Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan Curvature for all Matthias Kawski Dept. of Math & Statistics Arizona State University Tempe, AZ. U.S.A.

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 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 geodesics and Gauss curvature

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

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan What is the role of curvature? Key concept: Linearity Key concept: Derivative Key concept: Curvature quantifies “distance” from being linear “can be solved”, linear algebra, linear ODEs and PDEs, linear circuits, mechanics approximation by a linear object

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

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan Example: Graph of exponential function very straight, one gently rounded corner x Reparameterization by arc-length ?

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan Example: Graph of hyperbolic cosine very straight, one gently rounded corner s Almost THE ONLY nice nontrivial example

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan From physics to geometry Example of curves in the plane straightforward formulas are a means only objective: understanding, and new questions, Physics: parameterization by “time” components of acceleration parallel and perpendicular to velocity Geometry: parameterization by arc-length - what can be done w/ CAS? acc_2d_curv.mws

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

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

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

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

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan The Gauss map and Gauss curvature geodesics.mws

Matthias Kawski “Curvature for everyone” Asian Technology Conf. Mathematics Taiwan 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 1 st time w/ CAS –INVITES for true exploration & discovery