1. Let \(M\) and \(N\) be connected smooth manifolds. Let \(f\from M\to N\) be a smooth function. Show that \(Df_p=0\) for all \(p\in M\) if and only if \(f\) is a constant function.
2. Let \(M\) be a smooth \(d\)-manifold and let \(f\in C^\infty(M)\). Suppose that \(D_xf\ne 0\) for all \(x\in f^{-1}(0)\). Show that \(f^{-1}(0)\) is a smooth \(d-1\)-manifold.

3. Let \(M\) be a smooth manifold, let \(\gamma\from (-\epsilon,\epsilon)\to M\) be a smooth curve with \(\gamma(0)=p\). Let \(\varphi=(x^1, \dots, x^d)\from U\to \R^d\) be a chart defined on a neighborhood of \(p\); we can write \(\gamma\) in coordinates as \[\varphi\circ \gamma = (\gamma^1,\dots, \gamma^d) = (x^1\circ \gamma,\dots, x^d\circ \gamma).\]

   Let \(\varphi'=(y^1, \dots, y^d)\from U'\to \R^d\) be a second chart defined on a neighborhood of \(p\). This gives us a second way to write \(\gamma\) in coordinates: \[\varphi'\circ \gamma = (\bar{\gamma}^1,\dots, \bar{\gamma}^d) = (y^1\circ \gamma,\dots, y^d\circ \gamma).\]

   • Let \(f\in C^\infty(M)\). Explain the notation \(\frac{\partial f}{\partial x^j}\) and express \(\frac{d}{dt}f\circ \gamma\) in terms of \(\frac{d \gamma^i}{dt}\) and the \(\frac{\partial f}{\partial x^j}\)'s.
   • Express \(\frac{d \bar{\gamma}^i}{dt}\) in terms of \(\frac{d \gamma^i}{dt}\) and the partial derivatives \(\frac{\partial y^i}{\partial x^j}\).
   • Check your calculation by applying it to the case that \((x,y)\) are Cartesian coordinates on \(\R^2\) and \((r,\theta)\) are polar coordinates.
4. With notation as above, write a formula for \(\frac{d^2 \bar{\gamma}^i}{dt^2}\) in terms of \(\frac{d \gamma^i}{dt}\), \(\frac{d^2 \gamma^i}{dt^2}\) and partial derivatives of the \(y^i\)'s.

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 \(f^{-1}((0,\infty))\).

1. 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.)
2. Let \(M\) be a smooth manifold. Use a smooth partition of unity to construct a connection on \(M\). (It may help to show that if \(\nabla^1,\dots,\nabla^k\) are connections, \(a_1,\dots, a_k > 0\), and \(\sum_i a_i=1\), then \(\sum_i a_i \nabla^i\) is also a connection.)

3. 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 \[\nabla^T_{X_p} Y = \pi\left(\frac{d}{dt} \overline{Y}(p+tX_p)\right).\]

   • 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 the tangential connection is compatible with the metric induced by the embedding of \(M\) in \(\R^n\).
4. In class, we stated that the covariant derivative can be expressed in terms of parallel transport in the sense that if \(\gamma: I \to M\) is a smooth curve in \(M\), \(t_0\in I\), and \(p_t = P_{t_0+t,t_0} : T_{\gamma(t_0 + t)}M\to T_{\gamma(t_0)}M\), then for any \(X\in \mathcal{V}(\gamma)\), \[D_tX(t_0)=\left.\frac{d}{dt}\right|_{t=0} p_t(X(t)).\] Prove this identity.

5. Let \(X,Y\in\mathcal{V}(M)\). Show that the differential operator \(XY-YX\) is a derivation on \(M\), so \(XY-YX\) is a vector field on \(M\).

   (Note on notation: Working with vector fields can be tricky, because multiplication and application of operators look the same, i.e., \(X(Yf)=X[Y[f]]\) vs. \((Yf)X=Y[f]\cdot X\). It can help to use different notation, for instance, writing \(X[Yf]\) for \(X(Yf)\), \(Xf\cdot Yg\) for \((Xf)\times (Yg)\), and \(X[fg]\) for \(X(fg)\).)

6. (Optional) I haven't found a good source/elegant proof for this result, so I'm making this optional:

   One can show that for any \(X\in \mathcal{V}(M)\) and any \(p\in M\), there is an integral curve of \(X\) through \(p\), that is, a smooth curve \(\gamma_{X,p}\from (-\epsilon,\epsilon) \to M\) such that \(\gamma_{X,p}'(t)=X_{\gamma(t)}\) for all \(t\in (-\epsilon,\epsilon)\). Furthermore, if \(K\subset M\) is compact, one can choose \(\epsilon>0\) so that \(\gamma_{X,p}\) is defined on \((-\epsilon,\epsilon)\) for all \(p\in K\). For \(X\in \mathcal{V}(M)\), \(p\in M\), and sufficiently small \(t\), we define the flow of \(X\) on \(M\) to be \[\Phi_X^t(p) = \gamma_{X,p}(t).\]

   Let \(X,Y\in \mathcal{V}(\R^n)\) and let \[\alpha(t) = \Phi_{-Y}^{t}\circ \Phi_{-X}^{t}\circ \Phi_Y^{t}\circ \Phi_X^{t}(p).\] Show that \[\frac{d}{dt}|_{t=0} \alpha(\sqrt{t}) = [X,Y](p).\]

   For some starting points:

   • It's enough to consider the case \(M=\R^n\), in which case one can compute everything using the first-order Taylor expansions of \(X\) and \(Y\).
   • There's a Math.SE question along these lines at https://math.stackexchange.com/questions/1347657/lie-bracket-and-flows-on-manifold

1. Use the properties of the Lie bracket to show that if \(\phi\from U\to \R^n\) is a chart with coordinate fields \(\partial_1,\dots, \partial_n\), \(X=x^i\partial_i\), and \(Y=y^j\partial_j\), then \[[X,Y] = (x^j \cdot \partial_j y^i - y^j\partial_j x^i)\partial_i = Xy^i\cdot \partial_i - Yx^i\cdot \partial_i.\]
2. Let \(M\) be an embedded submanifold in \(\R^n\) and let \(p\in M\). Let \(p\in P\subset \R^n\) 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 \(\pi\from U\to P\) is a diffeomorphism. Show that for any constant vector field \(V\in \cV(P)\), the pushforward \(W=(\pi^{-1})_*(V)\in \cV(U)\) is locally parallel at \(p\) with respect to \(\nabla^T\). Use this to show that \(\nabla^T\) is torsion-free. Since \(\nabla^T\) is compatible with the Riemannian metric, this implies that \(\nabla^T\) is the Levi–Civita connection on \(M\).
3. Let \(\phi\in (0,\pi/2)\) and let \(M\subset \R^3\) be the cone \[M=\{(r \cos \theta \sin \phi ,r\sin \theta \sin \phi, r\cos \phi)\mid r>0, \theta\in [0,2\pi)\}.\] This can be parametrized by the map \[u(r, \theta)=(r \cos \theta \sin \phi ,r\sin \theta \sin \phi, r\cos \phi).\] This map has coordinate vector fields \(\partial_r=\frac{\partial}{\partial r}\) and \(\partial_\theta=\frac{\partial}{\partial \theta}\).
   • Let \(\nabla^T\) be the tangential connection on \(M\) and calculate \(\nabla^T_{\partial_r} \partial_r\), \(\nabla^T_{\partial_r} \partial_\theta\), \(\nabla^T_{\partial_\theta} \partial_r\), and \(\nabla^T_{\partial_\theta} \partial_\theta\).
   • Let \(\gamma\from [0,2\pi]\to M\) be the circle \(\gamma(t)=u(1,t)\), where \(u\) is as in the previous problem. Describe the parallel vector fields on \(\gamma\) and calculate the angle \(\angle(V(0),V(2\pi))\) when \(V\) is a parallel vector field.

4. 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 directly (without using the Gaussian curvature formula) 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?)

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 \(V_1,\dots, V_m\in \mathbf{V}(\alpha)\) 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. Suppose that \(M\) is a two-dimensional Riemannian manifold with curvature tensor \(R\).
   1. Use the symmetries of the curvature tensor to show that if \(p\in M\), \(V,W\in T_pM\), then \[K=\frac{\langle R(V,W)W,V\rangle}{\|V\|^2 \|W\|^2-\langle V,W\rangle^2}\] is independent of \(V\) and \(W\). This is the Gaussian curvature of \(M\) at \(p\).
   2. Prove that if \(K\) is as above, then \[R(X,Y)Z=K(\langle Y,Z\rangle X-\langle X,Z\rangle Y)\] for all \(X,Y,Z\in T_pM\), i.e., that \(R\) can be reconstructed from \(K\).
4. Let \(\phi\from M\to N\). Recall that if \(V\in \cV(M)\) and \(X\in \cV(N)\), then \(V\) and \(X\) are \(\phi\)–related if for all \(p\in M\), we have \[D\phi(V_p) = X_{\phi(p)}.\] Suppose that \(V, W \in \cV(M)\) are \(\phi\)–related to \(X, Y\in \cV(N)\) respectively. Show that \([V,W]\in \cV(M)\) is \(\phi\)–related to \([X,Y]\in \cV(N)\).

1. Prove the following properties of geodesics and the exponential map:
   • If \(\gamma\) is a geodesic and \(c\in \R\), then \(\lambda(t)=\gamma(ct)\) is a geodesic.
   • (Optional, added 2024-02-26) Let \(x\in M\). For any \(v\in T_x M\), let \(\gamma_v\from I\to M\) be a geodesic with \(\gamma_v'(0)=v\). Let \(c\in \R\). Show that \(\gamma_{cv}(t) = \gamma_v(ct)\) wherever they are both defined.
   • Let \(v\in T_pM\) and let \(\lambda(t) = \exp_x(tv)\). Then \(\lambda(t)\) is a geodesic with \(\lambda'(0)=v\).
   • If \(f\from M\to N\) is an isometry, \(x\in M\), and \(v\in T_x M\), then \[f(\exp_x(v)) = \exp_{f(x)}(D_xf(v)).\]
2. Let \(M\) be a Riemannian manifold and let \(f\from M\to N\) 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 \(M\). (In fact, a necessary condition for there to be a totally geodesic submanifold with tangent plane \(P\) is that \(P\) is closed under the curvature tensor.)
3. For \(x\in M\), the injectivity radius \(\operatorname{injrad}(x)\) is defined as \[\operatorname{injrad}(x) = \sup\{r > 0 : \exp_x \text{ is a diffeomorphism on } B_r(x)\}.\]
   • Show that any piecewise-smooth closed curve \(\gamma:[0,1]\to M\) with \(\gamma(0)=\gamma(1)=x\) and \(\ell(\gamma)<2\operatorname{injrad}(x)\) is null-homotopic.
   • Suppose \(M\) is compact. Show that there is an \(\epsilon>0\) such that every closed curve on \(M\) with length \(< \epsilon\) is null-homotopic.

1. In class, we defined distance using piecewise-smooth curves. Given a continuous \(\gamma\from [0,1]\to M\), one can also define \[\ell(\gamma) = \sup_{0=t_0 < \dots < t_n=1} \sum_{i=0}^{n-1} d(\gamma(t_i),\gamma(t_{i+1}))\] (which might be infinite). When \(\gamma\) is piecewise smooth, this agrees with the usual definition \(\ell(\gamma) = \int \|\gamma'(t)\| \,dt\).

   Show that if \(\gamma\from [0,1]\to M\) is a curve from \(p\) to \(q\), then there is a piecewise-smooth \(\lambda\) connecting \(p\) to \(q\) such that \(\ell(\lambda)\le \ell(\gamma)\) and \(\lambda\) is homotopic to \(\gamma\).

2. Show that \(M\) is complete as a metric space if and only if it is geodesically complete.
3. Let \(M, N\) be connected, not necessarily complete, Riemannian manifolds and let \(p\in M\). Show that if \(f,g\from M\to N\) are isometries (diffeomorphisms preserving the Riemannian metric) such that \(f(p)=g(p)\) and \(Df_p=Dg_p\), then \(f=g\).

4. Suppose that \(\gamma\from[0,1]\to M\) is a geodesic. The second variation formula states that if \(V=\frac{d\gamma}{dt}\) and \(W_1, W_2\in \cV(\gamma)\) are piecewise-smooth vector fields with \(W_i(0)=W_i(1)=0\), then \[H(E)(W_1,W_2)=-\sum_t \langle W_2\mid \Delta_t D_tW_1\rangle-\int_0^1 \langle W_2\mid D_t^2 W_1-R(V,W_1)V\rangle\;dt.\]

   Show that this can be rewritten in the more symmetric form \[H(E)(W_1,W_2)=\int_0^1 \langle D_t W_1\mid D_t W_2\rangle +\langle R(V,W_1)V\mid W_2\rangle\;dt.\] This expression is known as the index form.

   Conclude that if \(M=\R^n\) and \(\gamma\) is a straight line, then \(H(E)\) is positive definite, i.e., \(H(E)(W,W)\ge 0\) for all \(W\in T_\gamma \Omega\), with \(H(E)(W,W)=0\) if and only if \(W=0\).

The midterm exam will be in-class on March 14, covering everything up to March 7 (problem sets 1-6 plus next week's lectures). I'll post a few problems on March 7 covering material after problem set 6, but you don't have to turn in those problems.

I haven't written the midterm yet; I plan for the midterm to have some short questions followed by about five more involved problems, of which you can choose four to solve.

You can bring one sheet of handwritten notes (one letter/A4 sheet of paper, front and back). There will be somewhere around 10 problem sets in the course of the semester; the midterm will be worth about four problem sets and the final worth about six. Adjustments to the grading scale (dropping problem sets, etc.) will be made so that final grades align with departmental norms.

1. Suppose that \(f:M\to \R\) is a smooth function and that \(p\in M\) is a critical point of \(f\). If \(V,W\in T_pM\), let \(\alpha:(-\epsilon, \epsilon)\times (-\epsilon, \epsilon)\to M\) be a smooth map such that \(\alpha(0,0)=p\), \(\left. \frac{\partial \alpha}{\partial u_1}\right|_{(0,0)}=V\), and \(\left. \frac{\partial \alpha}{\partial u_2}\right|_{(0,0)} = W\). Define \[H(f)(V,W)=\left.\frac{\partial^2}{\partial u_1\partial u_2} f(\alpha(u_1,u_2))\right|_{(0,0)}.\] Prove that \(H(f)\) is a well-defined symmetric bilinear form. What if \(p\) is not a critical point of \(f\)?
2. Let \(\gamma\from \R \to M\) be a geodesic and let \(V=\gamma'\). Show that:
   • If \(J\) is a Jacobi field on \(\gamma\), then there are \(a,b\in \R\) such that \(\langle J(t),V(t)\rangle=at+b\). (It follows that if \(J(t_i)\) is orthogonal to \(V\) for some \(t_1\ne t_2\), then \(J(t)\) is orthogonal to \(V\) for all \(t\).)
   • Show that if \(J\) is a Jacobi field on \(\gamma\), then \(J\) can be decomposed as a sum \(J=J^\parallel+J^\perp\), where \(J^\parallel\) is a Jacobi field tangent to \(\gamma\) and \(J^\perp\) is a Jacobi field orthogonal to \(\gamma\).

3. Let \(a < b\) and let \(M\subset \R^3\) be the surface of revolution obtained by rotating the curve \[(x-b)^2+z^2=a^2, y=0\] around the \(z\)–axis (a torus with major radius \(b\), minor radius \(a\)).

   Show that the "equator" \(\gamma(t)=((a+b)\cos t, (a+b)\sin t,0)\) is a geodesic. Describe the Jacobi fields on \(\gamma\). (It may help to know that the Gaussian curvature along \(\gamma\) is \(K=\frac{1}{a(a+b)}\).) What does this suggest about geodesics that form a small angle with \(\gamma\)?

1. Let \(\gamma\) be a geodesic on \(S^n\). Characterize the set of Jacobi fields on \(\gamma\). (We described some of the Jacobi fields in class, but without proof. Start by letting \(E_1=\gamma',E_2,\dots, E_n\) be parallel orthonormal fields on \(\gamma\) and writing \(J\) as \(J=f^i E_i\). It may help to note that the formula \[R(X,Y)Z=K(\langle Y,Z\rangle X-\langle X,Z\rangle Y)\] from Problem Set 4 holds when \(M\) has constant sectional curvature.)
2. Suppose that \(\gamma_1,\gamma_2\from \R\to M\) are unit-speed geodesics such that \(\gamma_1(0)=\gamma_2(0)=p\). Use normal coordinates to show that for all sufficiently small \(t>0\), \[d(\gamma_1(t), \gamma_2(t)) = t \|\gamma_1'(0)-\gamma_2'(0)\| + O(t^2) = \sqrt{2-2\langle \gamma_1'(0)|\gamma_2'(0)\rangle} + O(t^2).\] (This is a key step in showing that any map \(M\to N\) that preserves the distance function also preserves the Riemannian metric.)
3. Suppose that \(M\) has nonpositive sectional curvature (i.e., for all \(p\in M\) and all linearly independent \(X,Y\in T_pM\), \(K(X,Y)\le 0\)). Show that no two points \(p\) and \(q\) are conjugate along any geodesic.
4. Suppose that \(M\) is geodesically complete and \(\exp_p\) is injective on the ball \(B_r(0)\). Show that \(\operatorname{injrad}(p)\ge r\) (i.e., that \(\exp_p\) is a diffeomorphism on \(B_r(0)\).)

1. Let \(M\) be a compact \(n\)–manifold and suppose that \(\Ric(U,U)\ge (n-1)k\) for all \(U\in TM\) such that \(\|U\|=1\). Prove that if \(\gamma\) is a geodesic in \(M\) of length greater than \(\frac{m\pi}{\sqrt{k}}\), then \(\gamma\) has index at least \(m\).
2. Let \(M\) be a compact connected manifold and let \(f\) be a Morse function on \(M\). For \(\lambda\ge 0\), let \(n_\lambda\) be the number of critical points of \(f\) with index \(\lambda\). Show that \(n_1 \ge n_0 - 1\).
3. (Optional, requires algebraic topology) Let \(M\) and \(f\) be as above and suppose that \(n_1=0\). Show that \(M\) is simply connected. Use any results from algebraic topology that you need.
4. (Optional, requires algebraic topology) Let \(f\) be a Morse function on \(S^k\) and let \(n_\lambda\) be as above. Show that \(\sum_i (-1)^i n_i = 2\) if \(k\) is even and \(0\) if \(k\) is odd. Use any results from algebraic topology that you need.
5. Let \(M\) be a compact connected manifold. For any closed curve \(\gamma:S^1\to M\), the free homotopy class of \(\gamma\) is the set of curves that are homotopic to \(\gamma\). Show that this set has a minimal-energy element and that this element is a closed geodesic (a smooth map \(S^1\to M\) that satisfies the geodesic equation).

1. Let \(M\) be a complete \(n\)–manifold with \(K_M\ge 1\). Show that the volume of \(M\) is at most the volume of the unit \(n\)–sphere.

   (Update: The following facts may be helpful: For \(p\in M\) and \(v\in T_p M\) is a unit vector, we define the cut time \(C(v)\) to be the supremal \(t_0\) such that \(\gamma(t)=\exp_p(tv)\) is a minimizing geodesic on \([0,t_0]\). If \(C\) is defined at \(v\), then \(C\) is Lipschitz on a neighborhood of \(v\) (see here for example).

   We define the cut locus \(E_p\) to be the set \[E_p = \{ w\in T_p M : \|w\| = C(\frac{w}{\|w\|})\}.\] Then \(E_p\) is homeomorphic to a sphere in \(T_pM\) (why is this?) and \(\exp_p(E_p)\) has measure zero. Furthermore, if \(D_p\) is the closed ball bounded by \(E_p\), then \(\exp_p(D_p) = M\) (why is this?))

2. Let \(X\) be a CAT(0) space and let \(f\from X\to X\) be an isometry. The displacement function \(\delta_f\from X\to \R\) is defined as \(\delta_f(x)=d(x,f(x))\). The translation length of \(f\) is defined as \(\tau_f=\inf_{x\in X} \delta_f(x)\).
   • Show that the (possibly empty) set \(M_f=\delta_f^{-1}(\tau_f)\) (called the min-set of \(f\) is convex and that \(f(M_f)=M_f\).
   • Show that if \(x\in M_f\) and \(\tau_f>0\), then the images \(f^n(x),n\in \Z\) lie on an infinite geodesic \(\gamma\). (This is called an axis of \(f\), and all axes of \(f\) are parallel.)
3. Let \(p\ge 1\) and let \(L_p^2\) be \(\R^2\) equipped with the \(L_p\) metric, i.e., \[d((x_1,y_1),(x_2,y_2)) = (|x_2-x_1|^p + |y_2-y_1|^p)^{\frac{1}{p}}.\] For what values of \(p\) is \(L_p^2\) a CAT(0) space?
4. Let \(X\) be a complete CAT(0) space and let \(y_1,\dots, y_n\in X\). Show that the function \(\sigma(x)=\sum_i d(x,y_i)^2\) has a unique minimum. (This is called the barycenter of \(\{y_1,\dots, y_n\}\), and it can be used to provide another proof that any isometry with a finite orbit has a fixed point.)

No problem set this week, but if you haven't seen homology before, here are a couple of exercises to try working with homology:

1. If you glue opposite sides of a square or a hexagon, you get a torus \(T\). This gives two different ways of realizing a torus as a cell complex. Calculate the homology of \(T\) using both decompositions.

2. If \(V_n,n\ge 0\) is a sequence of finite-dimensional vector spaces over \(\R\), with \(V_n=0\) for all but finitely many \(n\), define the Euler characteristic of the sequence \(V\) by \[\chi(V)=\sum_i (-1)^i \dim V_i.\] Suppose that \(V\) is also a chain complex (i.e., there are maps \(\partial_n: V_n\to V_{n-1}\) that satisfy \(\partial_{n}\partial_{n-1}=0\)). Then its homology groups are a sequence \[H_0(V),H_1(V),\dots, H_i(V),\dots=\ker \partial_i/\im \partial_{i+1}\] which we denote \(H_*(V)\). Prove that \(\chi(H_*(V))=\chi(V)\).

   Relate this to Euler's Formula that if \(S^2\) is triangulated with \(F\) faces, \(E\) edges, and \(V\) vertices, then \(F-E+V=2\). How should this formula read for the torus?