Lectures take place Wednesdays 14:15-15:45 EET in person at Physicum A102.
Spring 2025
Oldenburg - 5.04.4254
Winter 2023/2024
Lectures take place Wednesdays 12:15-13:45 CET online at BigBlueButton (via StudIP).
Summer 2024
Lectures take place Wednesdays 12:15-13:45 CET online at BigBlueButton (via StudIP).
Course materials
Mandatory course material is given by the Lecture notes, which are gradually extended during the course, and contain references as further, recommended materials.
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.
Homework
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.
It is also possible to improve the grade by one step. This means up to 10% of the total homework points or 7.5 points, which is the amount of 1.5 homework sheets. To get this bonus, please write a short essay (3-5 typewriter pages, including formulas) on one of the topics we have covered in this semester. That can be a section or chapter in the lecture notes, some particular example where the concepts of differential geometry are applied, some calculation or anything you find interesting in particular. If you are unsure or would like a suggestion, please send a mail with a brief what kind of topic would suit you. Essays should be submitted not later than January 28, 23:59 EST, the same way as the homework.
Manifolds and 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 \(\tilde{U} = (0, \pi) \times (−\pi, \pi)\) defined by \[\tilde{\psi}(\theta, \phi) = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\,.\]
Show that \(\tilde{\psi}(\tilde{U}) \subset M\) and that \(\tilde{\psi}\) is injective. (1 p.)
Show that \((U, \psi)\) with \(U = \tilde{\psi}(\tilde{U})\) and \(\psi = \tilde{\psi}^{-1}\) is a chart on \(M\). (1 p.)
Calculate the transition function \(\varphi \circ \psi^{-1}\), where \(\varphi: V \to \mathbb{R}^2\) with \(V = M \setminus (0, 0, 1)\) is the stereographic projection \(\varphi(X^1, X^2, X^3) = \left(\frac{X^1}{1 - X^3}, \frac{X^2}{1 - X^3}\right)\). (1 p.)
Functions on the sphere (2 p.)
Consider the sphere and the charts given in the previous exercise, as well as the additional chart \((V', \varphi')\) with \(V' = M \setminus (0, 0, -1)\) and \(\varphi'(X^1, X^2, X^3) = \left(\frac{X^1}{1 + X^3}, \frac{X^2}{1 + X^3}\right)\). Check whether the following functions on the chart \((U, \psi)\) can be completed to smooth maps on the whole sphere \(M\).
Hint: use the stereographic charts and check whether the given functions can be smoothly extended to both charts.
Deadline Tartu: 17. 09. 2024 23:59 EET
Deadline Oldenburg: 24. 10. 2023 23:59 CET
Fiber bundles
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.
Show that \((M, S^1, \pi, (−1, 1))\) is a fiber bundle.
Construct two sets of adapted coordinates \((x, y)\) and \((\tilde{x}, \tilde{y})\), such that they form an atlas on \(M\).
Show that if \(f : S^1 \to M\) and \(g : S^1 \to M\) are sections, there exists at least one \(p \in S^1\) such that \(f(p) = g(p)\).
Let \((\bar{M}, S^1, \bar{\pi}, (−1, 0) \cup (0, 1))\) with \(\bar{M} = M \setminus \{(X^1, X^2, 0) | (X^1)^2 + (X^2)^2 = R^2\}\) the Möbius strip without the zero section, whose fiber is the (disconnected) manifold \((−1, 0) \cup (0, 1)\). Show that the fiber bundle has no global section.
Deadline Tartu: 24. 09. 2024 23:59 EET
Deadline Oldenburg: 31. 10. 2023 23:59 CET
Vector and affine bundles
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 \(E = B \times \mathbb{R}\).
Show that \((E, B, \mathrm{pr}_1, \mathbb{R})\) is a vector bundle. (1 p.)
What is the rank of \(E\)? (1 p.)
Show that there is a one-to-one correspondence between sections \(\sigma \in \Gamma(E)\) and maps \(f \in C^{\infty}(B, \mathbb{R})\). (1 p.)
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. 2024 23:59 EET
Deadline Oldenburg: 07. 11. 2023 23:59 CET
Operations on vector bundles
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 \(o \in B\).
Show that \(f_1, f_2\) extend to smooth maps on \(E\). (1 p.)
Show that each of the images \(M_1 = f_1(E)\) and \(M_2 = f_2(E)\) is diffeomorphic to the infinite Möbius strip, with projection to \(B = S^1\) given by \(\pi\). (1 p.)
Show that \(f_1 : E \to M_1\) and \(f_2 : E \to M_2\) are vector bundle homomorphisms. (1 p.)
Show that \(f_1^{−1}(m)\) is diffeomorphic to \(\mathbb{R}\) for each \(m \in M_1\) (and analogously for \(M_2\)). (1 p.)
Show that \(E = M_1 \oplus M_2\). Is the direct sum of twice the Möbius strip trivial? (1 p.)
Deadline Tartu: 08. 10. 2024 23:59 EET
Deadline Oldenburg: 14. 11. 2023 23:59 CET
Tangent bundle and vector fields
Vector fields on the sphere (2 p.)
Consider a chart on the sphere \(S^2\) corresponding to the previously introduced coordinates \(\vartheta \in (0, \pi)\) and \(\varphi \in (0, 2\pi)\). Within this chart we define the function and vector fields
which can be uniquely smoothly extended to the complete sphere.
Calculate the Lie bracket \([X, Y]\). (1 p.)
Calculate \(Xf\), \(Yf\) and \([X, Y]f\). (1 p.)
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.
Describe the motion of the point by a curve \(\gamma\). Is this a smooth curve \(\gamma \in C^{\infty}(\mathbb{R}, \mathbb{R}^2)\)? (1 p.)
Calculate \(\dot{\gamma}(t)\). (1 p.)
Are there values of \(t\) for which \(\dot{\gamma}(t) = 0\)? How does this depend on the values of \(d, R, \omega\)? (1 p.)
Deadline Tartu: 15. 10. 2024 23:59 EET
Deadline Oldenburg: 21. 11. 2023 23:59 CET
Cotangent bundle and tensor bundles
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.
Calculate the total differential \(\mathrm{d}f\). (1 p.)
Calculate the tensor fields \(X \otimes \mathrm{d}f\), \(Y \otimes \mathrm{d}f\), \(X \otimes Y\) and \(Y \otimes X\). (1 p.)
Calculate the contractions \(\mathrm{tr}^1_1(X \otimes \mathrm{d}f)\) and \(\mathrm{tr}^1_1(Y \otimes \mathrm{d}f)\). Compare this result with your result from previous assignment. (1 p.)
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 \(A, B, C\).
Determine \(A, B, C\) such that \(\mathrm{tr}^2_1(G \otimes g) = \partial_a \otimes \mathrm{d}x^a\), i.e., such that \(G^{ac}g_{cb} = \delta_b^a\). (1 p.)
Deadline Tartu: 22. 10. 2024 23:59 EET
Deadline Oldenburg: 28. 11. 2023 23:59 CET
Differential forms
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
\(f(\theta, \phi) = \cos\theta\),
\(g(\theta, \phi) = \sin\theta\cos\phi\),
\(h(\theta, \phi) = \sin\theta\sin\phi\),
which can be uniquely smoothly extended to the complete sphere.
Calculate the 1-forms \(\mathrm{d}f\), \(\mathrm{d}g\) and \(\mathrm{d}h\). (1 p.)
Calculate the 2-forms \(\mathrm{d}f \wedge \mathrm{d}g\), \(\mathrm{d}g \wedge \mathrm{d}h\) and \(\mathrm{d}h \wedge \mathrm{d}f\). (1 p.)
Show that there exists a 2-form \(\omega\) such that \(\mathrm{d}f \wedge \mathrm{d}g = -h\omega\), \(\mathrm{d}g \wedge \mathrm{d}h = -f\omega\) and \(\mathrm{d}h \wedge \mathrm{d}f = -g\omega\), and write ω using the coordinates defined above. (1 p.)
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 \(Y = Y^a\partial_a\).
Calculate the coordinate expression for \(X(\iota_Y\omega) − Y(\iota_X\omega) − \iota_{[X,Y]}\omega\). (1 p.)
Calculate the coordinate expression for \(\iota_Y\iota_X\mathrm{d}\omega\) and compare with your previous result. (1 p.)
Deadline Tartu: 29. 10. 2024 23:59 EET
Deadline Oldenburg: 05. 12. 2023 23:59 CET
Pushforward and pullback
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 \[\psi: S^2 \to \mathbb{R}^3, \psi(\theta, \phi) = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\,.\]
Let \(v = v^{\theta}\partial_{\theta} + v^{\phi}\partial_{\phi} \in T_{(\theta, \phi)}S^2\) be a tangent vector at some point \((\theta, \phi) \in S^2\) and \(w = w^x\partial_x + w^y\partial_y + w^z\partial_z = \psi_*(v) \in T_{\psi(\theta, \phi)}\mathbb{R}^3\). Calculate the components of \(w\) as functions of the components of \(v\). (1 p.)
Calculate the pullback \(\psi^*(g)\) of the tensor field \(g = \mathrm{d}x \otimes \mathrm{d}x + \mathrm{d}y \otimes \mathrm{d}y + \mathrm{d}z \otimes \mathrm{d}z\). (1 p.)
Calculate the pullback \(\psi^*(\omega)\) of the 1-form \(\omega = x\,\mathrm{d}x + y\,\mathrm{d}y + z\,\mathrm{d}z\). (1 p.)
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. 2024 23:59 EET
Deadline Oldenburg: 12. 12. 2023 23:59 CET
Lie groups
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 \(AA^{\dagger} = \unicode{x1D7D9}_2\).
Show that every element \(g \in \mathrm{SU}(2)\) can uniquely be written in the form \[g = \begin{pmatrix} g_1 + ig_2 & g_3 + ig_4 \\ -g_3 + ig_4 & g_1 - ig_2 \end{pmatrix}\] with \(g_1, \ldots, g_4 \in \mathbb{R}\) and \(g_1^2 + g_2^2 + g_3^2 + g_4^2 = 1\), so that \(\mathrm{SU}(2)\) is diffeomorphic to \(S^3\). (1 p.)
Show that for \(g \in \mathrm{SU}(2)\) and \(x \in \mathbb{R}^3\) there exists \(x' \in \mathbb{R}^3\) such that \[g \begin{pmatrix} ix_1 & -x_2 + ix_3 \\ x_2 + ix_3 & -ix_1 \end{pmatrix} g^{-1} = \begin{pmatrix} ix'_1 & -x'_2 + ix'_3 \\ x'_2 + ix'_3 & -ix'_1 \end{pmatrix}\] (1 p.).
Determine the 3 × 3 matrix \(\varphi(g)\) for which holds \[\varphi(g) \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x'_1 \\ x'_2 \\ x'_3 \end{pmatrix}\,,\] where \(g, x, x'\) are as in the previous part of this exercise. (1 p.)
Show that \(\varphi(g) \in \mathrm{SO}(3)\). (1 p.)
Show that \(\varphi : \mathrm{SU}(2) \to \mathrm{SO}(3)\) is a Lie group homomorphism. What is \(\varphi^{−1}(\unicode{x1D7D9}_3) \subseteq \mathrm{SU}(2)\) (the kernel of \(\varphi\))? (1 p.)
Deadline Tartu: 12. 11. 2024 23:59 EET
Deadline Oldenburg: 19. 12. 2023 23:59 CET
Lie derivative and flow
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 \(X = \partial_{\phi}\).
What is the flow \(\psi: \mathbb{R} \times S^2 \to S^2\) of \(X\)? (1 p.)
Calculate the Lie derivative \(\mathcal{L}_X\omega\) of the 2-form \(\omega = \sin\theta\,\mathrm{d}\theta \wedge \mathrm{d}\phi\). (1 p.)
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
a function \(T = f \in C^{\infty}(M, \mathbb{R})\) by using \(\mathcal{L}_Xf = Xf\), (1 p.)
a vector field \(T = Z \in \mathrm{Vect}(M)\) by using \(\mathcal{L}_XZ = [X, Z]\), (1 p.)
a 1-form \(T = \omega \in \Omega^1(M)\) by using \(\mathcal{L}_X\omega = \mathrm{d}\iota_X\omega + \iota_X\mathrm{d}\omega\). (1 p.)
This can be shown without using coordinates, using only formulas that appear in the lecture notes.
Deadline Tartu: 19. 11. 2024 23:59 EET
Deadline Oldenburg: 09. 01. 2024 23:59 CET
Principal and associated bundles
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 \[\mathrm{SU}(2) = \left\{\left.\begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix} \right| a,b \in \mathbb{C}, a\bar{a} + b\bar{b} = 1\right\} \cong S^3\,,\]\[\mathrm{SO}(2) = \left\{\left.\begin{pmatrix} c & d \\ -d & c \end{pmatrix} \right| c,d \in \mathbb{R}, c^2 + d^2 = 1\right\} \cong S^1\,.\]
Use the action \(\rho: \mathrm{SU}(2) \times \mathbb{R}^3 \to \mathbb{R}^3\) defined by \(\rho(g, x) = x'\) where \[g \begin{pmatrix} ix_1 & -x_2 + ix_3 \\ x_2 + ix_3 & -ix_1 \end{pmatrix} g^{-1} = \begin{pmatrix} ix'_1 & -x'_2 + ix'_3 \\ x'_2 + ix'_3 & -ix'_1 \end{pmatrix}\,,\] to show that the coset space \(\mathrm{SU}(2)/\mathrm{SO}(2)\) is diffeomorphic to \(S^2\). (1 p.)
Calculate the fundamental vector field \(\tilde{X}\) of \[X = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \in \mathfrak{so}(2)\] on \(\mathrm{SU}(2)\). (1 p.)
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 \(F = \mathbb{R}^n\).
Show that the associated bundle \(\mathrm{GL}(M) \times_{\rho} \mathbb{R}^n\) is diffeomorphic to the tangent bundle \(TM\). (1 p.)
Show that there is a one-to-one correspondence between vector fields \(X \in \mathrm{Vect}(M)\) on \(M\) and equivariant maps \(\phi \in C^{\infty}_{\mathrm{GL}(n)}(\mathrm{GL}(M), \mathbb{R}^n)\). (1 p.)
Deadline Tartu: 26. 11. 2024 23:59 EET
Deadline Oldenburg: 16. 01. 2024 23:59 CET
Jet bundles
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 \(x = 0\).
Write out explicitly the coordinates \(y_{\Lambda}\) for all relevant \(\Lambda\) derived from \(x\) and \(y\) on \(J^2_p(M, N)\) and on \(J^2(M, N)\). What are the dimensions of these manifolds? (1 p.)
Let \(h \in C^{\infty}(M, N)\) be a smooth map, given in coordinates as \(y(x)\). What are the coordinates of \(j^2_ph \in J^2(M, N)\)? (1 p.)
Let \(f \in C^{\infty}(M, M)\) and \(g \in C^{\infty}(N, N)\) be diffeomorphisms, and \(\tilde{x} = f(x)\) and \(\tilde{y} = g(y)\) the corresponding new coordinates on \(M\) and \(N\). These induce new coordinates \((\tilde{x}, \tilde{y}_{\tilde{\Lambda}})\) on \(J^2_p(M, N)\), where the notation \(\tilde{\Lambda}\) indicates that derivatives are now taken with respect to \(\tilde{x}\) instead of \(x\). Rewrite \(h\) using the new coordinates \(\tilde{x}\) and \(\tilde{y}\). What are the new coordinates of \(j^2_ph \in J^2(M, N)\)? (1 p.)
Rewrite your result as a formula for the coordinates \(\tilde{y}_{\tilde{\Lambda}}\) on \(J^2_p(M, N)\) as functions of the coordinates \(y_{\Lambda}\). (1 p.)
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. 2024 23:59 EET
Deadline Oldenburg: 23. 01. 2024 23:59 CET
Connections
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. Denote by \(J^1(P)\) be the first jet bundle with coordinates \((t, \phi, \dot{\phi})\). Finally, let \(\alpha \in \mathbb{R}\) a constant and \(\omega: P \to J^1(P)\) be the Ehresmann connection which assigns to \((t, \phi)\) the element \((t, \phi, \dot{\phi}) = (t, \phi, \alpha)\).
Show that \(\omega\) is a principal Ehresmann connection, i.e., that it is \(G\)-equivariant. (1 p.)
Calculate the connection form \(\theta: TP \to VP\) of \(\omega\), using coordinates \((t, \phi, v^t, v^{\phi})\) on \(TP\) and \((t, \phi, v^{\phi})\) on \(VP\). (1 p.)
Calculate the principal \(G\)-connection \(\vartheta \in \Omega^1(P, \mathfrak{u}(1))\). Note that the Lie algebra \(\mathfrak{u}(1)\) is isomorphic as a vector space to \(\mathbb{R}\). (1 p.)
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 \(P \times_{\rho} F\).
Using coordinates \((t, z = x + iy, \dot{z} = \dot{x} + i\dot{y})\) on the first jet bundle \(J^1(P \times_{\rho} F)\), calculate the associated bundle connection \(\omega_{\rho}: P \times_{\rho} F \to J^1 (P \times_{\rho} F)\) induced by \(\omega\) from the first exercise. (1 p.)
Calculate the Koszul connection \(\nabla^{\omega_{\rho}}\). What is the covariant derivative of a section \(\epsilon: M \to P \times_{\rho} F\) given by \(\epsilon(t) = (t, z(t))\)? (1 p.)
Deadline Tartu: 10. 12. 2024 23:59 EET
Deadline Oldenburg: 30. 01. 2024 23:59 CET
Properties of connections
Connection on a bundle over the sphere (5 p.)
Consider the following geometric objects:
The base manifold \(M = \{\vec{X} \in \mathbb{R}^3, \|\vec{X}\| = 1\} \cong S^2\) is the sphere of vectors \(\vec{X}\) of unit length.
The total space \(P = \{(\vec{X}, \vec{Y}) \in \mathbb{R}^3 \times \mathbb{R}^3, \|\vec{X}\| = \|\vec{Y}\| = 1, \vec{X} \cdot \vec{Y} = 0\}\) attaches to each point of the sphere a circle of unit radius, consisting of vectors \(\vec{Y}\) of unit length perpendicular to \(\vec{X}\).
The projection \(\pi: P \to M, (\vec{X}, \vec{Y}) \mapsto \vec{X}\) discards the vector \(\vec{Y}\) and retains only the base point \(\vec{X}).
One can introduce coordinates \((x^a, a = 1, 2) = (\vartheta, \varphi)\) on \(M\) and \((y^{\alpha}, \alpha = 1) = (\psi)\) on each fiber, to define \((x^a, y^{\alpha}) = (\vartheta, \varphi, \psi)\) on \(P\), such that
Using the definitions above, we have \(M\) and \(P\) as submanifolds of \(\mathbb{R}^3\) and \(\mathbb{R}^3 \times \mathbb{R}^3\), respectively. This allows us to identify their tangent spaces at each point with the respective tangents to these embedded submanifolds. This leads us to the following structure of the tangent bundles:
The tangent bundle \(TM = \{(\vec{X}, \vec{U}) \in \mathbb{R}^3 \times \mathbb{R}^3, \|\vec{X}\| = 1, \vec{X} \cdot \vec{U} = 0\}\) contains for each \(\vec{X} \in M\) all vectors \(\vec{U}\) is perpendicular to \(\vec{X}\).
The tangent bundle \(TP = \{(\vec{X}, \vec{Y}, \vec{U}, \vec{V}) \in \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3, \|\vec{X}\| = \|\vec{Y}\| = 1, \vec{X} \cdot \vec{Y} = \vec{X} \cdot \vec{U} = \vec{X} \cdot \vec{V} + \vec{U} \cdot \vec{Y} = \vec{Y} \cdot \vec{V} = 0\}\) contains for each \((\vec{X}, \vec{Y}) \in P\) all pairs \((\vec{U}, \vec{V})\) of vectors such that \(\vec{U}\) is perpendicular to \(\vec{X}\) and \(\vec{V}\) describes the possible motion of \(\vec{Y}\) along the circle.
The pullback bundle \(\pi^*TM = \{(\vec{X}, \vec{Y}, \vec{U}) \in \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3, \|\vec{X}\| = \|\vec{Y}\| = 1, \vec{X} \cdot \vec{Y} = \vec{X} \cdot \vec{U} = 0\}\) contains the tangent space \(T_{\pi(\vec{X}, \vec{Y})}M\) for each \((\vec{X}, \vec{Y}) \in P\).
Alternatively, we can also write tangent vectors as
Finally, 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 \(\vec{V} = -\vec{Y} \times (\vec{X} \times \vec{U})\).
For a tangent vector \(u = u^{\vartheta}\frac{\partial}{\partial\vartheta} + u^{\varphi}\frac{\partial}{\partial\varphi} = u^a\frac{\partial}{\partial x^a} \in T_{\vec{X}}M\), calculate the components of \(\vec{U}\). (1 p.)
Determine \(v^{\psi}\), so that \(w = u^{\vartheta}\frac{\partial}{\partial\vartheta} + u^{\varphi}\frac{\partial}{\partial\varphi} + v^{\psi}\frac{\partial}{\partial\psi} = u^a\frac{\partial}{\partial x^a} + v^a\frac{\partial}{\partial y^a} \in H_{(\vec{X}, \vec{Y})}P\). (Hint: use \(\vec{U}\) from the previous exercise to find \(\vec{V}\), and compare with the formulas above.) (1 p.)
Determine the connection form \(\theta = (\theta^{\psi}_{\vartheta}(\vartheta, \varphi, \psi)d\vartheta + \theta^{\psi}_{\varphi}(\vartheta, \varphi, \psi)d\varphi + d\psi) \otimes \frac{\partial}{\partial\psi}\) which satisfies \(\theta(w) = 0\). (1 p.)
Calculate the curvature of the connection form. (1 p.)
Let \(\gamma: \mathbb{R} \to P, t \mapsto (\vartheta(t), \varphi(t), \psi(t))\) be a curve on \(P\), where \(\vartheta(t) = \Theta\) is constant, \(\varphi(t) = t\) and \(\psi(t)\) determined such that \(\dot{\gamma}(t)\) is everywhere horizontal, \(\theta(\dot{\gamma}(t)) = 0\). Calculate \(\psi(t)\) and \(\psi(2\pi) - \psi(0)\). (1 p.)
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. 2024 23:59 EET
Deadline Oldenburg: 06. 02. 2024 23:59 CET
Tensor densities and covariant derivative
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
the Lie derivatives \(\mathcal{L}_X(\mathfrak{Z} \otimes \mathfrak{A})\), \(\mathcal{L}_X\mathfrak{Z}\) and \(\mathcal{L}_X\mathfrak{A}\) with respect to a vector field \(X \in \mathrm{Vect}(M)\), (1 p.)
the covariant derivatives \(\nabla_X(\mathfrak{Z} \otimes \mathfrak{A})\), \(\nabla_X\mathfrak{Z}\) and \(\nabla_X\mathfrak{A}\) with respect to a vector field \(X \in \mathrm{Vect}(M)\) and an arbitrary connection. (1 p.)
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 \(\tau: TM \to M\).
Show that the components of these densities, given by the Levi-Civita symbols, do not change under a coordinate transformation. (1 p.)
Calculate the Lie derivatives \(\mathcal{L}_X\mathfrak{E}\) and \(\mathcal{L}_X\mathfrak{e}\) with respect to a vector field \(X \in \mathrm{Vect}(M)\), (1 p.)
Calculate the covariant derivatives \(\nabla_X\mathfrak{E}\) and \(\nabla_X\mathfrak{e}\) with respect to a vector field \(X \in \mathrm{Vect}(M)\) and an arbitrary connection. (1 p.)
Hint: You might need the total antisymmetry of the Levi-Civita symbol and its relation to the Kronecker symbol.
Deadline Tartu: 24. 12. 2024 23:59 EET
Deadline Oldenburg: 09. 04. 2024 23:59 CET
Integration
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 \(\omega = \sin\theta\,\mathrm{d}\theta \wedge \mathrm{d}\phi\).
Show that \(\omega\) is indeed a 2-form, i.e., a smooth section of \(\Lambda^2T^*M\). You may use the two charts given by stereographic projection we defined earlier for this purpose. (1 p.)
Calculate the integral of \(\omega\) over the \(2\)-cube \[\begin{array}{rcccc} c & : & [0, 1]^2 & \to & S^2 \\ && (x, y) & \mapsto & (\pi x, 2\pi y) \end{array}\,.\] (1 p.)
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 \[\begin{array}{rcccc} c & : & [0, 1]^3 & \to & M \\ && (x, y, z) & \mapsto & (x, y, z) \end{array}\,.\]
Determine the 2-chain \(\partial c\) (the boundary of \(c\)). (1 p.)
Calculate \(\mathrm{d}\omega\). (1 p.)
Calculate the integrals \(\int_{\partial c}\omega\) and \(\int_c\mathrm{d}\omega\). (1 p.)
Deadline Tartu: 22. 02. 2024 23:59 EET
Deadline Oldenburg: 16. 04. 2024 23:59 CET
Klein geometry and Cartan geometry
Hyperbolic Klein geometry (3 p.)
Consider the Klein geometry \((G,H)\) with \(G = \mathrm{SL}(2,\mathbb{R})\) and \[H = \left\{\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, t \in \mathbb{R}\right\}\,.\]
Show that there is a bijective mapping between \(G/H\) and \(\mathbb{R}^2 \setminus \{(0,0)\}\) such that \[G/H \ni \tilde{g}H = \begin{pmatrix} x & u \\ y & v \end{pmatrix}H \leftrightarrow (x,y) \in \mathbb{R}^2 \setminus \{(0,0)\}\,.\] (1 p.)
Calculate the action of an element \[g = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in G\] on an element of \(G/H\) represented by \((x,y)\). (1 p.)
What is the stabilizer of \((x,y) = (1,0)\)? (1 p.)
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 \[g = \begin{pmatrix} r\cos\theta & xr\cos\theta - \frac{\sin\theta}{r} \\ r\sin\theta & xr\sin\theta + \frac{\cos\theta}{r} \end{pmatrix}\,.\]
Show that \(A\) takes values in \(\mathfrak{sl}(2, \mathbb{R})\). (1 p.)
Consider a manifold \(M\) equipped with a non-linear connection. Show that the connection is…
…homogeneous if and only if the connection coefficients are homogeneous functions of degree 1 in the fiber coordinates: \(N^a{}_b(x, \lambda\bar{x}) = \lambda N^a{}_b(x, \bar{x})\). (1 p.)
…linear if and only if the connection coefficients are linear in the fiber coordinates: \(N^a{}_b(x, \bar{x}) = \Gamma^a{}_{cb}(x)\bar{x}^c\). (1 p.)
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 \(\bar{x}'^a = \bar{x}^b\frac{\partial x'^a}{\partial x^b}\):
The Liouville vector field: \(\bar{x}^a\bar{\partial}_a\). (1 p.)
The vertical lift: \(X^a(x)\bar{\partial}_a\). (1 p.)
The complete lift: \(X^a(x)\partial_a + \bar{x}^a\partial_aX^b(x)\bar{\partial}_b\). (1 p.)
Deadline Tartu: 07. 03. 2024 23:59 EET
Deadline Oldenburg: 07. 05. 2024 23:59 CET
Finsler geometry
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:
The order \(r\) of homogeneity, from \(\mathbf{c}F = rF\). (1 p.)
The Hilbert one-form \(\alpha\). (1 p.)
The Cartan one-form \(\theta\). (1 p.)
The Cartan two-form \(\omega\). (1 p.)
The geodesic spray \(\mathbf{S}\). (1 p.)
Deadline Tartu: 14. 03. 2024 23:59 EET
Deadline Oldenburg: 14. 05. 2024 23:59 CET
Affine connections
Affine connection on \(S^3\) (5 p.)
Consider the following geometric objects:
The manifold \(M = \{\vec{x} \in \mathbb{R}^4, \|\vec{x}\| = 1\}\).
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 \(TM\).
Calculate \((\vec{x}, \vec{y})\) in terms of \((x^a, \bar{x}^a)\). (1 p.)
Calculate the connection coefficients \(\Gamma^a{}_{bc}(x)\) such that \(\partial_a - \Gamma^b{}_{ca}\bar{x}^c\bar{\partial}_b \in HTM\). (1 p.)
Calculate the torsion tensor \(T\). (1 p.)
Calculate the curvature tensor \(R\). (1 p.)
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 geometry
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 \[g = r^2(\mathrm{d}\vartheta \otimes \mathrm{d}\vartheta + \sin^2\vartheta\,\mathrm{d}\varphi \otimes \mathrm{d}\varphi)\,.\]
Calculate the inverse metric \(g^{-1}\). (1 p.)
Calculate the components \(\Gamma^a{}_{bc}\) of the Levi-Civita connection. (1 p.)
Calculate the components \(R_{abcd}\) of the Riemann curvature tensor (with lower indices). (1 p.)
Calculate the components \(R_{ab}\) of the Ricci tensor. (1 p.)
Calculate the Ricci scalar \(R\). (1 p.)
Deadline Tartu: 28. 03. 2024 23:59 EET
Deadline Oldenburg: 28. 05. 2024 23:59 CET
Complex geometry and spin
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 \(\mathbb{R}^3\).
For \(x \in M\) and \(v \in T_xM\), define \(J(v) = x \times v\), via the cross product. Show that \(J: TM \to TM\) is an almost complex structure. (1 p.)
Calculate \(N_J\). (1 p.)
Show that the maps
\[\phi_1: M \setminus \{(0,0,1)\} \to \mathbb{C}, x \mapsto \frac{x_1 + ix_2}{1 - x_3}\]
and
\[\phi_2: M \setminus \{(0,0,-1)\} \to \mathbb{C}, x \mapsto \frac{x_1 - ix_2}{1 + x_3}\]
define a holomorphic atlas, i.e., that their transition function is holomorphic. (1 p.)
Show that \(J\) is the complex structure on the complex manifold obtained from the atlas above. (1 p.)
For \(x \in M\) and \(u, v \in T_xM\), define \(\omega(u, v) = x \cdot (u \times v)\). Show that this turns \(S^2\) into a Kähler manifold. (1 p.)
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
Variational problem
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 \(J^1(E)\).
Show that \(L = F\,\mathrm{d}t \in \Omega^1(J^1(E))\) is a Lagrangian. (1 p.)
Let \(\alpha = \bar{\partial}_aF\,\mathrm{d}x^a \in \Omega^1(J^1(E))\) be the Hilbert form (viewed as a form on \(J^1(E) \cong \mathbb{R} \times TQ\) via the pullback from \(TQ\) along the projection map). Show that \(L - \alpha\) is a contact form. (1 p.)
Let \(\gamma: M \to Q\) be a curve, which determines a section: \(\tilde{\gamma}: M \to E, t \mapsto (t, \gamma(t))\). Show that its first jet prolongation has the coordinate expression \(j^1\tilde{\gamma}: t \mapsto (t, \gamma^a(t), \dot{\gamma}^a(t))\). (1 p.)
Show that \[\int_{[a, b]}(j^1\tilde{\gamma})^*L = \int_{[a, b]}(j^1\tilde{\gamma})^*\alpha\]. (1 p.)
Calculate \(\mathrm{d}_HL\) and \(\mathrm{d}_VL\). (1 p.)
Deadline Tartu: 11. 04. 2024 23:59 EET
Deadline Oldenburg: 11. 06. 2024 23:59 CET
Action principle
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 \[L(t,q,\dot{q}) = \left(\frac{1}{2}g_{ab}(q)\dot{q}^a\dot{q}^b - V(q)\right)\mathrm{d}t \in \Omega^{1,0}(J^1(E))\,.\]
Calculate the vertical derivative \(\mathrm{d}_VL\). (1 p.)
Calculate the projection \(\mathcal{E}L = \varrho(\mathrm{d}_VL)\) onto \(\mathcal{F}^1(J^2(E))\). (1 p.)
Show that \(\mathcal{E}L\) is of the form \(\omega_a\theta^a \wedge \mathrm{d}t\), where \(\theta^a = \theta^a_{(0)}\). (1 p.)
Show that there exists \(\eta \in \Omega^{0,1}(J^1(E))\) such that \(\varrho(\mathrm{d}_VL) - \mathrm{d}_VL = \mathrm{d}_H\eta\). (1 p.)
Let \(\gamma \in C^{\infty}(\mathbb{R}, Q)\) be a curve, which determines a section \(\sigma = (\mathrm{id}_{\mathbb{R}},\gamma): \mathbb{R} \to \mathbb{R} \times Q\), and write it in coordinates in the form \(t \mapsto q^a(t)\). Derive the Euler-Lagrange equations for such curves. (1 p.)
Deadline Tartu: 18. 04. 2024 23:59 EET
Deadline Oldenburg: 18. 06. 2024 23:59 CET
Lepage forms
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.
Let \(\alpha = \bar{\partial}_aF\,\mathrm{d}x^a \in \Omega^1(J^1(E))\) be the Hilbert form. Calculate \(\mathrm{d}\alpha\). (1 p.)
Express \(\pi_{2,1}^*\mathrm{d}\alpha \in \Omega^2(J^2(E))\) using the contact basis \(\mathrm{d}t, \theta^a, \dot{\theta}^a\). (1 p.)
Calculate \(\mathrm{p}_1\mathrm{d}\alpha\) and show that \(\alpha\) is a Lepage form. (1 p.)
Show that \(\alpha\) is a Lepage equivalent of the Lagrangian \(L = F\,\mathrm{d}t \in \Omega^1(J^1(E))\). (1 p.)
Compare \(\mathrm{p}_1\mathrm{d}\alpha\) to the Euler-Lagrange form \(\mathcal{E}L\). (1 p.)
Deadline Tartu: 25. 04. 2024 23:59 EET
Deadline Oldenburg: -
First Noether theorem
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 \((t, x, y, \dot{x}, \dot{y})\).
Calculate \(\mathrm{d}_VL\) and \(\mathcal{E}L\). (1 p.)
Show that \(\mathcal{E}L - \mathrm{d}_VL = \mathrm{d}_H\eta\) is \(\mathrm{d}_H\)-exact and determine \(\eta\). (1 p.)
Calculate the prolongation \(\mathrm{pr}X\) of the evolutionary vector field \(X = -y\partial_x + x\partial_y\). (1 p.)
Show that \(X\) is a symmetry of the Lagrangian. (1 p.)
Calculate the conserved current \(\psi\) corresponding to \(X\), and use the Euler-Lagrange equations to check that \(\psi\) is indeed conserved. (1 p.)
Deadline Tartu: 02. 05. 2024 23:59 EET
Deadline Oldenburg: 25. 06. 2024 23:59 CET
Gauge theories I - Gauge transformations and gauge bundles
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.
Calculate the coordinates \((x',y',t')\) of an element \[g' = \begin{pmatrix} x' & t'x' - \frac{y'}{x'^2 + y'^2} \\ y' & t'y' + \frac{x'}{x'^2 + y'^2} \end{pmatrix} = gh\,,\] where \[h = \begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix} \in H\] is an element of the structure group \(H\). (1 p.)
Show that the natural projection \(G \to G/H, g \mapsto gH\) defines a principal \(H\)-bundle. (1 p.)
Show that for a function \(f: \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{R}\) the map \(\varphi_f: G \to G\) defined by \[\varphi_f(x, y, t) = (x, y, t + f(x, y))\] is a gauge transformation. (1 p.)
Consider \(f, f_1, f_2: \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{R}\) with \(\varphi_f = \varphi_{f_2} \circ \varphi_{f_1}\). Express \(f\) in terms of \(f_1\) and \(f_2\). (1 p.)
Show that \(\xi(x, y)\partial_t\) is an infinitesimal gauge transformation. (1 p.)
Deadline Tartu: 09. 05. 2024 23:59 EET
Deadline Oldenburg: 02. 07. 2024 23:59 CET
Gauge theories II - Matter fields
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 \(E = P \times_{\rho} \mathbb{C}\).
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 \(\beta \in C^{\infty}(M, \mathbb{R})\). Calculate the transformed gauge \(\epsilon' = \varphi \circ \epsilon\). (1 p.)
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 \(\phi \in C^{\infty}(M, \mathbb{C})\). Calculate the transformed matter field \(\Phi^{\epsilon'}\) in the gauge \(\epsilon'\) and express \(\phi'(x) = \Phi^{\epsilon'}(x)\) using \(\phi\) and \(\beta\). (1 p.)
Calculate the infinitesimal change \(\delta\phi'(x)/\delta\beta(x)\). (1 p.)
Show that \(\phi\phi^* = \phi'\phi'^*\) where the star \(*\) denotes the complex conjugate. (1 p.)
Let \(x \mapsto g^{\mu\nu}(x)\) be a symmetric tensor field and show that \[g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi^* \neq g^{\mu\nu}\partial_{\mu}\phi'\partial_{\nu}\phi'^*\,,\] so that this term is not gauge invariant. (1 p.)
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 theories III - Gauge fields
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 \(\phi \in C^{\infty}(M, \mathbb{C})\).
Show that \(\vartheta = i(a_{\mu}\,\mathrm{d}x^{\mu} + \mathrm{d}\alpha) \in \Omega^1(P, \mathfrak{u}(1))\) is a principal \(G\)-connection. (Note that \(\mathfrak{u}(1) \cong i\mathbb{R}\) are just the imaginary numbers.) (1 p.)
Let \(\chi: C \to M\) be the principal connection bundle of \(P\) and consider a connection \(\Omega: M \to C\) which is defined in the gauge \(\epsilon\) as \(\Omega^{\epsilon}(x) = iA_{\mu}(x)\,\mathrm{d}x^{\mu} \in \Omega^1(M, \mathfrak{u}(1))\). Calculate \(A_{\mu}\) in terms of \(a_{\mu}\). (1 p.)
Calculate \(\Omega^{\epsilon'}\) and use \(\Omega^{\epsilon'}(x) = iA'_{\mu}(x)\,\mathrm{d}x^{\mu}\) to express \(A'_{\mu}\) using \(A_{\mu}\) and \(\beta\). (1 p.)
Consider the covariant derivative \[D_{\mu}\phi = \partial_{\mu}\phi + iA_{\mu}\phi\] and \[D'_{\mu}\phi' = \partial_{\mu}\phi' + iA'_{\mu}\phi'\] and calculate \(D'_{\mu}\phi'\) in terms of \(D_{\mu}\phi\) and \(\beta\).
Show that \[g^{\mu\nu}D_{\mu}\phi(D_{\nu}\phi)^* = g^{\mu\nu}D'_{\mu}\phi'(D'_{\nu}\phi')^*\,,\] with the same definitions as in the previous homework. (1 p.)
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: -
Second Noether theorem
Gauged rotational symmetry (5 p.)
Consider the trivial fiber bundle \(\pi: M \times Q \to M\) with \(M = \mathbb{R}\) with coordinate \((t)\) and \(Q = \mathbb{R}^3\) with coordinates \((x, y, z)\) and a Lagrangian \(L = \mathcal{L}\,\mathrm{d}t \in \Omega^{1,0}(J^1(\pi))\), where \[\mathcal{L} = \frac{(\dot{x} + zy)^2 + (\dot{y} - zx)^2}{2} - V(x^2 + y^2)\] with a free function \(V\), and we wrote the coordinates on \(J^1(\pi)\) as \((t, x, y, z, \dot{x}, \dot{y}, \dot{z})\).
Calculate \(\mathrm{d}_VL\) and \(\mathcal{E}L\). (1 p.)
Show that \(\mathcal{E}L - \mathrm{d}_VL = \mathrm{d}_H\eta\) is \(\mathrm{d}_H\)-exact and determine \(\eta\). (1 p.)
Show that the evolutionary vector field \(X = f(t)(-y\partial_x + x\partial_y) + f'(t)\partial_z\) is a symmetry of the Lagrangian for any function \(f \in C^{\infty}(M, \mathbb{R})\). (1 p.)
Calculate the conserved current \(\psi\) corresponding to \(X\). (1 p.)
Show that \(\psi\) reduces to a superpotential, i.e., that it can be written as \(\psi = \mathrm{d}_H\alpha + \beta\), where \(\beta\) vanishes on the subspace \(\mathcal{E}L = 0\). (Note that \(\alpha\) or \(\beta\) can also be 0.) (1 p.)
Deadline Tartu: 30. 05. 2024 23:59 EET
Deadline Oldenburg: -
Diffeomorphism invariance and energy-momentum conservation
Deadline Tartu: -
Deadline Oldenburg: -
Symplectic geometry and Hamilton formalism
Classical spin (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 \(\mathbb{R}^3\).
For \(x \in M\) and \(v, w \in T_xM\), define \(\omega(v, w) = x \cdot (v \times w)\), via the cross product. Calculate \(\omega\) in spherical coordinates \(\vartheta \in (0, \pi)\) and \(\varphi \in (0, 2\pi)\). (1 p.)
Show that \(\omega\) is a symplectic form on \(M\). (1 p.)
Find a symplectic potential, i.e., a one-form \(\theta \in \Omega^1(M)\) such that \(\omega = -\mathrm{d}\theta\). (1 p.)
Calculate the Hamiltonian vector field \(X_H\) of the Hamiltonian \(H = c\cos\vartheta\) with \(c \in \mathbb{R}\) and the solution \(\gamma \in C^{\infty}(\mathbb{R}, M)\) for \(\gamma(0) = (\vartheta_0, \varphi_0)\). (1 p.)
Calculate the Poisson bracket \(\{f, H\}\) for \(f(\vartheta, \varphi) = \cos\varphi\) and confirm that \((f \circ \gamma)'(t) = \{f, H\}(\gamma(t))\) for all \(t \in \mathbb{R}\). (1 p.)