Curvature for all

1
Curvature for all

• Matthias Kawski
• Dept. of Math Statistics
• Arizona State University
• Tempe, AZ. U.S.A.

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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

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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

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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
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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

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Example Graph of exponential function very
straight, one gently rounded corner
Reparameterization by arc-length ?
x
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Example Graph of hyperbolic cosine very
straight, one gently rounded corner
Almost THE ONLY nice nontrivial example
s
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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?

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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!!!

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Invariants Curvature, torsion

• Easy exercise Frenet Frame animation
• a little trickyconstant 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)

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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

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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

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The Gauss map and Gauss curvature
geodesics.mws
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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