Given vector fields \(X_1,X_2,Y_1,Y_2 \in \mathfrak{X}(M)\) and \(f : M \to \mathbb{R}\) a smooth function, prove that the covariant derivative \(\nabla\colon \mathfrak{X}(M) \times \mathfrak{X}(M)\rightarrow \mathfrak{X}(M)\) satisfies the following properties.

1. \(\nabla_{X_1+X_2} Y_1=\nabla_{X_1}Y_1+\nabla_{X_2}Y_1.\)

2. \(\nabla_{X_1}(Y_1+Y_2)=\nabla_{X_1}Y_1+\nabla_{X_2}Y_2.\)

3. \(\nabla_{fX_1}Y_1=f\nabla_{X_1}Y_1.\)

4. \(\nabla_{X_1}(fY_1)=f\nabla_{X_1}Y_{1}+\mathrm{d}f(X_1)Y_1.\)

A local parametrization is said to be orthogonal if \(g\) is diagonal, i.e. \(g_{12}=g_{21}=0.\)

1. Derive expressions for the Christoffel symbols of an orthogonal parametrization.

2. For each of the following examples verify that the given local parametrization is orthogonal and then use the formulae you computed in part (a) to compute the Christoffel symbols.

1. The parametrization of the plane given by \(F\colon\mathbb{R}^2\rightarrow \mathbb{R}^3\) such that \((u,v)\mapsto p_0+u\overrightarrow{w}_1+v\overrightarrow{w}_2,\) where \(\overrightarrow{w}_1,\overrightarrow{w}_2\in\mathbb{R}^3\) are orthonormal vectors and \(p_0\in\mathbb{R}^3\) is a point.

2. The local parametrization of the sphere given by \(F\colon (0,\pi)\times(0,2\pi)\rightarrow\mathbb{R}^3\) such that \((u,v)\mapsto(\operatorname{sin}(u)\operatorname{cos}(v),\operatorname{sin}(u)\operatorname{sin}(v),\operatorname{cos}(u)).\)

3. The local parametrization of the catenoid given by \(F\colon\mathbb{R}\times(0,2\pi) \rightarrow\mathbb{R}^3\) such that \((u,v)\mapsto (\operatorname{cosh}(u)\operatorname{cos}(v),\operatorname{cosh}(u)\operatorname{sin}(v),u).\)

Let \(F\colon U\rightarrow \mathbb{R}^3\) be a local parametrization, and let \(p,q\in \operatorname{Im}(F).\) For each of the following, either prove the statement or find a counterexample.

1. If \(U\) is connected, then there exists a curve \(\gamma\colon [a,b]\rightarrow U\) with \(F\circ \gamma(a)=p\) and \(F\circ \gamma(b)=q,\) which is distance minimizing.

2. If there exists a curve \(\gamma\colon [a,b]\rightarrow U\) such that \(F\circ \gamma(a)=p\) and \(F\circ \gamma(b)=q\) which is distance minimizing, then it is unique.