• Syllabus
• Texts:

  [BC] Richard Bishop and Richard Crittenden Geometry of manifolds.

  [BG] Richard Bishop and Samuel Goldberg Tensor analysis on manifolds.

  [BT] Dennis Barden and Charles Thomas An Introduction to Differential Manifolds.

  [L] John Lee, Introduction to smooth manifolds. (Second edition)

  [Mat] Yukio Matsumoto, An introduction to Morse theory.

  [M] John Milnor, Topology from the Differentiable Viewpoint.

  [S] Michael Spivak, A comprehensive introduction to differential geometry. vol. I--V.

  [S'] Luiz San Martin, Lie Groups.

• HWA-12, due Thu, Nov 21.
  • Read [S', Chapter 5]
  • Exercises: 1, 3, 6, 9, and 17 from [S', 5.7]

Midterm Thu, Nov 14 in class; here is a list of questions; see also list of topics + some notes. You are more than welcome to use a cheat sheet.

• HWA-11, due Tue, Nov 12.
  • Think about the midterm; prepare at least 3 questions.
• HWA-10, due Tue, Nov 05.
  • Think about the midterm; prepare questions.
  • Skim thru [Mat, chapters 1--3] OR [BT, chapter 7].
  • Construct a Morse function on \(\mathbb{R}\mathrm{P}^2\) with 3 critical points. Describe the sublevel sets of this function. Try to do the same for \(\mathbb{C}\mathrm{P}^2\).
  • Let \(\pi\colon E\to B\) be a smooth locally trivial fiber bundle with fiber \(F\); that is, every \(b \in B\) admits an open neighborhood \(U \subset B\) with a diffeomorphism \(\varphi : U \times F\to \pi^{-1}(U)\) such that \(\pi\circ \varphi \colon U \times F\to U\) is the projection onto the first factor.
    Suppose that the base \(B\) admits a Morse function with \(m\) critical points and fiber \(F\) admits a Morse function with \(n\) critical points. Show that the total space \(E\) admits a Morse function with \(m{\cdot}n\) critical points.
    Hint: First do it for the trivial bundle; that is, \(E= B\times F\) and \(\pi\) is the projection to the first factor.
  • Let \(f\) be a Morse function on a closed smooth manifold \(M\). Suppose that \(a\) and \(b\) are regular values and there is a unique critical value \(s\) of index \(i\) in the interval \((a,b)\). Denote by \(M_t\) the sublevel set \(\{\,x\in M\mid f(x)\le t\}\). Show that either
    • \(\beta_i(M_b)=\beta_i(M_a)+1\) and \(\beta_j(M_b)=\beta_j(M_a)\) for \(j\ne i\) or
    • \(\beta_{i-1}(M_b)=\beta_{i-1}(M_a)-1\) and \(\beta_j(M_b)=\beta_j(M_a)\) for \(j\ne i-1\),
    where \(\beta_i(N)\) denotes the \(i\)-th Betti number of \(N\); that is \(\beta_i(N)=\dim H^i_{dR}(N)\).
    Hint: apply the Mayer--Vietoris sequence.
  • Let \(f\) be a Morse function on a closed smooth manifold \(M\). Recall that the Euler number \(\chi (M)\) is the index of self-intersection of \(M\) in its tangent bundle \(\mathrm{T}M\).
    Show that \(\chi (M)=n_0-n_1+n_2-\dots\), where \(n_i\) denotes the number of critical points of \(f\) with index \(i\).
    Use the previous problem to show that \(\chi (M)=\beta_0(M)-\beta_1(M)+\beta_2(M)-\dots\)
  • Suppose that a compact connected manifold \(M\) admits a Morse function with \(n\) critical points of index 1 and \(m\) critical points of index 2. Show that the fundamental group of \(M\) admits a presentation with at most \(n\) generators and \(m\) relations.
• HWA-09, due Tue, Oct 29.
  • Continue to read [L, Chapter 18]
  • Start to think about the midterm; prepare one question (at least); this is part of the HWA.
  • Problems 17-1, 17-5, 17-12, 17-13 from [L].
  • Let \(\alpha\) be a nowhere zero 1-form on a smooth compact manifold \(M\). Show that \(\alpha\wedge d\alpha=0\) if and only if there exists a 1-form \(\beta\) such that \(d\alpha= \alpha\wedge \beta\).
    Hint: First do it locally and then use a partition of unity.
• HWA-08, due Tue, Oct 22.
  • Read [L, Chapter 18]
  • Try to read "On the volume elements on a manifold" by J. Moser.
  • Let \(M\) and \(N\) be closed connected oriented \(n\)-dimesional manifolds and let \(\omega\) be an \(n\)-form on \(N\). Show that \(\int_M\varphi^*\omega=\deg\varphi\cdot\int_N\omega\), where \(\varphi\colon M\to N\) is a smooth map and \(\deg\varphi\) is its degree as defined in [M].
  • Calculate \(H^2_{dR}(\mathbb{S}^2\times \mathbb{S}^2)\) and \(H^2_{dR}(\mathbb{T}^4)\) using symmetries (here \(\mathbb{T}^4\) denotes 4-dimensional torus). Describe a homomorphism \(h:H^2_{dR}(\mathbb{S}^2\times \mathbb{S}^2)\to H^2_{dR}(\mathbb{T}^4)\) that cannot be realized as an induced homomorphism for a smooth map \(\mathbb{T}^4\to \mathbb{S}^2\times \mathbb{S}^2\).
    You may use [S, vol. V, chap. 13, sec. 6] as a reference.
  • Construct a closed 2-form \(\omega\) on \(M = (\mathbb{S}^2)^{\times n}\) for \(n \geqslant 2\) such that \(\omega^{\wedge n} \not= 0\) at any point of \(M\). Use it to show that any continuous map \(\mathbb{S}^{2\cdot n}\to M\) has vanishing degree.
  • Let \(f\colon \mathbb{S}^{2{\cdot}n-1}\to \mathbb{S}^n\) for \(n \geqslant 2\), be a smooth map, and let \(\omega\) be an \(n\)-form on \(\mathbb{S}^n\) such that \(\int_{S^n}\omega = 1\). Show that \(f^* \omega\) is exact, and if \(f^* \omega= d\alpha\), then the number \(\int_{S^{2n-1}}\alpha\wedge d \alpha\) is independent of the choice of \(\omega\) and \(\alpha\).
  • Show that any open star-shaped set in \(\mathbb{R}^n\) is diffeomorphic to \(\mathbb{R}^n\).
• HWA-07, due Tue, Oct 15.
  • Read [L, first 3 sections in Chapter 16].
  • Prove that the following identities
    • \(\mathcal{L}_Xd\omega = d\mathcal{L}_X \omega,\)
    • \(\mathcal{L}_X i_Y\omega- i_Y\mathcal{L}_X\omega = i_{[X,Y]}\omega,\)
    • \(i_X i_Y\omega+ i_Yi_X\omega = 0\)
    hold for any differential form \(\omega\) and any vector fields \(X\) and \(Y\). (Try to do it directly from the definitions.)
  • Let \(\alpha\) be a 1-form and let \(X\) and \(Y\) be vector fields on a smooth manifold. Express \((d\alpha)(X,Y)\) in terms of \(X\bigl(\alpha(Y)\bigr)\), \(Y\bigl(\alpha(X)\bigr)\), and \(\alpha([X,Y])\)
  • Show that the form \(\alpha = \frac{x\cdot dy - y\cdot dx}{x^2+y^2}\) on \(\mathbb{R}^2\backslash \{0\}\) is closed but not exact.
  • Let \(\omega\) be a volume form on a closed connected \(n\)-dimensional manifold \(M\); that is, \(\omega\) is a nonvanishing \(n\)-form. Show that for any \((n-1)\)-form \(\eta\) there is a unique vector field \(V\) on \(M\) such that \(i_V\omega=\eta\).
  • Problem 14-9 from [L] (submersion = all points are regular).
• HWA-06, due Tue, Oct 8.
  • Read [BC, 4.1--4.4]
  • [BG, Problem 3.6.7].
  • Describe the cross product as a tensor on \(\mathbb{R}^3\). Let \(\tau\) be the corresponding tensor field on \(\mathbb{R}^3\). Find the components of \(\tau\) in spherical coordinates.
  • Let \(\alpha_1,\dots, \alpha_k\) be linearly independent covectors. Suppose that \(\sum_i\alpha_i\wedge\beta_i=0\) for some covectors \(\beta_1,\dots, \beta_k\). Show that each \(\beta_i\) is a linear combination of \(\alpha_1,\dots, \alpha_k\).
  • Show that any 2-covector \(\vartheta\) can be written as a sum \(\sum_i\alpha_i\wedge\beta_i\) for linearly independent covectors \(\alpha_1,\dots, \alpha_k,\beta_1,\dots, \beta_k\).
  • Let \(\vartheta\) be a 2-covector. Show that \(\vartheta\wedge\vartheta=0\) if and only if \(\vartheta\) is decomposabe; that is, \(\vartheta=\alpha\wedge \beta\) for some covectors \(\alpha\) and \(\beta\).
• HWA-05, due Tue, Oct 1.
  • Read [BG, 3.2 + 3.6]; skim thru the rest of Chapter 3; prepare one question.
  • Problems from [BG]: 3.2.1, 3.2.2, 3.6.1, 3.7.1, 3.7.2, 3.8.1.
• HWA-04, due Tue, Sep 24.
  • Read [BC, Chapter 1] OR [L, Chapter 8].
  • Problem from [BC, Chapter 1]: 14.
  • Prove the Jacobi identity \([[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0\) for any vector fields \(X,Y\), and \(Z\).
  • Let \(f\) be a smooth function defined on a smooth manifold \(M\). Suppose \(p\) is a critical point of \(f\), and \(X,Y\) are vector fields on \(M\). Show that \((X(Yf))(p)\) depends only on \(X(p),Y(p)\in\mathrm{T}_pM\).
    Furthermore, the map \((X(p),Y(p))\mapsto X(Y(f))(p)\) is a symmetric bilinear form on \(\mathrm{T}_pM\).
  • Let \(X,Y\) be vector fields on a smooth 2-dimensional manifold \(M\). Suppose \(X(p)\) and \(Y(p)\) are linearly independent at \(p\in M\). Construct a local coordinate system at \(p\) so that the vector fields are tangent to its coordinate lines.
  • Show that one cannot do the same in higher dimensions. That is, find three vector fields \(X,Y,Z\) on a smooth 3-dimensional manifold \(M\) such that \(X(p)\), \(Y(p)\), and \(Z(p)\) form a basis in \(\mathrm{T}_p\), but there is no chart at \(p\) with these vector fields tangent to its coordinate lines.
    Upload to gradescope.com before 9:05am, Tue, Sep 24.
• HWA-03, due Tue, Sep 17.
  • Read and think [BC, Chapter 1] OR [L, Chapter 8].
  • Exercises from [M, §8]: 6, 13.
  • Suppose that \(f\colon M\to N\) is a smooth map between compact smooth oriented \(n\)-manifolds. Show that \(\mathrm{deg}f\) is the intersection number of the graph of \(f\) with a horizontal submanifold \(M\times x\subset M\times N\).
  • Given \(n\in\mathbb{Z}\), construct a smooth 4-manifold with an embedded 2-sphere that has self-intersection number \(n\).
  • Let \(X\) and \(Y\) be submanifolds of \(M\). Show that \(X\) is transversal to \(Y\) if and only if \(X\times Y\subset M\times M\) is transversal to the diagonal \(\Delta=\{\,(x,x)\in M\times M\,\}\).
    (ATTENTION: Upload to gradescope.com before 9:05am, Tue, Sep 17. Use this code to find the course: J78Y76. When uploading, split it into the separate problems.)
• HWA-02, due Tue, Sep 10.
  • Read and understand [M, §§3--5].
  • Read and think [M, §§6--7].
  • Exercises from [M, §8]: 4, 5.
  • Let \(f\) be a smooth function on \(\mathbb{R}^2\). Suppose \(S\) is a connected set such that \(d_xf=0\) for any \(x\in S\). Show that \(f\) is constant on \(S\).
  • Prove a Baire-category version of Sard's lemma: The set of critical values of a smooth map \(f\colon M\to N\) between smooth manifolds a countable union of compact nowhere dense sets.
  • Let \(\iota\) be a smooth involution of a smooth connected manifold \(M\). Assume \(\iota(x)=x\) and \(d_x\iota=\mathrm{id}_{\mathrm{T}_x}\) for some \(x\in M\). Show that \(\iota=\mathrm{id}_M\).
    (Scan to pdf and upload to CANVAS.)
• HWA-01, due Tue, Sep 3.
  • Read and understand [L, profs of C.34 and C.35] and [M, §§1--2].
  • Read and think [M, §§3--5 and Appendix].
  • Exercises from [M, §8]: 9, 10, 3.
  • Let \(\iota\) be a smooth involution of a smooth manifold \(M\). Show that each connected component of the fixed-point set \(S\subset M\) of \(\iota\) is a submanifold.
  • Construct a diffeomorphism \(f\colon \mathbb{R}^2\to \mathbb{R}^2\) with connected fixed-point set that is not a smooth submanifold.
    (Scan to pdf and upload to CANVAS.)