# Frankel 1.1 Problems

## (1)

The locus described by $$x^2 + y^2 - z^2 = c$$ is a 2 dimensional submanifold of $$\mathbb{R}^3$$ for $$c < 0$$ and $$c = 0$$ but not for $$c > 0$$. This can be seen by investigating the functional dependence

$$z = z(x,y) = \sqrt{x^2+y^2-c}.$$

For $$c < 0, z(x,y)$$ is differentiable over it's entire domain, by definition forming a 2D submanifold of $$\mathbb{R}^3$$. For $$c = 0, z(x,y)$$ is differentiable everywhere except at $$x = y = 0$$, and therefore forms a submanifold if and only if the origin is omitted. For $$c > 0, z(x,y)$$ is defined on the reals only for $$x^2+y^2 \ge c$$ and differentiable only for $$x^2+y^2 > c$$, so it only forms a submanifold for $$x^2+y^2 > c$$.

## (2)

SO($$n$$) is defined to be the set of all orthogonal $$n \times n$$ matrices $$x$$ with $$\det x = 1$$. The discussion on SO(3) in the book easily generalizes to show that SO($$n$$) is an $$n$$-dimensional submanifold of $$\mathbb{R}^{n^2}$$.

## (3)

The special linear group is the set

$$\mathrm{SL}(n) := \{ n\times n \text{ real matrices } x | \det x = 1 \}.$$

Clearly SL($$n$$) is a subset of the set of $$n \times n$$ matrices, which itself is isomorphic to $$\mathbb{R}^{n^2}$$. The condition on the determinant restricts SL($$n$$) to be a space of dimension $$n^2 - 1$$, but to show it's a submanifold we must demonstrate that $$\det x = 1$$ defines a differentiable map. To do this we first expand the determinant around the $$j$$'th row,

$$\det x = \sum_{i=1}^n (-1)^{i+j} M_{ij}x_{ij}.$$

Here, $$M_{ij}$$ is the minor.  Taking the derivative with respect to $$x_{kj}$$,

$$\frac{\partial}{\partial x_{kj}}\det x = \sum_{i=1}^n (-1)^{i+j}M_{ij}\frac{\partial x_{ij}}{\partial x_{kj}} = (-1)^{k+j}M_{kj}.$$

As long a $$M_{kj}$$ is non-zero, the implcit function theorem tells us that $$x_{kj}$$ can be expressed as a differentiable function of the other components. Therefore, by definition, SL($$n$$) is a $$n^2 - 1$$ submanifold of $$\mathbb{R}^{n^2}$$.