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Title: Differential Geometry Intro

1
Differential Geometry Intro
2
Catalog Description
• MAT 4140 3 hours On-Demand
• Differential Geometry
• Bulletin Description This is an introductory
course in the differential geometry of curves and
surfaces in space, presenting both theoretical
and computational components, intrinsic and
extrinsic viewpoints, and numerous applications.
The geometry of space-time will also be
considered. Prerequisite MAT 2130.

3
Background
• Geometry Riemannian Geometry PhD
• Physics Special relativity with Ralph Alpher,
one of the creators of the big bang.
• Research Geometry of Orbifolds, Mathematics and
Society Popular Culture, Women Minorities

4
Course Goals
• To develop geometric problem solving skills and
3-D spatial visualization skills.
• To develop a greater appreciation for connections
between various disciplines of mathematics,
including geometry, linear algebra, complex
analysis, and differential equations, along with
an introduction to these subjects as they apply
to differential geometry.
• To understand the importance of differential
geometry in various scientific fields, including
physics.

5
Topics
• Geometry of curves in space, including Frenet
formulas
• Theory of surfaces, including curvature,
geodesics, and metrics
• Geometry of space-time and applications to
general relativity (as time allows)

6
Culmination of Early Ideas
• At a very early age, children develop a very rich
visual intelligence' in terms of perception and
experiences. They have questions and lots of
these questions and explorations can be connected
to geometry if we use the right types of physical
and visual presentations. They have developed
advanced skills for which the precise vocabulary
is 3-D differential geometry and differential
topology 28 (Hoffmann, 1998), 36 (Koenderink,
1990)...we should connect with these abilities.
(Whiteley, 1999)

7
Rich History and New Relevance
• Machinery Design
• Classification of Spaces
• Fundamental Forces in the Universe
• Study of DNA
• Medical Imaging
• Computer Graphics

8
Approaches
• Intuitive
• Calculable
• Useful
• Interdisciplinary unification of many topics
including geometry, spatial visualization,
calculus, linear algebra, differential equations,
and complex variables along with various topics
from the sciences, including physics

9
Smokestack Problem
• Frank Morgan received a call from a company
constructing a huge smokestack, which required
the attachment of a spiraling strip or stake for
structural support.

10
• Strake cut out of flat metal pieces
• What inner radius would make it fit best?

11
Soap Bubbles
• Pressure versus surface tension
• What shape does it take on, and why?

12
Inconsistency in Mercurys Orbit
• Newton used solar gravitational attraction and
calculus to explain Keplers elliptical planetary
orbits.
• The orbit rotated (precessed) at an unexpected
rate
• General relativity and the geometry of space-time

13
Curves
• Some of the results were motivated by the desire
for more accurate clocks
• I have been occupied with a new discovery... In
order to make my clock even more exact.... What,
however, I never had expected I would discover, I
have now hit upon, the undoubtedly true shape of
curves... I determined it by geometric
reasoning. (C. Huygens December 1659)