the world is not flat: an introduction to modern geometry · 2018-09-06 · modern geometry richard...
TRANSCRIPT
![Page 1: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/1.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The World Is Not Flat:An Introduction to Modern Geometry
Richard G. Ligo
The University of Iowa
September 15, 2015
![Page 2: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/2.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The story of a hunting party
What color was the bear?
![Page 3: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/3.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The story of a hunting party
What color was the bear?
![Page 4: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/4.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The story of a hunting party
![Page 5: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/5.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Overview
I Introduction
I Euclid’s foundations
I Gauss and all his friends
I Riemann’s revolution
I Conclusion
![Page 6: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/6.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Euclid of Alexandria
I Active around 300 B.C.
I Authored several mathematical texts
I Transformed geometry into a rigorous discipline
![Page 7: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/7.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Euclid’s Elements
The five axioms:
1. A straight line segment can be drawn joining any twopoints.
2. Any straight line segment can be extended indefinitelyin a straight line.
3. Given any straight line segment, a circle can be drawnhaving the segment as radius and one endpoint ascenter.
4. All right angles are congruent.
5. Given any straight line and a point off the given line,there exists exactly one line through the point notintersecting the given line.
![Page 8: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/8.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The fathers of non-Euclidean Geometry
I Carl Frederich Gauss (1777-1855)
I Nikolai Lobachevsky (1792-1856)
I Janos Bolyai (1802-1860)
![Page 9: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/9.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Given any straight line and a point off the given line,
Euclidean: there exists exactly one line through the pointnot intersecting the given line.
Elliptic: there exist no lines through the point notintersecting the given line.
Hyperbolic: there exist at least two lines through the pointnot intersecting the given line.
“I created a new, different world out of nothing.”- Bolyai, in a letter to his father
![Page 10: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/10.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Straight lines
What makes a curve in a surface a “straight line”?
I It will (at least locally) represent the shortest distancebetween two points.
I It will have zero acceleration in the surface.
The generalization of a straight line in a non-Euclideanspace is called a geodesic.
![Page 11: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/11.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Straight lines
What makes a curve in a surface a “straight line”?
I It will (at least locally) represent the shortest distancebetween two points.
I It will have zero acceleration in the surface.
The generalization of a straight line in a non-Euclideanspace is called a geodesic.
![Page 12: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/12.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Straight lines
What makes a curve in a surface a “straight line”?
I It will (at least locally) represent the shortest distancebetween two points.
I It will have zero acceleration in the surface.
The generalization of a straight line in a non-Euclideanspace is called a geodesic.
![Page 13: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/13.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Euclidean: Given any straight line and a point off the givenline, there exists exactly one line through the point notintersecting the given line.
![Page 14: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/14.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Euclidean: Given any straight line and a point off the givenline, there exists exactly one line through the point notintersecting the given line.
![Page 15: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/15.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Euclidean: Given any straight line and a point off the givenline, there exists exactly one line through the point notintersecting the given line.
![Page 16: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/16.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Elliptic: Given any straight line and a point off the givenline, there exists no lines through the point not intersectingthe given line.
![Page 17: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/17.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Elliptic: Given any straight line and a point off the givenline, there exists no lines through the point not intersectingthe given line.
![Page 18: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/18.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Elliptic: Given any straight line and a point off the givenline, there exists no lines through the point not intersectingthe given line.
![Page 19: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/19.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Hyperbolic: Given any straight line and a point off the givenline, there exists at least two lines through the point notintersecting the given line.
![Page 20: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/20.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Hyperbolic: Given any straight line and a point off the givenline, there exists at least two lines through the point notintersecting the given line.
![Page 21: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/21.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
A family of parallel postulates
Hyperbolic: Given any straight line and a point off the givenline, there exists at least two lines through the point notintersecting the given line.
![Page 22: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/22.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Curve curvature
How can we formally describe curvature?
To calculate the curvature of a parametric curve, we find itssecond derivative.
Curvature is typically represented by the Greek letter κ.
zero κ low κ high κ
![Page 23: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/23.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Curve curvature
How can we formally describe curvature?
To calculate the curvature of a parametric curve, we find itssecond derivative.
Curvature is typically represented by the Greek letter κ.
zero κ low κ high κ
![Page 24: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/24.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Curve curvature
How can we formally describe curvature?
To calculate the curvature of a parametric curve, we find itssecond derivative.
Curvature is typically represented by the Greek letter κ.
zero κ low κ high κ
![Page 25: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/25.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Curve curvature
Circles have uniform curvature; specifically, κ = 1r .
κ = 12 κ = 1 κ = 3
![Page 26: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/26.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Curve curvature
In some sense, the curvature of a curve at a point is thecurvature of the largest circle one can make tangent to thecurve at that point.
![Page 27: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/27.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
There are several ways to determine the curvature at a pointon a surface, but the following process is the most intuitive:
1. Construct the tangent plane to the surface at the point.
2. Consider every plane perpendicular to the tangent plane.
3. Each of these perpendicular planes will intersect thesurface in some curve.
4. Select the two curves with the “most different”curvatures at the point. These are called the principalcurvatures.
5. Multiply their curvatures and choose the appropriatesign to obtain the surface’s curvature at that point.
![Page 28: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/28.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 29: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/29.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 30: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/30.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 31: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/31.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
K (p) = 0 · 1r = 0
![Page 32: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/32.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 33: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/33.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 34: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/34.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 35: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/35.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
K (p) = 1r ·
1r = 1
r2> 0
![Page 36: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/36.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 37: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/37.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 38: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/38.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
R
![Page 39: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/39.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
K (p) = −k1 · k2 < 0
![Page 40: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/40.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 41: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/41.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 42: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/42.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 43: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/43.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 44: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/44.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 45: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/45.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 46: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/46.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 47: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/47.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Gauss curvature
Thus, we can separate curvature at a point on a surface intothree cases:
Zero: the surface can be perfectly flattened.
Positive: the surface will rip when flattened.
Negative: the surface will fold when flattened.
![Page 48: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/48.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The Gauss-Bonnet Theorem
The Gauss-Bonnet Theorem illustrates a connection betweentopological and geometric properties.
Theorem: If M is a compact, two-dimensional, Riemannianmanifold with boundary ∂M, then∫
MK dA +
∫∂M
kg ds = 2πχ(M),
Where K is the Gauss curvature on M, kg is the geodesiccurvature on ∂M, and χ(M) is the Euler characteristic of M.
![Page 49: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/49.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
The Gauss-Bonnet Theorem
The Gauss-Bonnet Theorem illustrates a connection betweentopological and geometric properties.
Theorem: If M is a compact, two-dimensional, Riemannianmanifold with boundary ∂M, then∫
MK dA +
∫∂M
kg ds = 2πχ(M),
Where K is the Gauss curvature on M, kg is the geodesiccurvature on ∂M, and χ(M) is the Euler characteristic of M.
![Page 50: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/50.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
Conclusion
Extensions and applications:
I Spacetime models
I Medical imaging
I Industrial engineering
I Mathematical bridge-builder
![Page 51: The World Is Not Flat: An Introduction to Modern Geometry · 2018-09-06 · Modern Geometry Richard G. Ligo Introduction Euclid’s foundations Parallel Notions Beautiful curves Conclusion](https://reader033.vdocuments.us/reader033/viewer/2022050110/5f4784efdd6cf3004a010d3c/html5/thumbnails/51.jpg)
Modern Geometry
Richard G. Ligo
Introduction
Euclid’sfoundations
Parallel Notions
Beautiful curves
Conclusion
References
I Mlodinow, Leonard (2001). Euclid’s Window. NewYork, NY: The Free Press.
I Oprea, John (2007). Differential Geometry and itsApplications. Washington, DC: The MathematicalAssociation of America.
I Stewart, Ian (2001). Flatterland. Cambridge, MA:Perseus Publishing.
I Wallace, Edward and West, Stephen (2004). Roads toGeometry. Upper Saddle River, NJ: Pearson Education.