Topics in Differential Geometry

CU Boulder

R1. Nine Lectures on Exterior Differential Systems by R. L. Bryant
R2. Exterior Differential Systems by R. L. Bryant, et al.
R3. Cartan for Beginners, 2nd ed. by T. A. Ivey and J. M. Landsberg
R4. Notes on Exterior Differential Systems by R. L. Bryant
R5. The Method of Equivalence and Its Applications by R. B. Gardner

Lecture 16: Cartan on Cartan-Kähler.

8/26   The notion of an EDS; integral manifolds; basic examples and relation with ODE/PDE; Pfaffian systems; Pfaffian systems (distributions) in low dimensions/ranks; Theorems of Frobenius, Pfaff and Engel
Readings   [R3] 1.1-1.4, 1.9; [R1] Lecture 1

9/4     The notion of equivalence and examples; Jørgen's Theorem; Geometric examples (Weingarten surfaces); application of the Frobenius theorem: maps into a Lie group; Maurer-Cartan theorem
Readings   [R1] 1.5, 2.3

9/9     Bonnet's theorem; Cauchy characteristics, Cartan system and applications; theorem of Bianchi and Bäcklund
Readings   [R1] 2.4, 2.2

9/16   Grassmann manifolds; spaces of integral elements; ordinary and regular integral elements; examples
Readings   [R1] 3.1, 3.2

9/23   The Cartan-Kähler theorem; examples; [presentation T1]
Readings   [R1] 3.3

9/30   Cauchy problems; theorem of Cauchy-Kowalewski, Proof of the Cartan-Kähler theorem
Reading   [R1] 4.1, 4.2

10/7   Corollary of Cartan-Kähler; Cartan's test; generality of integral manifolds (solutions); Example: orthogonal coordinates on a Riemannian $3$-fold $(M^3,g)$
Reading   [R1] 4.3, 4.4, 5.2

10/14  Applications of Cartan-Kähler: Lie's 3rd Theorem, Weingarten surfaces, hyper-Kähler metrics
Reading   [R1] 5.1, 5.3, 5.4

10/21  Hyper-Kähler metrics (local existence); the concept of prolongation
Reading   [R1] 5.4, 6.1

10/28  Prolongation: properties; Example: restricted principal curvatures
Reading   [R1] 6.2

11/4    Theorem of Cartan-Kuranishi; $g$-orthogonal coordinates in dimension $n$
Reading   [R1] 6.3, 7.1

11/11  Isometric embedding with prescribed mean curvature; Bonnet pairs (local generality); [presentation T5]
Reading   [R1] 7.2

11/18  G-structures and equivalence; [presentations T4, T6]
Reading   [R1] 8.1, 8.2; further reading: [R5]

11/25-29  Fall & Thanksgiving break

12/2    Cartan's five-variable paper; [presentation T2]

12/9    [presentation T7]

E1. Let $\theta$ be a nonvanishing $1$-form defined on $U\subset \mathbb{R}^3$. Suppose that the EDS $(U,\langle \theta\rangle)$ admits an integral surface $S$. Explain why $\theta\wedge{\rm d}\theta|_p = 0$ for any $p\in S$.

E2. Write down some examples of $F$ and $G$ such that the EDS $(\mathbb{R}^3,\langle {\rm d}z - F(x,y,z){\rm d}x - G(x,y,z){\rm d}y\rangle)$
1) admits a unique integral surface;
2) admits a $1$-parameter family of integral surfaces that foliates $\mathbb{R}^3$;
3) admits no integral surface.

E3. The dual notion of a rank-$2$ Pfaffian system $I$ on a $4$-manifold $M$ is a rank-$2$ distribution $\mathcal{D}\subset TM$. Let $\mathcal{D}_0 = \mathcal{D}$ and $\mathcal{D}_{k+1}:=[\mathcal{D}_k, \mathcal{D}]$. Restate the classification results concerning local models of $I$ in terms of $\mathcal{D}$. In particular, convince yourself that the 3 cases that we've discussed in class can be distinguished by whether ${\rm rank}(\mathcal{D}_3)$ equals $2$, $3$ or $4$.

E4. Let $(M^4,\mathcal{I})$ be a rank-$2$ Pfaffian system on a $4$-manifold. Suppose that $\mathcal{I}$ is Engel. We have shown that locally there exist $1$-forms $\theta^1,\theta^2,\omega^1,\omega^2$, all linearly independent, such that $$\mathcal{I} = \langle \theta^1,\theta^2\rangle$$ and $$\left\{ \begin{array}{ll} {\rm d}\theta^1\equiv \theta^2\wedge\omega^1&\mod\theta^1,\\ {\rm d}\theta^2\equiv \omega^1\wedge\omega^2&\mod\theta^1,\theta^2. \end{array} \right. $$ Show that the rank-$3$ subbundle of $T^*M$ locally generated by $\theta^1,\theta^2,\omega^1$ is independent of the choice of the $1$-forms above. In other words, the line field $$\mathcal{W}:= \{\theta^1,\theta^2,\omega^1\}^\perp \subset \mathcal{D} = \ker\mathcal{I}^1$$ is canonically associated to an Engel distribution; $\mathcal{W}$ is said to induce the characteristic foliation associated to $(M,\mathcal{I})$.

E5. Complete the proof of the Maurer-Cartan theorem (p.12 of R1) by justifying:
(1) any $m$-dimensional ($m = \dim(G)$) integral manifold of $(N\times G, \langle\theta\rangle)$ submerses onto its image in $N$;
(2) restricted to a maximal $m$-dimensional integral manifold, the projection $\pi: N\times G \rightarrow N$ is a diffeomorphism.

E6. Let $\Sigma$ be a Riemannian surface with metric $g_\Sigma$. Let $\mathcal{F}_\Sigma$ and $\mathcal{F}_{\mathbb{E}}^3$ be the oriented orthonormal frame bundles over $\Sigma$ and $\mathbb{E}^3$ with projections $\pi$ and $\varpi$, respectively. Let $\eta^1, \eta^2$ be the canonical $1$-forms on $\mathcal{F}_\Sigma$; let $\eta^1_2$ be the unique $1$-form on $\mathcal{F}_\Sigma$ satisfying $${\rm d}\eta^1 = -\eta^1_2\wedge\eta^2, \quad {\rm d}\eta^2 = \eta^1_2\wedge\eta^2;$$ and let $\gamma$ be the Maurer-Cartan form on $\mathcal{F}_\Sigma$. Prescribing a quadratic form ${\rm II}$ on $\Sigma$, there uniquely exist functions $h_{11}, h_{12}, h_{22}$ on $\mathcal{F}_\Sigma$ such that $$\pi^*{\rm II} = h_{11}(\eta^1)^2+ 2h_{12}\eta^1\eta^2+h_{22}(\eta^2)^2.$$ Define $\eta^3_1 = h_{11}\eta^1+ h_{12}\eta^2$ and $\eta^3_2 = h_{12}\eta^1+ h_{22}\eta^2$. Prove that if $$f: \Sigma\rightarrow {\mathbb E}^3$$ is an isometric immersion that realizes ${\rm II}$ as the second fundamental form, then the associated map $\phi: \mathcal{F}_\Sigma\rightarrow \mathcal{F}_{{\mathbb E}^3}$ defined by $$\phi(p,u,v) = (f(p), f_*(u), f_*(v), f_*(u)\times f_*(v))$$ satisfies $$\phi^*\gamma = \left(\begin{array}{cccc} 0&0&0&0\\ \eta^1&0&\eta^1_2&-\eta^3_1\\ \eta^2&-\eta^1_2&0&-\eta^3_2\\ 0&\eta^3_1&\eta^3_2&0 \end{array}\right)=:\eta.$$ Deduce from this that $\eta$ satisfies ${\rm d}\eta = -\eta\wedge\eta$.

E7. Let $\eta^3_1$ and $\eta^3_2$ be as in E6. Explain why both $\Omega^3_1:={\rm d}\eta^3_1+\eta^1_2\wedge\eta^3_2$ and $\Omega^3_2:= {\rm d}\eta^3_2 -\eta^1_2\wedge\eta^3_1$ are scalar multiples of $\eta^1\wedge\eta^2$.

E8. Consider Exercises 2.12 and 2.13 on p.14 of R1.

E9. Bonnet's theorem tells us the type of data (i.e., ${\rm I}$ and ${\rm II}$ with Gauss and Codazzi equations satisfied) you need to specify on a surface that determines an isometric immersion into $\mathbb{E}^3$ up to Euclidean motion. What are analogous results concerning isometric immersions of a curve in $\mathbb{E}^2$ or $\mathbb{E}^3$? What if ${\mathbb E}^n$ is replaced by ${\mathbb S}^n$ or ${\mathbb H}^n$? Does the Maurer-Cartan theorem still apply?

E10. Let $\mathcal{F}_{{\mathbb E}^3}$ be the (oriented) orthonormal frame bundle over $\mathbb{E}^3$. Let $r>0$, $\sigma\in(0,\pi)$ be constants. Let $\phi_{r,\sigma}: \mathcal{F}_{{\mathbb E}^3}\rightarrow \mathcal{F}_{{\mathbb E}^3}$ be the right translation: $$\left(\begin{array}{cccc} 1&0&0&0\\ x&e_1&e_2&e_3 \end{array} \right) \mapsto \left(\begin{array}{cccc} 1&0&0&0\\ x&e_1&e_2&e_3 \end{array} \right)\left(\begin{array}{cccc} 1&0&0&0\\ r&1&0&0\\ 0&0&\cos\sigma&\sin\sigma\\ 0&0&-\sin\sigma&\cos\sigma \end{array}\right).$$ Let $\pi: \mathcal{F}_{{\mathbb E}^3}\rightarrow {\mathbb E}^3\times{\mathbb S}^2$ be the projection that sends $(x,e_1,e_2,e_3)$ to $(x,e_3)$. Suppose that $S\subset {\mathbb E}^3\times{\mathbb S}^2$ is an integral surface of $$\mathcal{J} =\left \langle u\cdot {\rm d}x, \Upsilon_2+\frac{\sin^2\sigma}{r^2}\Upsilon_0\right\rangle.$$ Show that, pulled back to $\pi^{-1}S\subset \mathcal{F}_{{\mathbb{E}}^3}$, the differential ideal $$\mathcal{B}_{r,\sigma}:= \langle\omega^3, \phi_{r,\sigma}^*\omega^3\rangle$$ is Frobenius. Conclude form this the second half of the Bianchi-Bäcklund theorem: For fixed constants $r,\sigma$ as above, any surface with negative constant Gauss curvature $K = -\sin^2\sigma/r^2$ is locally a focal surface of a $1$-parameter family of pseudo-spherical line congruences in $\mathbb{E}^3$.

E11. Let $(M,\mathcal{I})$ be an EDS. Show that the inclusion $\mathcal{V}^r_n(\mathcal{I})\subseteq \mathcal{V}^o_n(\mathcal{I})$ is open and dense. (c.f. Ex. 3.10 of R1)

E12. Let $M$ be a $3$-manifold with a coframing $(\omega^1,\omega^2,\omega^3)$. Let $\mathcal{I} = \langle\omega^1\wedge\omega^2,\omega^1\wedge\omega^3\rangle$. Show that any $1$-dimensional regular integral element of $(M,\mathcal{I})$ does not admit an extension to a $2$-dimensional integral element.

E13. Compare the proof of the Cartan-Kähler theorem with the equation ${\rm d}\alpha = *\alpha$ with $\alpha\in \Omega^1(\mathbb{R}^3)$ that we discussed in class.

E14. Use a simple argument to show that, locally, $g$-diagonalizing coordinates always exist on a Riemannian surface. Why do not we in general expect $g$-diagonalizing coordinates to exist when $\dim(M)>3$?

E15. Apply Cartan's test to the EDS $(\mathbb{R}^4, {\rm d}x^1\wedge {\rm d}x^3+{\rm d}x^2\wedge {\rm d}x^4)$ and compare the generality result with what you know about integral surfaces that may be expressed as $x^3 = f(x^1,x^2)$, $x^4 = g(x^1,x^2)$.

E16. Let $\frak{g}$ be an $n$-dimensional (real) Lie algebra. Suppose that $U\subset \frak{g}$ is an open neighborhood of $0\in \frak{g}$ on which a $\frak{g}$-valued $1$-form $\eta$ is defined, satisfying $${\rm d}\eta = -\frac{1}{2}[\eta,\eta]$$ and $\eta|_0 = {\rm id}: T_0\frak{g}\rightarrow \frak{g}$. Prove: There exists an open neighborhood $V\subset U$ of $0$ and a real analytic map $\mu: V\times V\rightarrow U$ satisfying:

     (1) $\mu(0,v) = \mu(v,0) = v$ for all $v\in V$;
     (2) for any $v\in V$, there exists a $v^*$ such that $\mu(v, v^*) =\mu(v^*, v) = 0$;
     (3) $\eta$ is left-invariant under the left-multiplication $L_u(v):= \mu(u,v)$;
     (4) for any $u,v,w\in V$, $\mu(u,\mu(v,w)) = \mu(\mu(u,v),w)$, whenever both sides are defined;
     (5) $\mu(x,y) = x+y+\frac{1}{2}[x,y]+ {\rm h.o.t.}$

(Hint: Let $\pi_i: U\times U\rightarrow U$ $(i = 1,2)$ be the projections to the two components. Consider the EDS $(U\times U, \pi_1^*\eta - \pi_2^*\eta)$. Verify that this EDS is Frobenius, hence, a neighborhood $V\times V$ is foliated by $n$-dimensional integral manifolds, each submersing onto both copies of $V$. Now define $\mu(x,y)$ to be the unique element such that $(0,x)$ and $(y,\mu(x,y))$ belong to the same leaf.)

E17. Let $\kappa_1, \kappa_2$ stand for the principal curvatures on a Weingarten surface without umbilic points. Pick your favorite relation $F(\kappa_1,\kappa_2) = 0$. Set up an EDS with this prescribed relation and study the existence of integral surfaces.

E18. Prove that, if a surface $S\subset \mathbb{E}^3$ has equal principal curvatures $\kappa_1 = \kappa_2$, then both $\kappa_i$ are constants. What does this tell you about $S$?

E19. Suppose that $(M^{4n},g;I,J,K)$ defines a hyper-Kähler structure with the convention $IJ = -K$, etc. Define the $2$-forms $\omega_I,\omega_J,\omega_K$ by $\omega_I(u,v) = g(Iu,v)$, etc.
(1) Prove that $\Omega = \omega_J - i\omega_K$ satisfies $\Omega^{n+1} = 0$ and $\Omega^n\ne 0$;
(2) Deduce that locally there exist $I$-holomorphic coordinates $z^1,...,z^{2n}$ such that $$\Omega = {\rm d}z^1\wedge {\rm d}z^{n+1}+\cdots+ {\rm d}z^n\wedge {\rm d}z^{2n}$$ and that there exist a positive definite Hermitian matrix $U:=(u_{i\bar\jmath})$ such that $$\omega_I = \frac{1}{2}u_{i\bar\jmath}{\rm d}z^i\wedge{\rm d}\bar{z^{j}};$$ (3) Prove that $U$ satisfies $U^T Q U = Q$ where $$Q = \left(\begin{array}{cc} 0 & I_n \\ -I_n & 0\end{array}\right).$$ E20. Suppose that $I,J,K$ are three almost complex structures on a Riemannian manifold $(M,g)$ satisfying the quaternionic identities $I^2 = J^2 = K^2 = -1$ and $IJ = -K = -JI$, etc. Let $\omega_I(u,v):= g(Iu,v)$, etc., be the canonical $2$-forms. Prove that if $\omega_I, \omega_J,\omega_K$ are all closed, then $I,J,K$ are integrable (Newlander-Nirenberg) and hence $g$-parallel.

E21. Consider Exercises 5.12 and 5.13 in R1.

E22. Consider Exercise 6.9 and 6.11 in R1.

E23. Assuming the theorem of persistence of involutivity, prove that, if $(M^{n+s},\mathcal{I})$ is involutive with Cartan characters $(s_0,\ldots,s_n)$ satisfying $s_0+\cdots+s_n = s$, then $$\dim M^{(k)} = n+ s_0 + \binom{k+1}{1} s_1+\cdots+ \binom{k+n}{n} s_n.$$ E24. As part of our discussion of $g$-orthogonal coordinates on a Riemannian manifold of dimension $n\ge 4$, prove that the following two conditions are equivalent:
(1) for all $i,j,k,l$ distinct, $R_{ijkl}\equiv 0$ on the orthonormal coframe bundle;
(2) the Weyl curvature of $g$ varnishes identically.

E25. Do "Bonnet pairs" occur when the ambient space is $\mathbb{S}^3$ with the round metric instead of $\mathbb{E}^3$? Consider this problem locally, then survey the literature for existing results.

T1. (Date: 09/27/2019; Presenter: Peter Rock) Proofs of the theorems of Frobenius and Pfaff.
T2. (Date: 12/04/2019; Presenter: Andrew Stocker) Present the Newlander-Nirenberg theorem and a proof in the analytic case.
T3. Survey some recent progress in the study of Engel structures (global and singular aspects, compatibility with other geometric structures, etc.) and give a report.
T4. (Date: 11/18/2019; Presenter: Evan Wickenden) Present the Cartan-Janet theorem.
T5. (Date: 11/15/2019; Presenter: Braden Balentine) Present a proof of the theorem of Cauchy-Kowalewski.
T6. (Date: 11/22/2019; Presenter: Albany Thompson) Present Hans Lewy's example, which shows that the theorem of Cauchy-Kowalewski cannot be extended to the $C^\infty$ case.
T7. (Date: 12/09/2019; Presenter: Taylor Klotz) TBA