1. If you haven't proved these facts before, prove them. Don't turn this in.
   • Show that the definitions of \(T_xM\) in terms of curves and in terms of derivations are equivalent.
   • State and prove the chain rule for maps between smooth manifolds.
2. 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.

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. Let \(M\) and \(N\) be smooth manifolds with \(\dim M = m < n = \dim N\). An embedding of \(M\) into \(N\) is a smooth map \(e\from M\to N\) such that \(e\) is a homeomorphism of \(M\) onto \(e(M)\) and \(D_xe\) is injective for every \(x\in M\).

   Let \(e\) be an embedding.

   • Show that if \(x\in e(M)\), there is a chart \(\phi \from U \to \R^n\), where \(x\in U\subset N\), such that \(\phi(U\cap M)\) is the intersection of an \(m\)–plane with \(\phi(U)\).
   • Give an example of a map \(e\from M\to N\) such that \(e\) is a homeomorphism of \(M\) onto \(e(M)\) and \(x\in e(M)\) but there is no chart with the property above.
5. 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 and describe the tangent bundle \(TM\).

1. 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 \(v\in T_xM\), \[\|\phi_*(v)\| = a(x) \|v\|.\]
2. Let \(H = \{x + y i \in \mathbb{C} \mid y > 0\}\) be the upper half-plane and equip \(H\) with the metric \[dg^2 = \frac{1}{y}(dx^2+dy^2).\] 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 \(H\).

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. 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 set of parallel vector fields on \(\gamma\) and calculate the angle \(\angle(V(0),V(2\pi))\) when \(V\) is a parallel vector field.

1. Let \(M\) be a smooth manifold and let \(\nabla\) be a connection on \(M\). We call \(X\) infinitesimally parallel (IP) at \(p\in U\) if \(\nabla_{V_p} X = 0\) for all \(V_p\in T_pM\).

   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 IP 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 \(M\).

2. Let \(M=\R^n\).

   • Construct a connection on \(M\) which is compatible with the Euclidean metric but not torsion-free.
   • Construct a connection on \(M\) which is torsion-free, but not compatible with the Euclidean metric.

   Both of these are possible without resorting to Christoffel symbols.

3. 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 \(X(fg)\).)

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. Let \(f\from M \to N\) be an isometry, let \(x\in M\) and let \(v\in T_x M\) be such that \(\exp_x(v)\) is defined. Show that \[f(\exp_x(v)) = \exp_{f(x)}(f_*(v)).\]
2. 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 \(M\). (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. Let \(M\) be an oriented \(2\)–dimensional manifold and let \(\gamma\from [0,\ell]\to M\) be a unit-speed closed curve. Let \(V=\gamma'\in \cV(\gamma)\). Since \(M\) is oriented, there is a notion of "clockwise" and "counterclockwise" rotation at each point. For \(\alpha\in \R\) and \(p\in M\), let \(R_\alpha \from T_p M \to T_p M\) be the counterclockwise rotation by angle \(\alpha\). Let \(N=R_{\frac{\pi}{2}}(V)\) be orthonormal to \(V\) and let \[\kappa = \langle N\mid D_t V\rangle.\]

   Show that if \[A(t) = \int_0^t \kappa(\tau)\ud \tau,\] then \(W(t) = R_{-A(t)}(V(t))\) is a parallel field on \(\gamma\). Conclude that if \(\gamma\from [0,\ell] \to M\) is a unit-speed closed curve with \(\gamma'(0)=\gamma'(\ell)\), then the holonomy \(P_\gamma\) satisfies \[P_\gamma = R_{-A(t)}.\]

4. Let \(\phi\from M\to N\) and let \(V\in \cV(M)\) and \(X\in \cV(N)\). We say that \(V\) and \(X\) are \(\phi\)–related if for all \(p\in M\), we have \[\phi_*(V_p) = X_{\phi(p)}.\]

   • 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 \([X,Y]\in \cV(N)\).

   Suppose that \(M\subset N\) is an embedded submanifold. Let \(U\subset M\), let \(\phi = (u^1,\dots, u^m) \from U\to \R^n\) be a chart on \(M\), and let \(\partial_1,\dots, \partial_m \in \cV(M)\) be the standard basis.

   Let \(W=\phi(U)\) and \(\alpha = \phi^{-1}\from W\to N\). Let \(\cV(\alpha)\) be the set of smooth vector fields on \(\alpha\), i.e., smooth maps \(X\from W\to TN\) such that \(X(w) \in T_{\alpha(w)}N\) for all \(w\in W\).

   • Use the Levi–Civita connection \(\nabla\) on \(N\) to define covariant derivatives \(D_{1},\dots, D_{m}\from \cV(\alpha)\to \cV(\alpha)\).
   • Show that \(D_i\partial_j = D_j\partial_i\) for all \(i\) and \(j\).

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 \(M, N\) 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 \(f=g\). (Problem 1 from PS#4 may be helpful.)
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 (i.e., there is a family of maps \(\gamma_t:[0,1]\to M\) for \(t\in [0,1]\) such that \(\gamma_0=\gamma\), \(\gamma_t(0)=\gamma_t(1)=x\), and \(\gamma_1(s)=x\) for all \(s\in [0,1]\).)
   • Suppose \(M\) is compact. Show that there is an \(\epsilon>0\) such that every closed curve on \(M\) with length \(< \epsilon\) is null-homotopic.
4. Suppose that \(M\) is a Riemannian manifold with curvature tensor \(R\).
   • Use the symmetries of the curvature tensor to show that if \(p\in M\), \(X,Y\in T_pM\), then \[K(X,Y)=\frac{\langle R(X,Y)Y\mid X\rangle}{\|X\|^2 \|Y\|^2-\langle X\mid Y\rangle^2}\] is independent of \(V\) and \(W\). This is the sectional curvature of the plane spanned by \(X\) and \(Y\).
   • Prove that if \(M\) is $2$–dimensional and \(K=K(X,Y)\), 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 \(K\). (This formula also holds for higher-dimensional manifolds with constant sectional curvature; there's also a more complicated formula for \(R\) in terms of \(K\) in the general case.)

1. Show that \(M\) is complete as a metric space if and only if it is geodesically complete.
2. 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 \(\alpha(0,0)=p\), \(\frac{\partial \alpha}{\partial u_1}=V\), and \(\frac{\partial \alpha}{\partial u_2}=W\). Define \[H(f)(V,W)=\frac{\partial^2}{\partial u_1\partial u_2} f(\alpha(u_1,u_2)).\] Prove that \(H(f)\) is a well-defined symmetric bilinear form. What if \(p\) is not a critical point of \(f\)?

3. Suppose that \(\gamma: [0,1]\to M\) 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.

   Show 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 \(W=0\). How does this match up with what we know about length-minimizing curves in \(\R^n\)?

The midterm exam will be in-class on March 19, covering everything up to March 12.

You can bring one index card of notes (one 3" x 5" card, 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 if necessary 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 \(\gamma\) be a geodesic in \(M\) and let \(V=\gamma'\). For any \(t\) and any \(X\in T_{\gamma(t)}M\), let \[\phi_t(X) = R(V(t),X)V(t).\]
   • Show that \(\phi_t\) is a linear map such that \(\langle \phi_t(X)\mid V(t)\rangle = 0\) and \[\langle \phi_t(X)\mid Y\rangle = \langle X\mid \phi_t(Y)\rangle.\] (That is, \(\phi_t\) is self-adjoint and thus has an orthonormal basis of eigenvectors.)
   • Suppose that \(E \in \cV(\gamma)\) is a parallel vector field on \(\gamma\) such that \(\phi_t(E) = -r^2 E\) for all \(t\), with \(r > 0\). Show that \(\gamma(0)\) is conjugate to \(\gamma(k \pi r^{-1})\) for any integer \(k>0\).
   • Extra: Construct a manifold \(M\) and \(\gamma\) so that there are \(n-1\) independent parallel vector fields \(E_1,\dots, E_{n-1}\) satisfying \(\phi_t(E_i) = -r_i^2 E_i\) for distinct \(r_i\)'s.