Curvature for all

1. Curvature for all Matthias Kawski Dept. of Math & Statistics Arizona State University Tempe, AZ. U.S.A. http://math.asu.edu/~kawskikawski@asu.edu

2. 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 http://math.asu.edu/~kawskikawski@asu.edu

3. 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/~kawskikawski@asu.edu

4. What is the role of curvature? Key concept: Linearity “can be solved”, linear algebra, linear ODEs and PDEs, linear circuits, mechanics Key concept: Derivative approximation by a linear object Key concept: Curvature quantifies “distance” from being linear http://math.asu.edu/~kawskikawski@asu.edu

5. 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/~kawskikawski@asu.edu

6. Example: Graph of exponential function very straight, one gently rounded corner Reparameterization by arc-length ? x http://math.asu.edu/~kawskikawski@asu.edu

7. Example: Graph of hyperbolic cosine very straight, one gently rounded corner Almost THE ONLY nice nontrivial example s http://math.asu.edu/~kawskikawski@asu.edu

8. From physics to geometry acc_2d_curv.mws • 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? http://math.asu.edu/~kawskikawski@asu.edu

9. Curvature as complete invariant serret.mws • 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 spaceexplorations  new questions, discoveries!!! http://math.asu.edu/~kawskikawski@asu.edu

10. 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/~kawskikawski@asu.edu

11. Curvature of surfaces, the beginnings meusnier.mws • 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 http://math.asu.edu/~kawskikawski@asu.edu

12. 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/~kawskikawski@asu.edu

13. The Gauss map and Gauss curvature geodesics.mws http://math.asu.edu/~kawskikawski@asu.edu

14. 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 http://math.asu.edu/~kawskikawski@asu.edu