for homogeneous and linear connections in the lecture notes.
Induced coordinate transformation (3 p.)
Show that the following coordinate expressions of vector fields are invariant under a change of the coordinates \((x^a) \to (x'^a)\) on the base manifold, which also changes the induced coordinates \((x^a, \bar{x}^a) \to (x'^a, \bar{x}'^a)\), where \(\bar{x}'^a = \bar{x}^b\frac{\partial x'^a}{\partial x^b}\):
Deadline Tartu: 07. 03. 2024 23:59 EET
Deadline Oldenburg: 07. 05. 2024 23:59 CET
Objects in Finsler geometry (5 p.)
Let \(g_{ab}\) be a positive definite metric on a manifold \(M\) and consider the Finsler function \[F(x, \bar{x}) = \sqrt{g_{ab}(x)\bar{x}^a\bar{x}^b}\] in induced coordinates \((x^a, \bar{x}^a)\) on \(TM\). Use the induced coordinates to calculate coordinate expressions for the following objects:
Deadline Tartu: 14. 03. 2024 23:59 EET
Deadline Oldenburg: 14. 05. 2024 23:59 CET
Affine connection on \(S^3\) (5 p.)
Consider the following geometric objects:
Consider further the coordinates \((x^a) = (\theta, \alpha, \beta)\) on \(M\), which are defined by \[\vec{x} = (\cos\theta\cos\alpha, \cos\theta\sin\alpha, \sin\theta\cos\beta, \sin\theta\sin\beta)\] on \(M\), and the corresponding induced coordinates \((x^a, \bar{x}^a)\) on \(TM\).
Hint: it may be useful to write \((\vec{u}, \vec{v})\) on \(HTM\) as \[u^a\partial_a + v^a\bar{\partial}_a = u^a(\partial_a - \Gamma^b{}_{ca}\bar{x}^c\bar{\partial}_b)\] and to solve for the components \((v^a) = (v^{\theta}, v^{\alpha}, v^{\beta})\).
Deadline Tartu: 21. 03. 2024 23:59 EET
Deadline Oldenburg: 21. 05. 2024 23:59 CET
Riemannian metric on the sphere (5 p.)
Let \(M = S^2\) be the sphere with coordinates \(\vartheta \in (0, \pi)\) and \(\varphi \in (0, 2\pi)\) and consider the metric \[g = r^2(\mathrm{d}\vartheta \otimes \mathrm{d}\vartheta + \sin^2\vartheta\,\mathrm{d}\varphi \otimes \mathrm{d}\varphi)\,.\]Deadline Tartu: 28. 03. 2024 23:59 EET
Deadline Oldenburg: 28. 05. 2024 23:59 CET
Riemann sphere (5 p.)
Consider the unit sphere \[M = S^2 = \{x \in \mathbb{R}^3, \|x\| = 1\}\,,\] and note that its tangent spaces can be written as \[T_xM = \{v \in \mathbb{R}^3, x \cdot v = 0\}\,,\] using the canonical scalar product on \(\mathbb{R}^3\).
Hint: One possible approach is to express everything in coordinates, either using one of the complex charts above, or the usual spherical coordinates.
Deadline Tartu: 04. 04. 2024 23:59 EET
Deadline Oldenburg: 04. 06. 2024 23:59 CET
Variation in Finsler geometry (5 p.)
Let \(Q\) be a manifold equipped with a Finsler function \(F: TQ \to \mathbb{R}\) and consider the trivial bundle \(E = \mathbb{R} \times Q\) over \(M = \mathbb{R}\), with \(\pi: E \to M\) the projection onto the first factor. Further, consider the first jet bundle \(J^1(E) \cong \mathbb{R} \times TQ\). Use coordinates \((t)\) on \(M = \mathbb{R}\), \((x^a)\) on \(Q\), \((x^a, \bar{x}^a)\) on \(TQ\), \((t, x^a)\) on \(E\) and \((t, x^a, \bar{x}^a)\) on \(J^1(E)\).
Deadline Tartu: 11. 04. 2024 23:59 EET
Deadline Oldenburg: 11. 06. 2024 23:59 CET
Derivation of Euler-Lagrange equations (5 p.)
Consider the example given in the previous lecture. Let \(M = \mathbb{R}\) and \(Q\) a manifold of dimension \(n\). Let \(E = \mathbb{R} \times Q\) be the trivial fiber bundle with projection \(\pi: \mathbb{R} \times Q \to \mathbb{R}\) onto the first factor. Sections of this bundle are uniquely expressed by maps \(\gamma \in C^{\infty}(\mathbb{R}, Q)\), i.e., by curves on \(Q\). We use the one-dimensional Euclidean coordinate \(t\) on \(\mathbb{R}\) and arbitrary coordinates \((q^a)\) on \(Q\), so that we have coordinates \((t,q^a)\) on \(\mathbb{R} \times Q\). From these coordinates we derive the coordinates \((t,q^a_{(0)},q^a_{(1)},\ldots)\) on \(J^r(E) \cong \mathbb{R} \times TQ\). For brevity, we write \(q^a_{(0)} = q^a\), \(q^a_{(1)} = \dot{q}^a\), \(q^a_{(2)} = \ddot{q}^a\) etc.
Let further \(g \in \Gamma(T^0_2Q)\) be a non-degenerate, positive definite, symmetric tensor field of type \((0,2)\) (the metric) and \(V \in C^{\infty}(Q, \mathbb{R})\) (the potential). Consider the Lagrangian given by \[L(t,q,\dot{q}) = \left(\frac{1}{2}g_{ab}(q)\dot{q}^a\dot{q}^b - V(q)\right)\mathrm{d}t \in \Omega^{1,0}(J^1(E))\,.\]
Deadline Tartu: 18. 04. 2024 23:59 EET
Deadline Oldenburg: 18. 06. 2024 23:59 CET
Hilbert form as a Lepage form (5 p.)
Let \(Q\) be a manifold equipped with a Finsler function \(F: TQ \to \mathbb{R}\) and consider the trivial bundle \(E = \mathbb{R} \times Q\) over \(M = \mathbb{R}\), with \(\pi: E \to M\) the projection onto the first factor. Further, consider the jet bundles \(J^r(E)\). Use coordinates \((t)\) on \(M = \mathbb{R}\), \((x^a)\) on \(Q\), \((x^a, \dot{x}^a)\) on \(TQ\), \((t, x^a)\) on \(E\), \((t, x^a, \dot{x}^a)\) on \(J^1(E)\), \((t, x^a, \dot{x}^a, \ddot{x}^a)\) on \(J^2(E)\), and so on.
Deadline Tartu: 25. 04. 2024 23:59 EET
Deadline Oldenburg: -
Rotational symmetry (5 p.)
Consider the trivial fiber bundle \(\pi: M \times Q \to M\) with \(M = \mathbb{R}\) with coordinate \((t)\) and \(Q = \mathbb{R}^2\) with coordinates \((x, y)\) and a Lagrangian \(L = \mathcal{L}\,\mathrm{d}t \in \Omega^{1,0}(J^1(\pi))\), where \[\mathcal{L} = \frac{\dot{x}^2 + \dot{y}^2}{2} - V(x^2 + y^2)\] with a free function \(V\), and we wrote the coordinates on \(J^1(\pi)\) as \((t, x, y, \dot{x}, \dot{y})\).
Deadline Tartu: 02. 05. 2024 23:59 EET
Deadline Oldenburg: 25. 06. 2024 23:59 CET
Gauge transformations on a coset bundle (5 p.)
Consider the groups \(G = \mathrm{SL}(2,\mathbb{R})\) and \[H = \left\{\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, t \in \mathbb{R}\right\}\,,\] as well as the coset space \(M = G/H\). Use coordinates \((x, y, t)\) on \(G\), such that \[g = \begin{pmatrix} x & tx - \frac{y}{x^2 + y^2} \\ y & ty + \frac{x}{x^2 + y^2} \end{pmatrix}\,,\] where \((x,y) \in \mathbb{R}^2 \setminus \{(0,0)\}\) are coordinates on \(M\) and \(t\) is a coordinate on the fibers.
Deadline Tartu: 09. 05. 2024 23:59 EET
Deadline Oldenburg: 02. 07. 2024 23:59 CET
Matter field with \(\mathrm{U}(1)\) symmetry (5 p.)
Let \(M\) be a manifold equipped with coordinates \((x^{\mu})\) and \(P = M \times S^1\) a trivial principal \(\mathrm{U}(1)\)-bundle over \(M\) with coordinates \((x^{\mu}, e^{i\alpha})\), where \(e^{i\alpha} \in \mathrm{U}(1)\). Consider further the canonical action \(\rho: \mathrm{U}(1) \times \mathbb{C} \to \mathbb{C}, (e^{i\alpha}, z) \mapsto e^{i\alpha}z\) on the complex numbers and the associated fiber bundle \(E = P \times_{\rho} \mathbb{C}\).
Note that the prime \('\) in this exercise denotes the second gauge - it is not a derivative! You can find helpful formulas in the lecture notes; see the chapter on gauge theories and the section on principal bundle connections.
Deadline Tartu: 16. 05. 2024 23:59 EET
Deadline Oldenburg: -
Gauge field with \(\mathrm{U}(1)\) symmetry (5 p.)
Let \(M\) be a manifold equipped with coordinates \((x^{\mu})\) and \(P = M \times S^1\) a trivial principal \(\mathrm{U}(1)\)-bundle over \(M\) with coordinates \((x^{\mu}, e^{i\alpha})\), where \(e^{i\alpha} \in \mathrm{U}(1)\). Consider further the canonical action \(\rho: \mathrm{U}(1) \times \mathbb{C} \to \mathbb{C}, (e^{i\alpha}, z) \mapsto e^{i\alpha}z\) on the complex numbers and the associated fiber bundle \(E = P \times_{\rho} \mathbb{C}\). Further, let \(\epsilon: M \to P\) be a global section (a gauge) defined by \[\epsilon: (x^{\mu}) \mapsto (x^{\mu}, 1)\] and consider a gauge transformation \(\varphi: P \to P\) defined by \[\varphi: (x^{\mu}, e^{i\alpha}) \mapsto (x^{\mu}, e^{i\alpha}e^{i\beta(x)})\] with an arbitrary smooth function \(\beta \in C^{\infty}(M, \mathbb{R})\). Finally, let \(\Phi: M \to E\) be a matter field, which is defined in the gauge \(\epsilon\) as \[\Phi^{\epsilon}(x) = \phi(x)\] by a complex function \(\phi \in C^{\infty}(M, \mathbb{C})\).
Note that the prime \('\) in this exercise denotes the second gauge - it is not a derivative! You can find helpful formulas in the lecture notes; see the chapter on gauge theories and the section on principal bundle connections.
Deadline Tartu: 23. 05. 2024 23:59 EET
Deadline Oldenburg: -
Deadline Tartu: 30. 05. 2024 23:59 EET
Deadline Oldenburg: -
Deadline Tartu: -
Deadline Oldenburg: -
Deadline Tartu: 06. 06. 2024 23:59 EET
Deadline Oldenburg: 09. 07. 2024 23:59 CET