The World Is Not Flat: An Introduction to Modern Geometry
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1 The World Is Not Flat: An to The University of Iowa September 15, 2015
2 The story of a hunting party
3 The story of a hunting party What color was the bear?
4 The story of a hunting party
5 Overview Gauss and all his friends Riemann s revolution
6 Euclid of Alexandria Active around 300 B.C. Authored several mathematical texts Transformed geometry into a rigorous discipline
7 Elements The five axioms: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 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 not intersecting the given line.
8 The fathers of non-euclidean Geometry Carl Frederich Gauss ( ) Nikolai Lobachevsky ( ) János Bolyai ( )
9 A family of parallel postulates Given any straight line and a point off the given line, Euclidean: there exists exactly one line through the point not intersecting the given line. Elliptic: there exist no lines through the point not intersecting the given line. Hyperbolic: there exist at least two lines through the point not intersecting the given line. I created a new, different world out of nothing. - Bolyai, in a letter to his father
10 Straight lines What makes a curve in a surface a straight line?
11 Straight lines What makes a curve in a surface a straight line? It will (at least locally) represent the shortest distance between two points. It will have zero acceleration in the surface.
12 Straight lines What makes a curve in a surface a straight line? It will (at least locally) represent the shortest distance between two points. It will have zero acceleration in the surface. The generalization of a straight line in a non-euclidean space is called a geodesic.
13 A family of parallel postulates Euclidean: Given any straight line and a point off the given line, there exists exactly one line through the point not intersecting the given line.
14 A family of parallel postulates Euclidean: Given any straight line and a point off the given line, there exists exactly one line through the point not intersecting the given line.
15 A family of parallel postulates Euclidean: Given any straight line and a point off the given line, there exists exactly one line through the point not intersecting the given line.
16 A family of parallel postulates Elliptic: Given any straight line and a point off the given line, there exists no lines through the point not intersecting the given line.
17 A family of parallel postulates Elliptic: Given any straight line and a point off the given line, there exists no lines through the point not intersecting the given line.
18 A family of parallel postulates Elliptic: Given any straight line and a point off the given line, there exists no lines through the point not intersecting the given line.
19 A family of parallel postulates Hyperbolic: Given any straight line and a point off the given line, there exists at least two lines through the point not intersecting the given line.
20 A family of parallel postulates Hyperbolic: Given any straight line and a point off the given line, there exists at least two lines through the point not intersecting the given line.
21 A family of parallel postulates Hyperbolic: Given any straight line and a point off the given line, there exists at least two lines through the point not intersecting the given line.
22 Curve curvature How can we formally describe curvature?
23 Curve curvature How can we formally describe curvature? To calculate the curvature of a parametric curve, we find its second derivative.
24 Curve curvature How can we formally describe curvature? To calculate the curvature of a parametric curve, we find its second derivative. Curvature is typically represented by the Greek letter κ. zero κ low κ high κ
25 Curve curvature Circles have uniform curvature; specifically, κ = 1 r. κ = 1 2 κ = 1 κ = 3
26 Curve curvature In some sense, the curvature of a curve at a point is the curvature of the largest circle one can make tangent to the curve at that point.
27 Gauss curvature There are several ways to determine the curvature at a point on 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 the surface in some curve. 4. Select the two curves with the most different curvatures at the point. These are called the principal curvatures. 5. Multiply their curvatures and choose the appropriate sign to obtain the surface s curvature at that point.
28 Gauss curvature R
29 Gauss curvature R
30 Gauss curvature R
31 Gauss curvature K(p) = 0 1 r = 0
32 Gauss curvature R
33 Gauss curvature R
34 Gauss curvature R
35 Gauss curvature K(p) = 1 r 1 r = 1 r 2 > 0
36 Gauss curvature R
37 Gauss curvature R
38 Gauss curvature R
39 Gauss curvature K(p) = k 1 k 2 < 0
40 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
41 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
42 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
43 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
44 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
45 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
46 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
47 Gauss curvature Thus, we can separate curvature at a point on a surface into three cases: Zero: the surface can be perfectly flattened. Positive: the surface will rip when flattened. Negative: the surface will fold when flattened.
48 The Gauss-Bonnet Theorem The Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties.
49 The Gauss-Bonnet Theorem The Gauss-Bonnet Theorem illustrates a connection between topological and geometric properties. Theorem: If M is a compact, two-dimensional, Riemannian manifold with boundary M, then K da + k g ds = 2πχ(M), M M Where K is the Gauss curvature on M, k g is the geodesic curvature on M, and χ(m) is the Euler characteristic of M.
50 Extensions and applications: Spacetime models Medical imaging Industrial engineering Mathematical bridge-builder
51 References Mlodinow, Leonard (2001). Window. New York, NY: The Free Press. Oprea, John (2007). Differential Geometry and its Applications. Washington, DC: The Mathematical Association of America. Stewart, Ian (2001). Flatterland. Cambridge, MA: Perseus Publishing. Wallace, Edward and West, Stephen (2004). Roads to Geometry. Upper Saddle River, NJ: Pearson Education.
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