These exercises are meant to review your knowledge of smooth manifolds. You should be able to do them, but you don't need to turn them in.

1. Suppose that \(M\) is a smooth manifold and let \(p\in M\).
   • Show that the definitions of \(T_p M\) in terms of curves and in terms of derivations are equivalent.
   • Give two proofs of the chain rule for maps between smooth manifolds, one using each definition.

If \(F: \R^m \to \R^n\) is a smooth function, we say that \(x\in \R^n\) is a regular value of \(F\) if \(\rank DF_y = n\) for every \(y\in F^{-1}(x)\). In class, we showed that this implies that \(F^{-1}(x)\) is a smooth submanifold of \(\R^m\).

2. Let \(M\subset \R^n\) be a \(d\)–submanifold and let \(p\in M\).
   • Show that there is a neighborhood \(U\) containing \(p\) and a diffeomorphism \(F\) from \(U\) to an open subset of \(\R^n\) such that \(F\) sends \(U\) to the unit ball in \(\R^n\) and sends \(M\cap U\) to the intersection of the ball with a \(d\)–plane.
   • Conclude that there is a smooth function \(G: \R^n \to \R^{n-d}\) such that \(0\) is a regular value of \(G\) and \(G^{-1}(0) \cap U = M \cap U\). That is, \(M\) can locally be written as the preimage of a regular value of a smooth function.
3. Construct an example of a submanifold \(M\subset \R^n\) that cannot be written as the preimage of a regular value of a smooth function. (Hint: Suppose that \(n=3\) and \(F: \R^3\to \R\) is a smooth map such that \(0\) is a regular value of \(F\). Show that \(F^{-1}(0)\) has two "sides." Construct a smooth submanifold \(M\subset\R^3\) that does not have two sides.)
4. Construct two different charts \(\psi_1\) and \(\psi_2\) for the sphere using map projections of your choice. (The simplest are probably equirectangular and orthographic.) Suppose \(p\) lies in both charts. Then each projection gives a basis for \(T_p S^2\). Use the transition map \(\psi_2\circ \psi_1^{-1}\) to write down the change of basis matrix for the two bases.

Due at 11pm on Thursday, February 5 on Gradescope.

• Reading: Chapter 2 of Lee.
• If Lee is too verbose for your taste, more concise introductions can be found in Milnor, Morse Theory and Petersen, Riemannian Geometry.

1. Let \(M\) be a smooth manifold and let \(U\subset M\) be an open subset. Show that if \(f\in C^\infty(U)\) and \(p\in U\), then there are an open subset \(\hat{U}\) and a \(\hat{f}\in C^\infty(M)\) such that \(p\in \hat{U} \subset U\) and \(f|_{\hat{U}}= \hat{f}|_{\hat{U}}\). Why can't we take \(\hat{U}=U\) here?

A smooth partition of unity for \(M\) is a set of smooth, nonnegative functions \(\{\rho_{\alpha}\}_{\alpha\in A}\) such that for every \(x\in M\):

• there is a neighborhood of \(x\) where all but finitely many of the \(\rho_\alpha\)'s are zero
• \(\sum_{\alpha} \rho_\alpha(x)=1\).

It's a standard and useful result in differential topology that for any \(M\) and any open cover \(M=\bigcup_{\alpha\in A} U_\alpha\), there is a partition of unity \(\{\rho_\alpha\}_{\alpha\in A}\) such that \[\supp \rho_\alpha \subset U_\alpha\] for all \(\alpha\), where \(\supp(\rho_\alpha)\) is the closure of \(f^{-1}((0,\infty))\).

2. Let \(M\) be a smooth manifold. Use a smooth partition of unity to construct a Riemannian metric on \(M\). (It may help to show that if \(g_1,\dots,g_k\) are positive-definite inner products and \(a_1,\dots, a_k > 0\), then \(\sum_i a_i g_i\) is also a positive-definite inner product.)
3. Suppose that \(M\) is a smooth manifold. Let \(G\) be a compact group that acts on \(M\) by diffeomorphisms. Let \(\rho: G\to \Diff(M)\) be the action and suppose that \(\rho\) is continuous. Show that there is a Riemannian metric \(h\) such that \(G\) acts on \((M,h)\) by isometries. (Hint: Use the fact that \(G\) has a Haar measure, i.e., a probability measure which is left- and right-invariant.)
4. Let \(n>0\) and let \(M,N\) be Riemannian manifolds of dimension \(n\). Suppose that \(\phi \from M\to N\) is a conformal map, i.e., that \(\phi_* \from T_xM\to T_{\phi(x)}N\) is injective for all \(x\in M\) and for all \(x\in M\) and all \(v,w\in T_xM\) with \(v,w\ne 0\), \[\angle(\phi_*(v), \phi_*(w)) = \angle(v,w).\] Show that there is a conformal factor \(a\in C^{\infty}(M)\) such that for all \(x\in M\) and all \(v\in T_xM\), \[\|\phi_*(v)\| = a(x) \|v\|.\]
5. Let \(H = \{x + y i \in \mathbb{C} \mid y > 0\}\) be the upper half-plane and equip \(H\) with the metric \[g = \frac{1}{y^2}(dx^2+dy^2).\] Show that for any \(a,b,c,d\in \R\) such that \(ad-bc = 1\), the map \(f\from H\to H\), \[f(z) = \frac{az + b}{cz + d}\] is an isometry of \(H\).

Reading: Chapter 3 of Lee

1. Lee, 2-15
2. Lee, 2-31
3. Consider \(S^3\) as the unit sphere \(\{(z,w)\in \mathbb{C}^2 : |z|^2 + |w|^2 = 1\}\).
   • Show that \(S^3\) is diffeomorphic to the group \(\operatorname{SU}(2)\) (viewed as a submanifold of the space of \(2\times 2\) complex matrices).
   • Then \(\operatorname{SU}(2)\) acts on \(S^3\) by left multiplication. Show that you can choose the diffeomorphism so that the round metric is invariant under this action.
   • Describe the family of metrics on \(S^3\) that are invariant under the left action. (We don't have the tools to show this yet, but most of these metrics are homogeneous but not isotropic.)

4. \(\gamma''\) and the energy of curves: Let \(\gamma: [0,1] \to \R^n\) be a smooth curve. Define the energy \(E(\gamma)\) of \(\gamma\) by \[E(\gamma) = \frac{1}{2} \int_0^1 \|\gamma'(t)\|^2 \,dt.\] Suppose that \(h: [0,1] \times (-\epsilon, \epsilon) \to \R^n\) is a smooth map. For \(u\in (-\epsilon, \epsilon)\), let \(\gamma_u(t) = h(t,u)\). Suppose that \(\gamma_0 = \gamma\) and that \(\gamma_u(0) = \gamma(0)\) and \(\gamma_u(1) = \gamma(1)\) for all \(u\in (-\epsilon, \epsilon)\). We call \(h\) a variation of \(\gamma\).

   Let \(W(t) = \frac{\partial h}{\partial u}(t,0)\). Show that \[\frac{d}{du}[E(\gamma_u)]_{u=0} = \int_0^1 - \langle W(t), \gamma''(t)\rangle\,dt.\] Suppose that \(\gamma\) is a smooth curve from \(p\) to \(q\) that has the smallest energy among all smooth curves from \(p\) to \(q\). Show that \(\gamma\) is a straight line.

5. Let \(\nabla\) be a connection on \(TM\) and let \(p\in M\).
   • Use the definition of a connection to show that if \(A|_p = 0\), then \(\nabla_A X(p) = 0\) for all \(X\in \mathcal{V}(M)\). Conclude that if \(V|_p = W|_p\), then \(\nabla_V X(p) = \nabla_W Y(p)\).
   • Let \(U\subset M\) be an open set. Let \(Z \in \mathcal{V}(M)\) be a vector field such that \(Z|_p = 0\) for all \(p\in U\). Use the definition of a connection to show that \(\nabla_V Z(p) = 0\) for any \(V\in \mathcal{V}(M)\) and \(p\in U\). Conclude that if \(X|_p = Y|_p\) for all \(p\in U\), then \(\nabla_V X(p) = \nabla_V Y(p)\).

Reading: Chapter 4-5 of Lee

1. Let \(M\) be a smooth manifold and let \(X,Y\in \mathcal{V}(M)\) be vector fields.
   • Prove that \([X,Y] = XY - YX\) is a vector field, i.e., that it satisfies the Leibniz rule \[[X,Y](fg) = [X,Y](f)\cdot g + f\cdot [X,Y](g).\]
   • Let \(f\from M\to \R\) be a smooth function. Prove the product rule for the Lie bracket, i.e., \[[X,fY] = Xf\cdot Y + f[X,Y].\]

For the following problems, let \(M\subset \R^n\) be a smooth manifold and for \(p\in M\), let \(\pi_p\from \R^n \to T_p M\) be the orthogonal projection.

Recall that the tangential connection \(\nabla^T\) on \(M\) is the connection such that if \(p\in M\); \(X, Y\) are smooth vector fields; and \(\overline{Y}\) is a smooth extension of \(Y\) to a neighborhood of \(M\) in \(\R^n\), then \[\nabla^T_{X_p} Y = \pi\left(\frac{d}{dt} \overline{Y}(p+tX_p)\right).\]

The next problems show that the tangential connection is compatible with the metric on \(M\) and torsion-free.

2. • Let \(\gamma\from (-\epsilon,\epsilon) \to M\) be a smooth curve and let \(V\in \mathcal{V}(\gamma)\) be a smooth vector field. Show that \[D^T_t V(t) = \pi_{\gamma(t)}(\frac{d}{dt} V).\]
   • Show that any parallel field along \(\gamma\) has constant length. Conclude that the tangential connection is compatible with the metric induced by the embedding of \(M\) in \(\R^n\).

3. Let \(p\in M\) and let \(P\subset \R^n\) be the plane tangent to \(M\) at \(p\). By the implicit function theorem, there is a neighborhood \(p\in U\subset M\) such that the orthogonal projection \(\phi \from U\to P\) is a diffeomorphism, i.e., \(\phi\) is a chart.

   For any vector \(v\in T_pM\), there is a constant vector field \(V_v\in \cV(P)\) with \(V_v(q) = v\) for all \(q\in P\). Let \(W_v\) be the pullback \(W_v = \phi^*(V_v)\in \cV(U)\).

   • Show that \(\nabla^T_{X} W_v = 0\) for all \(X\in T_pM\). We say that \(W_v\) is infinitesimally parallel at \(p\).
   • Use the \(W_v\)'s to show that the tangential connection is torsion-free.
4. Suppose \(M\subset \R^m\) and \(N\subset \R^n\) are submanifolds and that \(F\from M\to N\) is a Riemannian isometry. Let \(\nabla^M\) be the tangential connection on \(M\) and let \(\nabla^N\) be the tangential connection on \(N\). Use the Fundamental Theorem of Riemannian Geometry and the previous two exercises to show that \(F\) takes \(\nabla^M\) to \(\nabla^N\).
5. Let \(M = \{(x,y,z)\in \R^3 : x^2 + y^2 = 1\}\) be the unit cylinder in \(\R^3\).
   • Use a direct computation to show that curves of the form \(\gamma(t) = (\cos(at),\sin(at),bt)\) are geodesics on \(M\) with respect to the tangential connection.
   • Show that \(M\) is locally isometric to \(\R^2\). Use this to give an alternate proof that the curves above are geodesics.

Reading:

• Milnor, Morse Theory, Chapter 10, pages 55-58
• The "potato-peel" model of curvature: Tristan Needham, Visual Differential Geometry and Forms

Let \(M\) be a 2-dimensional Riemannian manifold and let \(\gamma:[0,1]\to M\) be a unit-speed smooth curve. Let \(R_\theta:T_{\gamma(t)} M \to T_{\gamma(t)} M\) be rotation by \(\theta\), and let \(V = \gamma'(t)\) and \(N=R_{\frac{\pi}{2}}(V)\) be an orthonormal frame on \(\gamma\). In class, we showed that there is a function \(\kappa: [0,1] \to \R\) such that \(D_tV = \kappa N\).

1. Show that \(D_tN = - \kappa V\) by calculating \(\langle D_t N | V\rangle\) and \(\langle D_t N | N\rangle\).
2. Let \(\theta(t) = \int_0^t \kappa(\tau)\,d\tau\) and let \(E\in \cV(\gamma)\), \(E(t) = R_{-\theta(t)}(V(t))\). Show that \(E\) is a parallel vector field. Find a parallel orthonormal frame on \(\gamma\).

3. Let \(S^2\) be the unit sphere. Show that great circles are geodesics (i.e., curves with \(D_t \gamma'=0\)). Describe the set of parallel fields along a great circle.

   Let \(\Delta\) be a triangle on the unit sphere \(S^2\) whose edges are great circles and whose angles are \(\alpha\), \(\beta\), \(\gamma\). Show that parallel transport \(P_{\partial \Delta}\) around the boundary of \(\Delta\) is rotation by angle \(\alpha+\beta+\gamma-\pi.\) (Consider a parallel field \(W\) on \(\partial \Delta\). How does the angle between \(W\) and \(\partial \Delta\) change along each edge? At each corner?)

Reading:

• Lee, Chapter 9, 196-202
• Milnor, Morse Theory, Chapter 10

1. In class, we stated that the curvature tensor can be expressed in terms of parallel transport. Specifically, if \(q\in M\) and if \(\alpha\from \R^2\to M\) is a smooth map such that \(\frac{\partial \alpha}{\partial x}=X\), \(\frac{\partial \alpha}{\partial y}=Y\), then for all \(Z\in T_qM\), we have \[R(X,Y)Z=\lim_{s\to 0} \frac{Z-p_{\gamma_s}(Z)}{s^2},\] where \(\gamma_s:[0,4]\to M\) is the image under \(\alpha\) of the boundary of an \(s\times s\) square, i.e., \[\gamma_s(t)=\begin{cases} \alpha(st,0) & t\in [0,1]\\ \alpha(s,s(t-1)) & t\in [1,2]\\ \alpha(s(3-t),s) & t\in [2,3]\\ \alpha(0,s(4-t)) & t\in [3,4]. \end{cases}\] Prove this fact.

   (Hint: Construct a frame of vector fields \(V_1,\dots, V_m\in \cV(\alpha)\) such that \(\nabla_X V_i(x,0)=0\) and \(\nabla_Y V_i=0\). Any vector field \(W\) along \(\gamma_s\) can be expressed as a linear combination of the \(V_i\) — when is \(W\) parallel?)

2. Let \(M\) be an \(n\)–dimensional Riemannian manifold, let \(k < n\), and let \(S\subset M\) be a smooth \(k\)–dimensional submanifold. Suppose that \(V\in \cV(M)\) is a vector field which is parallel along \(S\), i.e., for any vector \(V_p\in T_pS\subset T_pM\), we have \(\nabla_{V_p} V = 0\). What can we conclude about the Riemann curvature tensor of \(M\)?
3. Let \(M, N\) be connected Riemannian manifolds and let \(p\in M\). Show that if \(f,g\from M\to N\) are isometries such that \(f(p)=g(p)\) and \(Df_p=Dg_p\), then \(f=g\). (You can start by using an exponential coordinate system to show that \(f\) and \(g\) are equal in a neighborhood of \(p\).)
4. Let \(M\) be a Riemannian manifold and let \(f\from M\to M\) be an isometry. Let \(G=\{x\in M : f(x)=x\}\) be its fixed point set.
   • Show that any connected component \(C\) of \(G\) is a manifold and that \(T_xC' = \{v\in T_x M : Df(v)=v\}\).
   • Let \(C\) be a connected component of \(G\) and \(x\in C\). Show that \(\exp_x(v)\in C\) for any \(v\in T_x C\) such that \(\exp_x(v)\) is defined. (In this case, we say that \(C\) is totally geodesic.)
   • Show that \(TC\) is closed under the Riemannian curvature tensor, that is, for \(X,Y,Z \in TC\), we have \(R(X,Y)Z\in TC\), where \(R\) is the curvature tensor of \(M\).