Summary

1. Existence and uniqueness of integral curves: For every smooth vector field $X$ on $M$ and every $p \in M$, there exists a unique maximal integral curve through $p$.

2. Levi-Civita Theorem: On a Riemannian manifold, there exists a unique metric-compatible, torsion-free connection.

3. Stokes’ Theorem: For a compact oriented $n$-manifold $M$ with boundary and an $(n-1)$-form $\omega$ on $M$: $\int_{\partial M} \omega = \int_M d\omega$.

4. Gauss-Bonnet Theorem: For a compact oriented Riemannian 2-manifold: $\int_M K\, dA = 2\pi\chi(M)$.

5. Frobenius Theorem: A distribution $\mathcal{D} \subseteq TM$ is integrable if and only if it is involutive: $[X, Y] \in \Gamma(\mathcal{D})$ for all $X, Y \in \Gamma(\mathcal{D})$.

Christoffel symbols: $\Gamma^k_{ij} = \frac{1}{2}g^{k\ell}(\partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij})$

Geodesic equation: $\ddot\gamma^k + \Gamma^k_{ij}\dot\gamma^i\dot\gamma^j = 0$

Riemann curvature tensor: $R^\ell_{ijk} = \partial_i\Gamma^\ell_{jk} - \partial_j\Gamma^\ell_{ik} + \Gamma^m_{jk}\Gamma^\ell_{im} - \Gamma^m_{ik}\Gamma^\ell_{jm}$

Lie bracket (coordinates): $[X, Y] = \left(X^j\frac{\partial Y^i}{\partial x^j} - Y^j\frac{\partial X^i}{\partial x^j}\right)\frac{\partial}{\partial x^i}$

Exterior derivative: $d\omega = d\left(\frac{1}{k!}\omega_{i_1\ldots i_k} dx^{i_1} \wedge \cdots \wedge dx^{i_k}\right) = \frac{1}{k!}\frac{\partial\omega_{i_1\ldots i_k}}{\partial x^j} dx^j \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k}$

Lie derivative: $\mathcal{L}_X Y = [X, Y], \quad \mathcal{L}_X f = X(f), \quad \mathcal{L}_X \omega = \frac{d}{dt}\big|_{t=0} (\Phi_t^*\omega)$

The following diagram shows how the core concepts of differential geometry relate:

• Manifold $\xrightarrow{\text{smooth structure}}$ Tangent bundle $\xrightarrow{\text{section}}$ Vector field
• Vector field $\xrightarrow{\text{integral curve}}$ Flow $\xrightarrow{\text{Lie bracket}}$ Lie algebra
• Metric $\xrightarrow{\text{Levi-Civita}}$ Connection $\xrightarrow{\text{parallel transport}}$ Curvature
• Curvature $\xrightarrow{\text{trace}}$ Ricci $\xrightarrow{\text{trace}}$ Scalar curvature
• Connection $\xrightarrow{\text{geodesic equation}}$ Geodesic $\xrightarrow{\text{exponential map}}$ Normal coordinates
• Manifold $\xrightarrow{\text{cotangent bundle}}$ Differential forms $\xrightarrow{\text{exterior derivative}}$ de Rham cohomology

Every compact, connected, oriented 2-manifold is homeomorphic to a sphere with $g$ handles, where $g$ is the genus. The Euler characteristic is $\chi = 2 - 2g$.

1. Cartan’s magic formula: $\mathcal{L}_X \omega = d(i_X \omega) + i_X(d\omega)$.
2. Poincaré lemma: If $d\omega = 0$ (closed form) and the domain is contractible, then $\omega = d\eta$ (exact form).
3. Hodge star: $\star : \Omega^k(M) \to \Omega^{n-k}(M)$ satisfies $\star\star = (-1)^{k(n-k)}$.
4. Kodaira vanishing: On a Kähler manifold with positive line bundle, certain cohomology groups vanish, giving restrictions on the topology.

1. The Levi-Civita connection depends on the metric, not just the smooth structure. Two metrics on the same smooth manifold generally give different connections, geodesics, and curvatures.

2. Not every smooth distribution is integrable. The Frobenius theorem gives necessary and sufficient conditions; involutivity ($[X, Y] \in \mathcal{D}$ for all $X, Y \in \mathcal{D}$) must be checked.

3. The Lie bracket is not the commutator of flows. $[X, Y] = 0$ means flows commute, but $[X, Y] \neq 0$ does not mean they do not flow at all — only that the composition order matters.

4. Stokes’ theorem requires compact support or compact manifold with boundary. For non-compact manifolds, additional decay conditions are needed for the integral to be well-defined.

5. The exponential map is not globally defined. It is only defined on a neighborhood of zero in $T_p M$, unless the manifold is geodesically complete (Hopf-Rinow theorem).

1. Show that $S^2$ and $S^1 \times S^1$ (the torus) are not homeomorphic by comparing their Euler characteristics. Compute $\chi(S^2)$ and $\chi(T^2)$ explicitly using a triangulation.

2. Prove that the Lie bracket satisfies the Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$.

3. Compute the Christoffel symbols for the sphere $S^2$ with the round metric $g = d\theta^2 + \sin^2\theta\,d\phi^2$. Then derive the geodesic equations and verify that great circles are geodesics.

4. For the 2-torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ with the flat metric inherited from $\mathbb{R}^2$, compute the Riemann curvature tensor. Show that it vanishes identically.

5. Let $M$ be a compact oriented Riemannian 2-manifold with $\chi(M) = 0$. Use the Gauss-Bonnet theorem to prove that $\int_M K\,dA = 0$. Give an example of such a manifold.

6. Show that the exterior derivative $d$ satisfies $d^2 = 0$. Use this to prove that every exact form is closed. Give a counterexample to the converse on a non-contractible manifold.

7. Compute the de Rham cohomology groups $H^1_{\text{dR}}(S^1)$ and $H^1_{\text{dR}}(T^2)$. What do they tell you about the topology of these manifolds?