MTH-696A: Topics in Geometric Mechanics Assignment 4 (by Dr. Leo Butler)

Navigation

MTH-696a

Course outline

Assignments & Solutions

Schedule

Lecture Notes

Sources

Home Page of Leo Butler

Let $\Liegp{G} = \Orth{n;\R}$ and let $\liealg{g} = \orth{n;\R}$ be the Lie algebra of $\Liegp{G}$.
1. Show that $\liealg{g}$ can be identified with the skew-symmetric real $n \times n$ matrices with Lie bracket $\liebracket{x}{y} = xy-yx$.
2. Show that the Cartan-Killing trace form $\kappa(x, y) := -\mathrm{Tr}\, (xy)$ is positive definite on $\liealg{g}$.
Let us continue with the notation of the previous question. Let $\Liegp{G}$ act on $V=\Mat{n}{\R}$ by \begin{align*} g \cdot x &:= g x && g \in \Liegp{G}, x \in V. \end{align*}
1. Show that this defines a smooth action of $\Liegp{G}$ on $V$.
2. Let $$\mathbf{x}_k = \mathbf{diag}(\underbrace{1, \ldots, 1}_{k \text{ times}},\underbrace{0, \ldots, 0}_{n-k \text{ times}})$$ and let $V_k = \orbit{\Liegp{G}}{\mathbf{x}_k}$. Show that the columns of each $x \in V_k$ form an orthonormal basis of a $k$-dimensional subspace of $\R^n$. Conversely, for each orthonormal basis $\hat{x} = [x_1, \ldots, x_k]$ of a $k$-dimensional plane in $\R^n$, there is a unique $x \in V_k$ whose columns provide the same orthonormal basis. Conclude that $V_k$ can be identified with the set of all orthonormal bases of all $k$-dimensional subspaces of $\R^n$. [This manifold is known as the Stiefel manifold of $k$-frames in $\R^n$.]
3. Show that $V_1$ is diffeomorphic to $\sphere{n-1}$ the unit sphere in $\R^n$.
4. Show that $V_2$ is diffeomorphic to the unit tangent bundle of $\sphere{n-1}$.
Let $e_1, \ldots, e_n$ be the standard basis of $V=\R^n$ and let $f_1, \ldots, f_n$ be the dual basis of $V^*$. Let $A \in \hom{V}{V}$ be a linear transformation.
1. For an ordered subset $I \subset \set{1, \ldots, n}$, of cardinality $k$, let $e_I = e_{i_1} \wedge \cdots \wedge e_{i_k}$. For $k=n-1$, note that there is a bijection between the subsets $I$ and elements $i$ ($\set{i}=I^c$). Let $\phi_i = e_I$, using this bijection. Relative to the basis $\phi_1, \ldots, \phi_n$, show that he induced linear transformation $\extalg{n-1}{A} : \extalg{n-1}{V} \to \extalg{n-1}{V}$ has the matrix \begin{align*} C &= \begin{bmatrix} C_{11} & \cdots & C_{1n} \\ \vdots & & \vdots \\ C_{n1} & \cdots & C_{nn} \end{bmatrix} \end{align*} where $C_{ij}$ is the determinant of the $(i, j)$ minor of the matrix of $A$. Let $D_{ij} = (-1)^{i+j}C_{ji}$.
2. Show that $AD = (\det A) I$.