Left-Invariant Vector Fields

Let \(V,W:\mathbb {R}^4\to \mathbb {R}^4\) be vector fields on \(\mathbb {R}^4\) defined by

and

Is there a nonempty surface \(S\subset \mathbb {R}^4\) such that for every \(p\in S\) both \(V(p)\) and \(W(p)\) are tangent to \(S\)? Find such a surface or show it does not exist.

Let \(V(x,y,z)=\frac {\partial }{\partial x}+y\frac {\partial }{\partial z}\) and \(W(x,y,z)=\frac {\partial }{\partial y}-x\frac {\partial }{\partial z}\) be vector fields on \(\mathbb {R}^3\). Show that there does not exist a surface \(S\subset \mathbb {R}^3\) such that both \(V\) and \(W\) are tangent to \(S\).

Let \(V\) be a smooth vector field on \(\mathbb {R}^2\) and assume that outside of a

compact set \(V=\frac {\partial }{\partial x}\) where \(\frac {\partial }{\partial x}\) is the standard horizontal vector field. Show that the flow for \(V\) exists for all time.

Let \(\mathcal {D}\) be the 2-dimensional distribution in \(\mathbb {R}^3\) spanned by the vector

fields \[ V=\frac {\partial }{\partial x} \quad \text {and}\quad W=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Let \(M\) be a smooth 2-manifold and \(\phi :M\to \mathbb {R}^3\) a smooth map such that \(\phi _*(p)(T_pM)\subset \mathcal {D}_{\phi (p)}\) for all \(p\in M\). Show that the derivative \(\phi _*(p):T_pM\to T_{\phi (p)}\mathbb {R}^3\) is not injective for any \(p\in M\).

Let \(X\) and \(Y\) be smooth vector fields on smooth manifolds \(M\) and \(N\) that are \(f\)-related by a smooth map \(f:M\to N\). If \(\psi _t\) is the flow generated by \(X\) and \(\phi _t\) is the flow generated by \(Y\) show that

Let \(G_0\) and \(G_1\) be Lie groups. Then the product \(G_0\times G_1\) is a Lie group and we let \(p_i:G_0\times G_1\to G_i\) be the projection to each coordinate. Let \(X\) be a vector field on \(G_0\times G_1\) and \(Y_0\) and \(Y_1\) vector fields on \(G_0\) and \(G_1\) such that \(X\) and the \(Y_i\) are \(p_i\)-related. Show that \(X\) if left-invariant if and only if both \(Y_{0}\) and \(Y_{1}\) are left-invariant.

Let \(M\) be a smooth manifold and \(V\) a vector field on \(M\) such that flow for \(V\) is defined for all time. Let \(W\) be another vector field on \(M\) such that \(V\) and \(W\) agree outside of a compact set in \(M\) . Show that \(W\) is defined for all time.

Let \(\phi : G \to H\) be a Lie group homomorphism between Lie groups \(G\) and \(H\). Let \(X\) and \(Y\) be \(\phi \)-related vectors fields on \(G\) and \(H\), respectively, and assume that \(Y\) is left invariant. Is \(X\) left invariant? Either prove this or provide a counterexample.

Define two vector fields \(V\) and \(W\) on \(\mathbb {R}^3\) as follows: at \(p=(a,b,c)\) let

Prove that there is no nonempty surface \(S\subset \mathbb {R}^3\) such that both \(V\) and \(W\) are tangent to \(S\) at each of its points.

Let \(M\) be a smooth manifold and \(\gamma :S^1\to M\) be a smooth embedding. Show that there exists a smooth vector field \(X\) on \(M\) that does not vanish on \(\gamma (S^1)\) and which is tangent to \(\gamma (S^1)\) at each point. (The last condition means that for each point \(p=\gamma (t)\), the vector \(X_p\) is a multiple of \(\gamma '(t)\).)

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3:x^2+y^2+z^2=1\}\) be the two sphere in three space. Consider the vector fields \[ V=y\frac {\partial }{\partial z}-z\frac {\partial }{\partial y}, \qquad W=z\frac {\partial }{\partial x}-x\frac {\partial }{\partial z}. \]

1. Show \(V\) and \(W\) are tangent to \(S^2\), and hence by restriction define vector fields on \(S^2\).
2. Find the flow \(\Theta _t^V(x,y,z)\) on \(S^2\) generated by \(V\).
3. Compute \((\Theta _h^V)_*W\) on \(S^2\).
4. Compute the Lie Derivative \(\mathcal {L}_VW\) on \(S^2\) in two ways.

Let \(V\) be a continuous vector field on the unit ball \(B^n\subset \mathbb {R}^n\) which is nowhere zero. Prove that there are points \(x,y\in S^{n-1}=\partial B^n\) and positive numbers \(a,b>0\) such that \(V(x)=ax\) and \(V(y)=-by\) (so the vector field points outward at \(x\) and inward at \(y\)).

Consider the Lie Group \(G=\Ogrp (n)=\{A\in \mathcal {M}_n:A^TA=I\}\) where \(\mathcal {M}_n\cong \mathbb {R}^{n^2}\) is the collection of real \(n\times n\) matrices and \(A^T\) is the transpose of \(A\).

1. Describe the set of matrices which form the tangent space \(T_PG\) at the point \(P\in G\).
2. Given \(V\in T_IG\) and \(P\in G\), find \(X_V(P)\in T_PG\) where \(X_V\) is the left invariant vector field on \(G\) such that \(X_V(I)=V\). Show that your \(X_V\) is left invariant.
3. Let \(\theta _t^X\) be the flow for the left invariant vector field \(X=X_V\) for some \(V\in T_IG\). Find and verify a formula for \(\theta _t^X(P)\) in terms of \(P\), \(t\) and \(V\). Compute \(\theta _t^X(I)\) in case \(G=O(2)\) and \[ V=\begin {pmatrix}0&1\\-1&0\end {pmatrix}. \]

Suppose the three sphere \(S^3=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2+z^2+w^2=1\}\) is embedded into four space the standard way. Two independent vector fields are given by \(U=(y,-x,w,-z)\) and \(V=(w,z,-y,-x)\). Determine whether it is possible to find a smooth embedded two dimensional submanifold \(N\subset S^3\) such that the vector fields \(U\) and \(V\) are tangent to \(N\) at every point of \(N\), i.e., \(U(p),V(p)\in T_pN\) for every \(p\in N\)?

Let \(\Sigma \) be a closed, smooth surface. Show that if \(\Sigma \) admits a nonvanishing vector field, then the Euler characteristic of \(\Sigma \) equals 0.

Let \(M\) be a compact manifold with a vector field \(X\in \Gamma (TM)\) and resulting one-parameter group \(\{\theta _t^X\}_{t\in \mathbb {R}}\le \operatorname {Diff}(M)\). For any \(f\in \operatorname {Diff}(M)\), let \(f_*X\in \Gamma (TM)\) be the pushforward of \(X\). Show that the one-parameter group for \(f_*X\) is \(\{f\circ \theta _t^X\circ f^{-1}\}_{t\in \mathbb {R}}\).

Let \(B\) be the open ball in \(\mathbb {R}^n\) that is centered at \(0\) and has radius \(1\). If \(y\in B\) and \(\lVert y\rVert <1/2\), define a vector field on \(B\) and use the resulting flow to find a diffeomorphism \(h:B\to B\) with \(h(0)=y\) such that \(h\) is homotopic to the identity map on \(B\). Conclude that if \(N\) is a smooth connected manifold and \(p,q\in N\), then there is a diffeomorphism \(H:N\to N\) such that \(H(p)=q\) and such that \(H\) is homotopic to the identity map on \(N\).

Show that \(S^n\) has a continuous field of nonzero tangent vectors iff \(n>0\) is odd. Conclude that the even-dimensional spheres are not Lie groups.

Let \(U\subset \mathbb {R}^n\) be a connected, open subset. For all \(p,q\in U\) show that there exists a diffeomorphism \(\phi :\mathbb {R}^n\to \mathbb {R}^n\) with \(\phi (p)=q\) and \(\phi \) the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

Let \(V\) be a smooth vector field on \(\mathbb {R}^2\) and assume that outside of a compact set \(V=\frac {\partial }{\partial x}\). Show that the flow for \(V\) is defined for all time.

Let \(f:M\to N\) be a differentiable map and \(V\) and \(W\) vector fields on \(M\) and \(N\), respectively, with \(f_*(x)V(x)=W(f(x))\) for all \(x\in M\). Assume that \(\phi _t\) and \(\psi _t\) are the respective flows of \(V\) and \(W\) and show that \(f\circ \phi _t=\psi _t\circ f\).

Let \(U\subset \mathbb {R}^n\) be a connected open set with \(p,q\in U\). Show that there exists a diffeomorphism \(\phi \) of \(\mathbb {R}^n\) such that \(\phi (p)=q\) and \(\phi \) is the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

Let \(W\) be a vector field on a smooth manifold \(M\) and assume that \(V\) has a flow that is defined on all of \(M\) and for all time. Let \(V\) be another vector field on \(M\) such that \(V-W\) has compact support. Show that \(V\) has a flow on all of \(M\) defined for all time.

Let \(\Delta \) be a plane field on a manifold \(M\) and let \(X,Y\) be two vector fields tangent to \(\Delta \). Show that if for some \(p\in M\) we have \(Y_p=0\) then \([X,Y]_p\in \Delta \).

Let \(A\) be an element of the Lie algebra of \(\mathrm {GL}_n(\mathbb {R})\). What is the left-invariant vector field on \(\mathrm {GL}_n(\mathbb {R})\) that is associated to \(A\)? Show that it is left-invariant.

Let \(h:\mathbb {R}^3\to \mathbb {R}\) be the function \(h(x,y,z)=xyz\). Let \(s\) be the vector field on \(\mathbb {R}^3\) given by \[ s(x,y,z)=xy\frac {\partial }{\partial x} +(y-z^3)\frac {\partial }{\partial y} +3\frac {\partial }{\partial z}. \] Find \(L_s(f)(1,1,2)\), the Lie derivative of \(f\) in the direction of \(s\) at the point \((1,1,2)\).

Find the following bracket of two vector fields in \(\mathbb {R}^3\) \[ \left [ (x+y)\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, yz\frac {\partial }{\partial y}+x^2\frac {\partial }{\partial z} \right ]. \]

Let \(M\) be a smooth compact manifold and let \(X\) be a smooth vector field with a corresponding 1-parameter flow group \(\{\theta _t^X:M\to M\}_{t\in \mathbb {R}}\). Let \(f\) be a diffeomorphism of \(M\) and let \(f_*X\) be its pushforward. Show that the flow group for \(f_*X\) is \(f\circ \theta _t^X\circ f^{-1}\).

Give a nowhere vanishing vector field on \(\mathrm {SL}_n(\mathbb {R})\).

Suppose \(M\) is a smooth manifold with a smooth plane field. If the plane field is integrable, then prove that the bracket of any pair of smooth vector fields on \(M\) that are tangent to the plane field is another smooth vector field on \(M\) that is tangent to the plane field.

Let \(X\) be a smooth vector field on \(S^2\). Prove that \(X(p)=0\) for some \(p\in S^2\).

Find the Lie bracket \[ \left [2\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, x\frac {\partial }{\partial x}+y^2\frac {\partial }{\partial z}\right ]. \] Is the plane field \[ \Delta _{(x,y,z)} =\operatorname {Span}\left \{2\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, x\frac {\partial }{\partial x}+y^2\frac {\partial }{\partial z}\right \} \] integrable?

Draw a nowhere vanishing vector field on \(S^1\). Explain why there is a nowhere vanishing vector field on \(S^3\).

1. Let \(X\) be a vector field on a smooth manifold \(M\) and let \(f:M\to N\) be a diffeomorphism. Define \(f_*(X)\) – the pushforward of \(X\) on \(N\).
2. Let \(f:\mathbb {R}^3\to \mathbb {R}\times (-\frac {\pi }{2},\frac {\pi }{2})\times \mathbb {R}\) be defined by \(f(x,y,z)=(e^x,\tan ^{-1}(y),-2z)\). Calculate the bracket \[ \left [ f_*\left (\frac {\partial }{\partial x}+\frac {\partial }{\partial z}\right ), f_*\left (\frac {\partial }{\partial y}-\frac {\partial }{\partial z}\right ) \right ]. \]

Let \(V\) and \(W\) be vector fields on \(\mathbb {R}^3\) given by \[ V(x,y,z)=\frac {\partial }{\partial x} -y\frac {\partial }{\partial z} \quad \text {and}\quad W(x,y,z)=\frac {\partial }{\partial y} +x^2\frac {\partial }{\partial z}. \]

1. Find \([V,W](x,y,z)\).
2. What is the definition of an integrable, \(k\)-dimensional plane field on a manifold \(M\)?
3. Prove or disprove: \(\Delta (x,y,z)=\operatorname {Span}\{V(x,y,z),W(x,y,z)\}\) is an integrable plane field.
4. Give an example of three foliations on a two dimensional torus: one with every leaf compact; one with every leaf noncompact; and one with some compact and some noncompact leaves.

Let \(U\) be the open set in \(\mathbb {R}^3\), \(U=\{(x,y,z):x>0,\ y>0,\ z>0\}\) and let \(D\) be the distribution in \(U\) spanned by the vector fields: \[ V_1=y\frac {\partial }{\partial z}-z\frac {\partial }{\partial y} \qquad V_2=z\frac {\partial }{\partial x}-x\frac {\partial }{\partial z} \] Show that \(D\) is involutive. What does the Frobenius Theorem imply about \(D\)?

Show that the group of diffeomorphisms of a compact manifold \(M\) acts transitively on \(M\). That is, if \(p,q\in M\) there exists a diffeomorphism \(\phi :M\to M\) such that \(\phi (p)=q\). Hint: First assume \(p\) and \(q\) are in the same coordinate neighborhood \(U\) and construct \(\phi \) as the flow of a vector field supported in \(U\).

Let \(T=S^1\times S^1\subset \mathbb {C}\times \mathbb {C}\) be the torus. Let \(X\) be the vector field on \(T\) defined by \(X(z_1,z_2)=(iz_1,aiz_2)\), where \(a\in \mathbb {R}\). Compute the flow \(\Phi (t,p)\), \(p=(p_1,p_2)\) of \(X\). For \(a\) rational, show that \(t\to \Phi (t,p)\) is periodic.

Consider the vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad \text {and}\qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

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.

1. State the Frobenius integrability theorem.
2. Show that the plane field in \(\mathbb {R}^3\) spanned by the vector fields \[ X=\frac {\partial }{\partial x}+z\frac {\partial }{\partial y} \qquad \text {and}\qquad Y=\frac {\partial }{\partial z} \] is not integrable on any nonempty open subset of \(\mathbb {R}^3\).

Let \(V,W:\mathbb {R}^4\to \mathbb {R}^4\) be vector fields on \(\mathbb {R}^4\) defined by \[ V(x,y,z,w)=(y,-x,w,-z) \] and \[ W(x,y,z,w)=(w,z,-y,-x). \] Is there a nonempty surface \(\Sigma \subset \mathbb {R}^4\) such that for every \(p\in \Sigma \), we have \(V(p),W(p)\in T_p\Sigma \)? Find such a surface or prove that it does not exist.

Let \(M\) be a manifold and \(\Delta \) a smooth \(k\)-plane field on \(M\). Suppose that \(X\) and \(Y\) are smooth vector fields on \(M\) with values in \(\Delta \). Show that if for some \(p\in M\) we have \(X_p=0\) then \([X,Y]_p\in \Delta \).

Let \[ X=y\frac {\partial }{\partial x}-(x+1)\frac {\partial }{\partial y} \] and \[ Y=y\frac {\partial }{\partial x}-(x-1)\frac {\partial }{\partial y} \] be two vector fields in \(\mathbb {R}^2\). Compute \([X,Y]\).

Let \(V\) be the vector field in the complex plane \(\mathbb {C}\) given by \[ V(z)=z^m \] for some \(m=1,2,\cdots \), where we use the standard identification \(T_z\mathbb {C}=\mathbb {C}\). Compute the index of \(V\) at the origin.

Consider vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

Consider the following vector fields in \(\mathbb {R}^3\): \[ X=\frac {\partial }{\partial x},\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Here \(x,y,z\) are the coordinates on \(\mathbb {R}^3\) and \(\frac {\partial }{\partial x}\), \(\frac {\partial }{\partial y}\), \(\frac {\partial }{\partial z}\), are the three standard coordinate vector fields. Show that there is no (nonempty) surface \(M\subset \mathbb {R}^3\) which is tangent to both vector fields at each of its points.

Define the index of a vector field on a smooth manifold. Construct an explicit vector field on the 2-sphere \(S^2\) and compute the Euler characteristic of \(S^2\) by computing the index of this vector field.

Define the index of a singularity of a vector field and state the Poincaré-Hopf index theorem.

Consider the vector fields \[ X=\frac {\partial }{\partial x}\qquad \text {and}\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z} \] on \(\mathbb {R}^3\). Prove or disprove: There is a nonempty surface \(M^2\subset \mathbb {R}^3\) such that for every \(p\in M\) the tangent space \(T_pM\) is spanned by \(X_p\) and \(Y_p\).

Prove that the \(n\)-sphere admits a nowhere vanishing vector field if and only if \(n\) is odd.

Consider the two vector fields in \(\mathbb {R}^3\) given by \(V(x,y,z)=(x^2+y^2,0,-y)\) and \(W(x,y,z)=(0,x^2+y^2,x)\). Find a two dimensional surface \(M^2\subset \mathbb {R}^3\) containing the point \(p=(1,0,0)\) which is tangent to the vector fields \(V\) and \(W\) in a neighborhood of \(p\), or show that such a surface \(M\) doesn’t exist.

Let \(M^n\) be a smooth oriented manifold. Suppose \(f:M\to \mathbb {R}\) is a smooth function such that \(df\ne 0\) at each point of \(M\). Show that there is a smooth vector field \(V\) on \(M\) so that \(df(V)\le -1\). Show also, that if \(\phi _t\) is the flow generated by \(V\), then \(f(\phi _1(x))\le f(x)-1\) for all \(x\in M\) such that \(\phi _1(x)\) exists.

Suppose the three sphere \[ S^3=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2+z^2+w^2=1\} \] is embedded into four space the standard way. Two independent vector fields on \(S^3\) are given by \(V=(y,-x,w,-z)\), \(W=(w,z,-y,-x)\). Determine whether it is possible to embed a closed smooth two dimensional surface \(N\subset S^3\) so that the vector fields are tangent to \(N\) at every point of \(N\), i.e. \(V(p),W(p)\in T_pN\) for every \(p\in N\).

1. Show that \(\Phi :\mathbb {R}^2\to \mathbb {R}^2\) given by \[ (p,q)=\Phi (x,y)=(x+y^2+2y,x+y^2-2y) \] is a diffeomorphism. Compute \(d\Phi (\partial /\partial y)\) and the pullback \(\Phi ^*(p^2\,dp\wedge dq)\).
2. Let \(X\) be a smooth vector field and \(\omega \) a smooth \(n\)-form on the smooth compact boundaryless manifold \(M^n\). Let \(M\times \mathbb {R}\ni (x,t)\mapsto \phi _t(x)\in M\) be the flow generated by \(X\). Let \[ A(t)=\int _M\phi _t^*\omega \] be the integral of the pulled back \(n\)-form. Find \(dA/dt\).

Let \(S^n\) denote the unit sphere of \(\mathbb {R}^{n+1}\) and let \(a:S^n\to S^n\) denote the antipodal map \(a(x)=-x\). Show that \(S^n\) admits a nonvanishing tangent vector field if and only if \(a\) is homotopic to the identity. For which values of \(n\) does this happen?

Let \(M\subset \mathbb {R}^{n+1}\) be the graph \(x^{n+1}=f(x^1,\ldots ,x^n)\) where \(f\in C^\infty (\Omega )\) and \(\Omega \subset \mathbb {R}^n\) is a convex open set. Suppose the Hessian of \(f\) has constant rank i.e. for some \(k\) such that \(0\le k<n\) and every \(x\in \Omega \), \[ \operatorname {rank}\left (\frac {\partial ^2 f(x)}{\partial x^i\partial x^j}\right )=k. \] Show that \(M\) is foliated by \((n-k)\) dimensional manifolds whose tangent vectors lie in \(\ker (dG)\) where \(G:M\to \mathbb {R}^n\) is given by \[ G(x)=\left (\frac {\partial f}{\partial x^1},\ldots , \frac {\partial f}{\partial x^n}\right ). \] Show that a connected component \(L\) of such a “leaf” manifold locally coincides with points where \(G\) is constant. In fact, the graph of \(L\) is an affine subspace of \(\mathbb {R}^{n+1}\) restricted to \(\Omega \times \mathbb {R}\).