• 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.

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.

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.

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 \(\R^n\).

• 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.

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 \(M\).

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.

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 \(X(fg)\).)

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?)

• 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.)

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 \[P_\gamma = R_{-A(t)}.\]

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 \(j\).