Local Geometry and Differential Equations

Abraham D. Smith

San José State University

February 18, 2015

What is Geometry?

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

\( \simeq \)
\( \simeq \)
\( \simeq \)
\( \simeq \)

Allowed by rotation:


\[ A=ab \]

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

What is Geometry?

The study of shapes using properties
that are unchanged by rotation.

The study of shapes using properties
that are unchanged by rotation.

But, what about this picture?

Maybe allow reflections, too.

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

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 \( =\{ A : \cdots~\text{some condition}~\cdots \} \subset GL\)

Is this enough to describe our world?

Lines might seem to be the shortest route nearby

but not for long.

On Earth, lines are nonsense.

In the Universe, even light can't travel straight.

How can we reconcile these?

We need to make geometry local

We need to allow geometry to change.

You are heading for a comet.

You are closing toward a comet.

You are approaching a comet.

You are landing on a comet.

You are touching a comet.

You have landed on comet \(K\) at location \(p\).

Near \(p\), comet \(K\) looks like \(\mathbb{R}^2\).
Can explore in two dimensions.

Near \(p\), comet \(K\) looks like \(\mathbb{R}^2\).
Can make a circle.

Near \(p\), comet \(K\) looks like \(\mathbb{R}^2\).
Can reach a distance and measure angles.

Near \(p\), comet \(K\) looks like \(\mathbb{R}^2\).
Can rotate (and reflect).

Defining local geometry

\(K\) is made of points like \(p\).

Each \( p \in K\) has a tangent plane \( T_p K\), approximates region near \(p\) by \( \mathbb{R}^2\)

Draw a circle in each \(T_p K\).

Each \(T_p K\) can be rotated/reflected individually.

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 \( T_p K\).
    • (get the other for free)

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.

Algebraic equations come in algebraic families.

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

Differential equation \(=\) a choice of an algebraic equation in each \( T_pK \)

What if we choose \(\Delta=B^2-4AC = 0\)?

Differential equations come in differential families!

Differential equations come in differential families!

What are the sub-types?

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