Differential Forms and Exterior Calculus

Let \(\omega =dx\wedge dy\) be the 2-form on \(\mathbb {R}^3\). Let \(h:\mathbb {R}^3\to \mathbb {R}\) be smooth so that 0 is a regular value. Let \(Z=h^{-1}(0)\). Assume that \(\frac {\partial h}{\partial z}\ne 0\) at all points of \(Z\). Show that the restriction of \(\omega \) to \(Z\) is a volume form on \(Z\).

Let \(\omega \) be a compactly supported \(n\)-form on \(\mathbb {R}^n\), \(n\ge 1\). Show that the following statements are equivalent:

1. \[ \int _{\mathbb {R}^n}\omega =0. \]

2. There exists a compactly supported \((n-1)\)-form \(\eta \) on \(\mathbb {R}^n\) such that \(\omega =d\eta \).

   You are allowed to use computations of de Rham cohomology of Euclidean spaces and spheres but not of the compactly supported cohomology of \(\mathbb {R}^n\).

Let \(\omega \) be a closed \(n\)-form on a smooth manifold \(M\) . Let \(N\) be a compact, orientable \(n\)-manifold without boundary and \(f:N\to M\) a smooth map with

Show that deRham cohomology group \(H^n_{dR}(M)\) is not trivial.

1. Construct an explicit \(n\)-form \(\omega \) on \(\mathbb {R}^n\) with compact support such that \(\omega \) is closed but there does not exists an \((n-1)\)-form \(\eta \) with compact support and \(d\eta =\omega \). You can use the existence of bump functions in your construction but you need to prove that your construction works.
2. Let \(M\) be a smooth, compact orientable \(n\)-manifold without bound- ary and let \((U,\phi )\) be a chart for \(M\). Show that there exists a closed \(n\)-form on \(M\) with support in \(U\) that is not exact.

In this problem you are allowed to use the fact that every closed 1-form on the 2-sphere \(S^2\) is exact and every 2-form \(\omega \) on \(S^2\) such that \(\int _{S^2}\omega =0\) is exact. You are also allowed to use that \(\int _{S^2} a^*(\omega )=-\int _{S^2}\omega \) for every 2-form \(\omega \) on \(S^2\), where \(a:S^2\to S^2\) is the antipodal map. Consider the map \(p:S^2\to \mathbb {R}P^2\) that identifies the antipodal points. Show that all closed 1- and 2-forms on \(\mathbb {R}P^2\) are exact by considering their pullbacks to \(S^{2}\) and applying the above facts.

Find an explicit closed 1-form \(\omega \) on \(\mathbb {R}^2\setminus \{0\}\) which is not exact, and prove both statements.

Let \(M\) be a closed, compact manifold and show that if \(\omega \) is a nowhere zero 1-form on \(M\) then the de Rham cohomology group \(H^1_{dR}(M)\) is non-trivial. Use this to show that if there is a submersion from \(M\) to the circle then \(H^1_{dR}(M)\) is non-trivial.

Let \(\omega \) and \(\eta \) be two differential forms on a manifold \(M\) .

1. Define the wedge product \(\omega \wedge \eta \).
2. Prove that if \(\omega \) and \(\eta \) are both closed then so is \(\omega \wedge \eta \).
3. Prove that the de Rham cohomology class of \(\omega \wedge \eta \) is unchanged if \(\omega \) and \(\eta \) are replaced by different representatives of the same cohomology classes. You are allowed to use the identity relating exterior differentiation and wedge product, but you should state it carefully.

Let \(f:\mathbb {R}^3\to \mathbb {R}^2\) be the function \(f(x,y,z)=(xy,z)\) and let \(\omega =dx\wedge dy\) be the area form on \(\mathbb {R}^2\). Compute the pull-back \(f^*(\omega )\).

Let \(\alpha \) be a closed two form on \(S^4\). Prove that \(\alpha \wedge \alpha \) vanishes at some point.

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3:x^2+y^2+z^2=1\}\) be the unit sphere. Let \[ \omega = \begin {cases} \dfrac {dy\wedge dz}{x}, & \text {if } x\ne 0;\\[1ex] \dfrac {dz\wedge dx}{y}, & \text {if } y\ne 0;\\[1ex] \dfrac {dx\wedge dy}{z}, & \text {if } z\ne 0. \end {cases} \]

1. Show that \(\omega \) is a well defined two form on \(S^2\)
2. Find \(\displaystyle \int _{S^2}\omega \).
3. Is \(\omega \) exact? Why?

Let \(M\) be a smooth compact manifold without boundary. Prove that if \(M\) has a smooth nowhere vanishing closed one form, then \(H^1_{\operatorname {dR}}(M)\ne \{0\}\). Deduce that if there is a smooth submersion of \(M\) onto \(S^1\), then \(H^1_{\operatorname {dR}}(M)\ne \{0\}\).

Let \(X\) be the wedge sum of two copies of \(\mathbb {R}\mathrm {P}^2\). Show that there is only one connected degree three cover of \(X\) and determine if it is regular or irregular.

Let \(M^m\) be a smooth compact orientable manifold and \(\omega \in \Omega ^m(M)\) a smooth, nonvanishing \(m\)-form that determines the orientation of \(M\). Show that \(\int _M\omega >0\).

Let \(X\) be the wedge product of a circle and a projective plane. Find all connected three sheeted covers of \(X\).

Let \(M\) and \(N\) be smooth manifolds, and let \(f:M\to N\) be smooth. Prove that the set of all \(p\in M\) for which the differential \(D_pf:T_pM\to T_{f(p)}N\) is injective is an open set.

Let \(M\) and \(N\) be smooth manifolds, and let \(f:M\to N\) be smooth. Prove that if \(\theta \) and \(\psi \) are closed differentiable \(n\)-forms on \(N\) (so that \([\theta ]\) and \([\psi ]\) are classes in de Rham cohomology) then \([\theta ]=[\psi ]\) implies \([f^*\theta ]=[f^*\psi ]\).

Let \(A:\mathbb {R}^2\to \mathbb {R}^3\) be given by \[ A= \begin {pmatrix} 3 & 2\\ 1 & 4\\ 0 & -1 \end {pmatrix} \] Find \(\lambda \) such that \(A^*(3dy_1\wedge dy_2-2dy_2\wedge dy_3+7dy_1\wedge dy_3) =\lambda dx_1\wedge dx_2\).

Suppose \(f:M\to N\) is a smooth function of smooth manifolds. Suppose \(Q\) is an embedded submanifold of \(N\), and that \(f\) is transverse to \(Q\). Prove that if \(p\in f^{-1}(Q)\), then \[ T_p(f^{-1}(Q))=(D_pf)^{-1}[T_{f(p)}Q]. \] (The above line says that the tangent space for \(f^{-1}(Q)\) at \(p\) is the inverse image of the tangent space of \(Q\) at \(f(p)\) under the differential of \(f\) at \(p\).)

Let \(\omega =xdy+dz\) be a 1-form on \(\mathbb {R}^3\). Is the 2-plane field \(\operatorname {Ker}(\omega )\) integrable?

1. State Stokes theorem.
2. If \(\omega \in \Omega ^n(\mathbb {R}^n)\) has compact support and \(\int _{\mathbb {R}^n}\omega \ne 0\) show that there does not exist an \(\alpha \in \Omega ^{n-1}(\mathbb {R}^n)\) with compact support and \(d\alpha =\omega \).
3. Now assume that \(n=1\) and that \(\int _{\mathbb {R}}\omega =0\). Find an \(\alpha \in \Omega ^0(\mathbb {R})\) with compact support and \(d\alpha =\omega \).

Let \(\omega \) and \(\eta \) be closed forms on a manifold \(M\). Show that the de Rham cohomology class of \(\omega \wedge \eta \) only depends on the cohomology classes of \(\omega \) and \(\eta \).

Define \(r:\mathbb {R}^2\backslash \{0\}\to \mathbb {R}^2\) by \[ r(x,y)=\left (\frac {x}{\sqrt {x^2+y^2}},\frac {y}{\sqrt {x^2+y^2}}\right ). \] Calculate \(r^*(-y\,dx+x\,dy)\) and \(r^*(x\,dx+y\,dy)\). Show that for any two form \(\omega \) on \(\mathbb {R}^2\), \(r^*\omega =0\) and conclude that for any 1-form \(\alpha \) on \(\mathbb {R}^2\), \(r^*\alpha \) is closed (even if \(\alpha \) is not).

Define \(\phi :\mathbb {R}^2\to \mathbb {R}^2\backslash \{0\}\) by \(\phi (x,y)=(e^x\cos y,e^x\sin y)\). Let \(\omega \) be a 1-form on \(\mathbb {R}^2\backslash \{0\}\) such that \(dy=\phi ^*\omega \). Show that \(\omega \) is closed but not exact on \(\mathbb {R}^2\backslash \{0\}\).

Define \(\omega :\mathbb {R}^n\times \mathbb {R}^n\to \mathbb {R}\) by \(\omega (x,y)=x_1y_1+\cdots +x_{n-1}y_{n-1}-x_ny_n\) and define \(f:\mathbb {R}^n\to \mathbb {R}\) by \(f(x)=\omega (x,x)\). Show that \(M=f^{-1}(-1)\) is a smooth manifold. Viewing \(\omega \) as a symmetric 2-tensor on \(\mathbb {R}^n\) show that the restriction of \(\omega \) to \(M\) is positive definite. (Hint: First assume \(y\in T_xM\) and that \(x_n+y_n=0\) and show that \(\omega (x+y,x+y)=-1+\omega (y,y)\ge 0\).)

Let \(S^1=\{(x,y):x^2+y^2=1\}\) and define \(f:\mathbb {R}\to S^1\) by \(f(t)=(\cos 2\pi t,\sin 2\pi t)\). Show there exists a unique 1-form \(\omega \) on \(S^1\) such that \(f^*\omega =dt\). Show that \(\omega \) is closed but not exact.

For \((x,y)\in \mathbb {R}^2-\{(0,0)\}\), let \[ \omega (x,y)=\frac {-y}{x^2+y^2}\,dx+\frac {x}{x^2+y^2}\,dy. \] Find \(\int _{C(r)}\omega \) where \(C(r)\) is the circle of vectors in \(\mathbb {R}^2\) of norm equal to \(r>0\) oriented counterclockwise. Find a smooth function \(f:\mathbb {R}^2-\{(0,0)\}\to \mathbb {R}\) such that \(\omega =df\) or explain why there is no such \(f\).

\(13x\,dx+y^2\,dy+xyz\,dz\) is a form on \(\mathbb {R}^3\). What’s its exterior derivative?

Let \(f:\mathbb {R}^3\to \mathbb {R}^2\) be the function \(f(x,y,z)=(xy,z)\). Find the pullback form \(f^*(dy\wedge dz+x^2\,dx\wedge dy)\).

Let \(S^2\) be the vectors of length 1 in \(\mathbb {R}^3\). Find \(\int _{S^2}dx\wedge dy\).

Suppose that \(a\) is a smooth, real valued, compactly supported function on \(\mathbb {R}^n\) so that \(\theta =a\,dx_1\wedge \cdots \wedge dx_n\) is an \(n\)-form on \(\mathbb {R}^n\). What is the definition of \(\int _{\mathbb {R}^n}\theta \)? What’s the definition of \(\int _M\omega \) where \(\omega \) is a compactly supported \(k\)-form on an oriented \(k\)-dimensional smooth manifold \(M\)? Show that this definition is well-defined.

Find \(a,b\in C^\infty (\mathbb {R}^3)\) such that \[ d(xyz\,dx\wedge dy+3x^3\,dy\wedge dz)=a\,dx\wedge dy\wedge dz \] and \[ dx\wedge (14x\,dz)\wedge dy=b\,dx\wedge dy\wedge dz. \]

Let \(f:T^2\to \mathbb {R}^3\) be an embedding. Find \(\displaystyle \int _{T^2} dx\wedge dy\).

1. State Stokes’ Theorem.
2. Given two 3-forms on \(S^7\), denoted \(\theta \) and \(\omega \), prove that \[ \int _{S^7}d\theta \wedge \omega =\int _{S^7}\theta \wedge d\omega . \]

Let \(S^n=\{v\in \mathbb {R}^{n+1}\mid \lVert v\rVert =1\}\) and let \(f:S^n\to S^n\) be the restriction of the map \(\mathbb {R}^{n+1}\to \mathbb {R}^{n+1}\) given by \(v\mapsto -v\). Suppose \(\omega \) is an \(n\)-form on \(S^n\) with \(\int _{S^n}\omega =\frac {\pi ^2}{6}\). What is \(\int _{S^n}f^*\omega \). Why?

Let \(M=T^2\times S^2\), and suppose \(\omega \) is a 4-form on \(M\) such that \(\int _M\omega =7\).

1. Define a smooth map \(f:M\to M\) that has degree 2.
2. Define a 4-form \(\theta \) on \(M\) such that \(\int _M\theta =56\), using both \(f\) and \(\omega \) nontrivially.

Let \(\Lambda ^k(M)\) denote the space of smooth \(k\)-forms on a manifold \(M\). Suppose that \(\omega \in \Lambda ^p(M)\) and \(\eta \in \Lambda ^q(M)\) are closed forms. Show that the de Rham cohomology class of \(\omega \wedge \eta \) depends only on the de Rham cohomology classes of \(\omega \) and \(\eta \).

Let \(M\) and \(N\) be compact orientable \(n\)-dimensional manifolds and \(f:M\to N\) a smooth map. Suppose that \[ \int _M f^*\Omega \ne 0 \] for some \(n\)-form \(\Omega \). Prove that \(f\) is surjective.

Let \(M\) be a compact orientable manifold with boundary. Use differential forms to show that there is no smooth retraction \(r:M\to \partial M\).

Let \(\omega \) be an \(n\)-form on \(\mathbb {R}^n\) with compact support. Show that there exists a compactly supported \((n-1)\) form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to use the standard results about deRham cohomology of \(\mathbb {R}^n\) and spheres.

Compute \(\pi _2(X)\) where \(X\) is the wedge sum of \(S^1\) and \(S^2\).

Give an explicit example of a closed 1-form on \(\mathbb {R}^2-\{0\}\) which is not exact, and prove both properties.

Verify the formula \[ d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]) \] for any 1-form \(\omega \) on \(\mathbb {R}^n\) and any two vector fields \(X,Y\) on \(\mathbb {R}^n\). This is using Spivak’s normalization conventions \[ dx\wedge dy\left (\frac {\partial }{\partial x},\frac {\partial }{\partial y}\right )=1; \] using Guillemin-Pollack’s one should multiply the left-hand side by 2.

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Prove that the following two statements are equivalent.

1. There exists a compactly supported smooth \((n-1)\)-form \(\eta \) such that \(\omega =d\eta \).
2. \(\displaystyle \int _{\mathbb {R}^n}\omega =0\).

Note: You are allowed to use the fact that \(H^n(S^n)=\mathbb {R}\), but you are not allowed to quote the fact that \(H_c^n(\mathbb {R}^n)=\mathbb {R}\).

1. Let \(a:S^n\to S^n\) be the antipodal map. Suppose that \(\omega \) is a smooth form on \(S^n\) such that \(a^*\omega =\omega \). Prove that if \(\omega \) is exact, then there is a smooth form \(\eta \) with \(\omega =d\eta \) and \(a^*\eta =\eta \).
2. Use (a) and the fact that \(H^k_{\mathrm {DR}}(S^n)=0\) for \(0<k<n\) to deduce that \(H^k_{\mathrm {DR}}(\mathbb {R}P^n)=0\) for \(0<k<n\).

Let \(f:\mathbb {R}^2\to \mathbb {R}^3\) be the function given by \(f(x,y)=(x^2-y,x+y^2,\sin (x))\). Let \(\omega \) be the 2-form on \(\mathbb {R}^3\) given by \(\omega (u,v,w)=u\,dv\wedge dw\). Compute the pull-back \(f^*\omega \).

Let \[ \omega =(x^2+y^2+z^2)^\alpha (x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy) \] be a 2-form in \(\mathbb {R}^3\setminus \{0\}\).

1. Find \(\alpha \in \mathbb {R}\) such that \(\omega \) is closed.
2. Show that \(\omega \) is not exact.

Let \(X\) and \(Y\) be two closed oriented manifolds. Prove that Euler characteristic is multiplicative, i.e. that \[ \chi (X\times Y)=\chi (X)\times \chi (Y). \] You may not appeal to the Kunneth formula.

Find a non-regular cover of \(S^1\vee S^1\), the wedge of 2 circles. You should both explicitly describe the cover and the corresponding subgroup of \(\pi _1(S^1\vee S^1)\).

Let \(\omega \) be the 1-form on \(\mathbb {R}^2-\{0\}\) given by \[ \omega (x,y)=\frac {-y\,dx}{x^2+y^2}+\frac {x\,dy}{x^2+y^2}. \]

1. Calculate \(\int _C\omega \) for any circle \(C\) centered at the origin.
2. Prove that in the half-plane \(x>0\) there is a smooth function \(f\) such that \(\omega =df\).
3. Prove that there is no smooth function \(f\) defined on \(\mathbb {R}^2-\{0\}\) such that \(\omega =df\).

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Show that there exists a compactly supported smooth \((n-1)\)-form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to quote facts about \(H^k(X)\) but not about \(H_c^k(X)\).

Let \(f:\mathbb {R}^2\to \mathbb {R}^2\) be given by \[ f(x,y)=(x^2+y,x-y^2). \] Explicitly compute \(f^*dx\), \(f^*dy\), \(f^*(dx\wedge dy)\).

Consider the 1-form on \(\mathbb {R}^2\setminus \{0\}\) \[ \omega =\frac {x\,dy-y\,dx}{x^2+y^2}. \]

1. Prove that \(\omega \) is closed, i.e. \(d\omega =0\).
2. Prove that \(\omega \) is not exact, i.e. there is no smooth function \(f:\mathbb {R}^2\setminus \{0\}\to \mathbb {R}\) such that \(\omega =df\).

Compute the pull-back of the differential form \[ \omega =x\,dx\wedge dy+z\,dy\wedge dz \] in \(\mathbb {R}^3\) under the map \(f:\mathbb {R}^2\to \mathbb {R}^3\) given by \[ f(u,v)=(u^2,u+v,\sin (u+v)). \]

Compute the pullback \(f^*\omega \) of the differential form \[ \omega =dx\wedge dy \] by the map \(f:\mathbb {R}^3\to \mathbb {R}^3\), \(f(x,y,z)=(xyz,y+x,z)\).

Let \(\omega \) be a smooth closed \(p\)-form on \(S^n\) with \(p>0\). Show that there is a smooth \((p-1)\)-form \(\alpha \) so that \(d\alpha =\omega \) if \(p<n\) but that this need not be the case when \(p=n\). You are allowed to quote standard calculations in deRham cohomology.

Prove that \(\mathbb {C}P^2\) and the wedge sum (one point union) \(S^2\vee S^4\) are not homotopy equivalent.

For \((x,y,z)\in \mathbb {R}^3\) let \(\eta =dx\otimes dx+dy\otimes dy-dz\otimes dz\) denote a symmetric 2-form on \(\mathbb {R}^3\). Let \[ M=\{(x,y,z)\in \mathbb {R}^3:1+x^2+y^2=z^2,\ z>0\} \] be one nappe of the hyperboloid.

1. Compute \[ \int _\Omega dx\wedge dy+(4-2z)dx\wedge dz+(z-2)dy\wedge dz \] where \(\Omega \subset M\) is the region inside \(x^2+y^2\le 3\).
2. Show that the restriction \(\eta |_M\) is a Riemannian metric for \(M\).
3. What is the length of the curve \(t\mapsto (\sinh t,0,\cosh t)\) for \(a\le t\le b\) in this metric?

True or false: Any closed differential form of degree \(\ge 1\) on \(\mathbb {R}P^2\times \mathbb {R}P^2\) is exact. Justify your answer.

Let \(F:\mathbb {R}^2\to \mathbb {R}^3\) be a mapping given by the formula \[ F(u,v)=(e^u\cos (v),e^u\sin (v),0). \] Compute the pull-back \(F^*(\omega )\) of the differential form \[ \omega =xy\,dx\wedge dy+x\,dy\wedge dz\in \Omega ^*(\mathbb {R}^3). \]

Let \((M^n,g)\) be a Riemannian manifold. Show that the form given locally by \[ \omega =\sqrt {g}\,dx^1\wedge dx^2\wedge \cdots \wedge dx^n \] represents a globally-defined, nowhere vanishing \(n\)-form. (Here \(\sqrt {g}\) is the standard shorthand for the square root of the determinant of the matrix \([g_{ij}]=[\langle \frac {\partial }{\partial x^i},\frac {\partial }{\partial x^j}\rangle ]\) which gives the metric \(g\) in local coordinates.)

Let \(M\) be a differentiable manifold, \(\omega \) a closed \(p\)-form satisfying \[ \int _c \omega =0 \] for all smooth \(p\)-cycles \(c\). Prove \(\omega \) is exact.

Suppose a smooth oriented manifold \(M^n\) has a Riemannian metric which is given in any local coordinate system by \[ ds^2=\sum _{i,j=1}^{n}g_{ij}(x)\,dx^i\otimes dx^j. \] Then the form given locally in the same coordinate systems by \[ \omega =\det (g_{ij}(x))\,dx^1\cdots dx^n \] represents a globally defined nowhere vanishing \(n\)-form on \(M\).

Let \(f\in C^\infty (S^n)\) be a smooth function and let \(x_1,x_2\in S^n\) be two points such that \(f(x_1)<0<f(x_2)\). Let \(\omega \) be a smooth nonvanishing \(n\)-form. Then there is a diffeomorphism \(\varphi :S^n\to S^n\) such that \[ \int _{S^n}(\varphi ^*f)\,\omega =0. \]

II. Ten point questions.