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 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 \[\frac{d}{du}[E(\gamma_u)]_{u=0} = \int_0^t - \langle W(t), \gamma''(t)\rangle\,dt.\] 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 \(\nabla_V X(p) = \nabla_V Y(p)\).