http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Curvature for all

Matthias Kawski

Dept. of Math & StatisticsArizona State UniversityTempe, AZ. U.S.A.

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Outline

• The role of curvature in mathematics (teaching)?• Curves in plane: From physics to geometry• Curvature as complete set of invariants

– recover the curve from the curvature (& torsion)

• 3D: Frenet frame. Integrate Serret-formula

• Euler / Meusnier: Sectional curvature• Gauss: simple idea, huge formula

– interplay between geodesicsand Gauss curvature

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Focus• Clear concepts with simple, elegant definitions

• The formulas rarely tractable by hand, yet straightforward with computer algebra

• The objectives are not more formulas but understanding, insight, and new questions!

• Typically this involves computer algebra, some numerics, and finally graphical representations

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

What is the role of curvature?

Key concept: Linearity

Key concept: Derivative

Key concept: Curvaturequantifies “distance” from being linear

“can be solved”, linear algebra, linear ODEs and PDEs, linear circuits, mechanics

approximation by a linear object

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Lots of reasons to study curvature

• Real life applications– architecture, “art”, engineering design,….

– dynamics: highways, air-planes, …

– optimal control: abstractions of “steering”,…

• The big questions– Is our universe flat? relativity and gravitational lensing

• Mathematics: Classical core concept– elegant sufficient conditions for minimality

– connecting various areas, e.g. minimal surfaces (complex …)

– Poincare conjecture likely proven! “Ricci (curvature) flow”

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Example: Graph of exponential function

very straight, one gently rounded corner

x

Reparameterization by arc-length ?

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Example: Graph of hyperbolic cosine

very straight, one gently rounded corner

s

Almost THE ONLY nice nontrivial example

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

From physics to geometry

• Example of curves in the planestraightforward formulas are a means onlyobjective: understanding, and new questions,

• Physics: parameterization by “time”components of accelerationparallel and perpendicular to velocity

• Geometry: parameterization by arc-length - what can be done w/ CAS?

acc_2d_curv.mws

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Curvature as complete invariant

• Recover the curve from the curvature (and torsion)- intuition- usual numerical integration

• For fun: dynamic settings:curvature evolving according to some PDE- loops that “want to straighten out”- vibrating loops in the plane, in space

explorations new questions, discoveries!!!

serret.mws

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Invariants: {Curvature, torsion}

• Easy exercise: Frenet Frame animation – a little tricky:constant speed animation– most effort: auto-scale arrows, size of curve…..

• Recover the curve: integration on SO(3)(“flow” of time-varying vector fields on manifold)

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Curvature of surfaces, the beginnings

• Euler (1760) – sectional curvatures, using normal planes – “sinusoidal” dependence on orientation

(in class: use adaped coordinates )

• Meusnier (1776)– sectional curvatures, using general planes

• BUT: essentially still 1-dim notions of curvature

meusnier.mws

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Gauss curvature, and on to Riemann

• Gauss (1827, dissertation) – 2-dim notion of curvature– “bending” invariant, “Theorema Egregium”– simple definition– straightforward, but monstrous formulas

• Riemann (1854)–intrinsic notion of curvature, no “ambient space” needed

• Connections, geodesics, conjugate points, minimal

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

The Gauss map and Gauss curvaturegeodesics.mws

http://math.asu.edu/~kawski kawski@asu.edu

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 2003

Summary and conclusions

• Curvature, the heart of differential geometry– classical core subject w/ long history

– active modern research: both pure theory and many diverse

applications

– intrinsic beauty, and precise/elegant language

– broadly accessible for the 1st time w/ CAS

– INVITES for true exploration & discovery