• 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.

  [CE] J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry.

  [K] Alexander Kupers, Lectures on diffeomorphism groups of 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.

  [P] Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms.

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

  [S'] Luiz San Martin, Lie Groups.

• HWA-14, due Thu, Dec 12.
  • Read [P, 1.1--1.4] and [CE, sections 0--2 in chapter 1].
  • Suppose that a symplectomorphism \(\varphi\) is generated by a time-independent Hamiltonian function \(H\colon M \to \mathbb{R}\). Show that every isolated fixed point \(p \in M\) of \(\varphi\) is a critical point of \(H\).
  • Let \(f_t\) and \(g_t\) be a Hamiltonian flows generated by time-dependent Hamiltonians \(F_t\) and \(G_t\) respectively. Show that \(f_t \circ g_t\) is the Hamiltonian flow generated by \(H_t= F_t + G_t \circ f_t\).
  • Show that \(T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]\) is a tensor for any affine connection \(\nabla\).
  • Show that \(\mathrm{grad}\,(fh)=h{\cdot}\mathrm{grad}\,f+f{\cdot}\mathrm{grad}\,h\) for any two smooth \(f\) and \(h\) functions on a Riemannian manifold.
  • Show the Riemannian distnace between any two disticnt points of Riemannian manfold is strictly positive.
• HWA-13, due Thu, Dec 5.
  • Describe a nonorientable 1-dimensional foliation on the 2-torus. Sketch its leaves on the square diagram.
  • Show that the Reeb foliation in the 3-sphere cannot be obtained from a submersion.
  • Show that a vector field transverse to the Reeb foliation in the 3-sphere has a periodic orbit.
  • Let \(\alpha\) be a nonvanishing 1-form on a smooth manifold \(M\). Note that the kernel of \(\alpha\) defines a codimension-one distribution on \(M\). Show that this distribution is involutive if and only if \(\alpha\wedge d\alpha=0\).
  • The position of a car in the plane is determined by three parameters: two coordinates \(x\) and \(y\) for the location (a good choice is the location of the midpoint between the back wheels) and an angle \(\alpha\) which describes the orientation of the car. The motion of the car satisfies the equation \(x'{\cdot}\sin\alpha=y'{\cdot}\cos \alpha\).
    Describe the motion of the car via horizontal curves of a plane distribution on a 3-dimensional manifold. Is this distribution integrable, nonintegrable, and/or completely nonintegrable? What does it mean for the car?
  • Try to read [K, Lecture 7].

• 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.)