Abraham D. Smith

The study of shapes using properties like length, angle, area, etc.

$$\simeq$$
$$\simeq$$
$$\simeq$$
$$\simeq$$

Allowed by rotation:

$A=ab$

$c^2 = 4 \left(\frac12 ab\right) + (b-a)^2 = 2ab + b^2 -2ab + a^2 = a^2 + b^2$

Comparing different sizes is useful.

$\pi = \frac{A}{R^2} = \frac{C}{2R}$
$$\overset{?}{\simeq}$$ $$\simeq$$ $$\overset{?}{\simeq}$$

the study of shapes using properties that are unchanged by rotations (and maybe reflection) (and maybe scaling)

 can get rotate rotate, reflect rotate, reflect, scale angle, length, orient'n angle, length angle $$SO(\mathbb{R}^2)$$ $$O(\mathbb{R}^2)$$ $$CO(\mathbb{R}^2)$$
 can get rotate rotate, reflect rotate, reflect, scale angle, length, orient'n angle, length angle $$SO(\mathbb{R}^2)$$ $$O(\mathbb{R}^2)$$ $$CO(\mathbb{R}^2)$$

Other shapes give other geometries!

$x^2 + y^2 = 1$

$xy = 1$

$xy = 1, \text{(squish/stretch,but preserve area)}$
In higher dimensions, get more intricate transformations.

Geometry is...

the study of shapes using properties that are unchanged by a family of transformations, $$G$$ usually given as a matrix group.

Actions Name Formal Definition
Rotations $$SO(\mathbb{R}^n)$$ $$=\{ A : A^TA=I, \det(A)=1\} \subset GL$$
Rotations & Reflections $$O(\mathbb{R}^n)$$ $$=\{ A : A^TA =I, \det(A)=\pm 1\} \subset GL$$
Rotations & Reflections & Scalings $$CO(\mathbb{R}^n)$$ $$=\{ rA : A \in O(n), r \in \mathbb{R} \} \subset GL$$
Volume-preserving $$SL(\mathbb{R}^n)$$ $$=\{ A : \det(A)=1\} \subset GL$$
All (linear) $$GL(\mathbb{R}^n)$$ $$=\{ A : \det(A)\neq 0\}$$
$$\cdots$$ G $$=\{ A : \cdots~\text{some condition}~\cdots \} \subset GL$$

Defining local geometry

General Approach:

• Work on $$K$$ of dim $$n$$
• Each $$p \in K$$ has a $$T_p K \cong \mathbb{R}^n$$
• Choose a shape $$\Sigma_p \subset T_p K$$
• or transformations $$G_p$$ acting on $$T_p K$$.

This is a $$G$$-Structure on $$K$$.

Have and . Choose or .

What is a differential equation?

We know algebraic equations:

• equalities with unknown numbers
• “solve” means “find all numbers that make it true”
• Examples: $$x^2 + y^2 = 1$$ or $$xy=1$$
• Solutions: or

Differential equations:

• equalities with unknown functions
• “solve” means “find all functions that make it true”
• Example: $\frac{\mathrm{d}y}{\mathrm{d}x} = (y-x)$
• $$\{ f:\mathbb{R}\to\mathbb{R} ~\text{such that}~ f'(x) = f(x)-x \ \text{for all x} \}$$ = ???
$\mathrm{d}y = (y-x) \mathrm{d}x$
 $$(7.5,6.5) \in K$$ $$\{ (\mathrm{d}x,\mathrm{d}y), \mathrm{d}y = -1 \mathrm{d}x\}$$
$\mathrm{d}y = (y-x) \mathrm{d}x$
 $$p \in K$$ $$\Sigma_p \subset T_p K$$
$\mathrm{d}y = (y-x) \mathrm{d}x$
 $$\{(x,f(x))\} \subset K$$ $$\{\mathrm{d}y=f'(x)\mathrm{d}x\} \subset T_p K$$

General Approach:

• Write a diff.eq. on $$K$$ of dim $$n$$
• Each $$p \in K$$ has a $$T_p K \cong \mathbb{R}^n$$
• diff.eq. specifies a shape $$\Sigma_p \subset T_p K$$
• so transformations $$G_p$$ act on $$T_p K$$ to preserve $$\Sigma_p$$.

A differential equation gives a $$G$$-Structure on $$K$$.

Have and . Get , so .

The Geometry of Geometry

$\{ Ax^2 + Bxy + Cy^2 +Dx + Ey = F \}$

Compute $$\Delta = B^2 - 4AC$$:

 $$x^2+y^2=1$$ $$xy=1$$ $$x^2=y$$

Algebraic equations come in algebraic families.

$$\{ Ax^2 + Bxy + Cy^2 +Dx + Ey = F \} =$$

Thanks!

Document Credits:

• Desargues theorem image: public domain via Wikipedia
• Earth image: NASA
• p67 images: ESA
• slide tech: reveal.js, MathJax, TikZ/TeXLive

All other content: Abraham D. Smith, 2015. www.curieux.us/abe/talks/local_geometry_and_differential_equations