Examples in Mathematics

For matrix groups we have the following explicit formulas:

If \(\dot{h} = X\) then \(\left.\frac{d}{dt}\right|_{t=0} ghg^{1} = g\dot{h}g^{-1}\). Thus,

\begin{align} \Ad_g(X) = gXg^{-1}. \end{align}

Furthermore, if \(\dot{g} = Y\) then \(\left.\frac{d}{dt}\right|_{t=0} gXg^{-1} = \dot{g}X - X\dot{g}\). Thus,

\begin{align} \ad_Y(X) = [Y,X]. \end{align}

Given a quadratic form \(q\) on an \(n\)-dimensional vector space over \(\mathbb{k}\) we have the orthogonal group \(O(q, \mathbb{k})\) that preserves it. It is an algebraic group; however for the case where \(\mathbb{k} = \mathbb{R}, \mathbb{C}, \mathbb{Q}_p\), it is a Lie group. \(O(n, \mathbb{k})\) acts transitively on the sphere \(S^{n-1}_{\mathbb{k}}\) by the Cartan-Dieudonné theorem.

The \(n\)-dimensional sphere \(S^n\) may be viewed as an algebraic variety by the polynomial equation

\begin{align} \label{eq:unit-sphere} x_1^2 + \cdots + x_n^2 = 1. \end{align}

As a CW-complex, the sphere may be obtained as \(e^0\sqcup e^n/{\sim}\) gluing by \(\varphi : S^{n-1} \to e^0\).¹ Differentially, it has spherical coordinates obtained by a simple inductive argument as \(\varphi_j \in [0,\pi]\) for \(j=1,\dots,n-1\) and \(\varphi_n \in [0,2\pi]\), where each \(\varphi_j\) measures the zenith angle from the \(j\)-th axis, and the final \(\varphi_n\) coordinate belongs to \(S^1\): fixing \(\varphi_1, \dots, \varphi_{j-1}\), we obtain a sphere dimension \(n-j+1\) and of radius \(\sin\varphi_1\cdots\sin\varphi_{j-1}\) on which we define the \(j\)-th zenith angle \(\varphi_j\).

The Riemannian metric \(g\) inherited from \eqref{eq:unit-sphere} is given in spherical coordinates by the diagonal tensor (here \(\langle \cdot, \cdot\rangle\) denoting the standard metric of \(\mathbb{R}^{n+1}\)),

\begin{align} g & = \langle \partial /\partial \varphi_i, \partial / \partial \varphi_k\rangle d\varphi^i\otimes d\varphi^k \\ & = \sum_{j=1}^n (\sin\varphi_1\dots\sin\varphi_{j-1})^2d\varphi_j^2. \end{align}

The non-zero Christoffel symbols are given by (modulo permutations of lower indices),

\begin{align} \begin{cases} \Gamma^i_{jj} & = -\cot\varphi_i(\sin\varphi_i\cdots\sin\varphi_{j-1})^2, & i < j, \\ \Gamma^j_{ij} & = \cot\varphi_i & i < j. \end{cases} \end{align}

From this there's a few ways to compute the Riemann curvature tensor:

1. Compute the sectional curvature (a constant) and recover using Theorem 6.5 (and subsequent results) of Chapter 6 of Kuhnel.
2. Compute as in Petersen's Chapter 3.2 using formulas of ambient geometry (second fundamental form).
3. View the sphere as a symmetric space and apply the formula \(R(X,Y)Z = -[[X, Y], Z]\).
4. View the sphere as a wrapped product \(S^n \cong S^{n-1}\times_{\sin\theta_n} I\) and apply related formulas.
5. Direct computation using the formula for \(R^i_{jkl}\) in terms of the Christoffel symbols.

We proceed with a direct computation.² We list all the non-zero cases modulo permutations of the lower indices of \(\Gamma\):

\begin{align} \begin{cases} \partial_i \Gamma^i_{jj} & = (2\sin^2\varphi_i - 1)(\sin\varphi_{i+1}\cdots\sin\varphi_{j-1})^2, & i < j, \\ \partial_j \Gamma^i_{kk} & = -2\cot\varphi_i\cot\varphi_j (\sin\varphi_i\cdots\sin\varphi_{k-1})^2, & i < j < k, \\ \partial_i \Gamma^j_{ij} & = -\csc^2\varphi_i, & i < j. \end{cases} \end{align}

Furthermore,

\begin{equation} \begin{aligned} \begin{cases} (\Gamma^i_{jj})(\Gamma^j_{kk}) & = \cot\varphi_i\cot\varphi_j (\sin\varphi_i\cdots\sin\varphi_{k-1})^2, & i < j < k \\ (\Gamma^i_{kk})(\Gamma^k_{jk}) & = -\cot\varphi_i\cot\varphi_j (\sin\varphi_i\cdots\sin\varphi_{k-1})^2, & i, j < k \\ (\Gamma^j_{ij})(\Gamma^j_{kk}) & = -\cot\varphi_i\cot\varphi_j (\sin\varphi_j\cdots\sin\varphi_{k-1})^2, & i < j < k \\ (\Gamma^k_{ik})(\Gamma^k_{jk}) & = \cot\varphi_i\cot\varphi_j, & i, j < k. \end{cases} \end{aligned} \end{equation}

We have from \eqref{eq:curvature-coefficients} that:

The \(n\)-dimensional real projective space \(\mathbb{PR}^n\) is the real Grassmannian of lines in \(\mathbb{R}^{n+1}\), i.e. \(\Gr_{\mathbb{R}}(n+1,1)\). Equivalently it's the quotient \(S^n/{\mathbb{Z}_2}\) of the antipodal action. As a CW complex, it is given as a quotient of \(e^0\sqcup e^1\sqcup \cdots \sqcup e^n\): the idea is that inductively \(\mathbb{PR}^n\) is obtained from \(\mathbb{PR}^{n-1}\) by attaching an \(n\)-cell \(e^n\) via the map

There are only two such bundles, \(S^1\times \mathbb{R}\) and the Möbius strip: By the classification theorem for bundles, since the maximal compact subgroup of \(\GL(1, \mathbb{R})\) is \(O(1)\), we have that they are classified by \(H^1(S^1; \mathbb{Z}_2)\) which is simply \(\mathbb{Z}_2\), and hence there are only two such bundles. A distinguished element that disambiguates between the two is the first Stiefel-Whitney class.

Consider \(\mathbb{R}^3\) with Christoffel symbols defined by \(-\Gamma^z_{yx} = \Gamma^z_{xy} = 1\), \(-\Gamma^y_{zx} = \Gamma^y_{xz} = -1\), and \(-\Gamma^x_{zy} = \Gamma^x_{yz} = 1\), and all others set to zero. The geodesic equation is given by:

\begin{align} \nabla_{\partial_t} \dot{\gamma} = 0 \implies \ddot{\gamma}^i - \Gamma^i_{jk}\dot{\gamma}^j\dot{\gamma}^k = 0. \end{align}

Because of the antisymmetry in the lower indices, the geodesic equation ends up being \(\ddot{\gamma}^i = 0\), which is the same for the Levi-Civita connection of \(\mathbb{R}^3\). Now consider the parallel translation equation for a vector field \(V\) over \(\gamma\):

\begin{align} \nabla_{\partial_t} V = 0 \implies \dot{\gamma}^j\partial_j V^i - \Gamma^i_{jk}\dot{\gamma}^jV^k = 0. \end{align}

This equation may be rewritten as \(\dot{V} = MV\) where \(M\) is skew-symmetric; in particular, \(\dot{V} = \dot{\gamma}\times V\). The solution will preserve \(\|V\|\) as \(\frac{d}{dt}\|V\|^2 = 2V\cdot \dot{V} = 0\) and hence \(V(t) = R(t)V(0)\) for some \(R(t)\in SO(3)\) solving \(\dot{R} = MR\) with \(M\in\mathfrak{so}(3)\). Now if \(\gamma\) is not just any general loop but a geodesic, according to the previous observation, it will be an arclength parametrized straight line, so \(\dot{\gamma}\) is constant (and so is \(M\)), and therefore \(R = \exp(tM)\) and \(V(t)\) will rotate about \(\dot{\gamma}\) with angle \(t\).

Let \(\operatorname{Prin}(X, G)\) be the isomorphic classes of principal \(G\)-bundles over a paracompact topological space \(X\).

1. There exists a bijection with the first Čech cohomology group \(H^1(X; G)\).
2. There exists a bijection³ with the homotopy classes of maps \(X \to BG\) which often reduces further. This bijection works as follows, given any map \(f : X \to BG\), we may form the principal \(G\)-bundle \(P \coloneqq f^*(EG)\). The bundle \(EG \to BG\) is called the classifying space of principal \(G\)-bundles. Its construction as an infinite join \(G*G*\cdots\) is due to Milnor.⁴

The easiest calculation comes from

\begin{align} [\nabla_k, \nabla_l], \end{align}

where, in the above expression, only the terms \(\nabla \Gamma\) survive:

\begin{equation} \label{eq:curvature-coefficients} \begin{aligned} R^i_{jkl} & = \nabla_l \Gamma^i_{jk} - \nabla_k \Gamma^i_{jl} \\ & = \partial_l\Gamma^i_{jk} - \partial_k\Gamma^i_{jl} + \Gamma^i_{mk}\Gamma^m_{jl} - \Gamma^i_{ml}\Gamma^m_{jk}. \end{aligned} \end{equation}