Objective Parallel transport along a hyperbolic triangle Compare - - PowerPoint PPT Presentation

Hyperbolic Geometry and Parallel Transport in R2

+

Tia Burden1 Vincent Glorioso2 Brittany Landry3 Phillip White2

1Department of Mathematics

Southern University Baton Rouge, LA, USA

2Department of Mathematics

Southeastern Louisiana University Hammond, LA, USA

3Department of Mathematics

University of Alabama Tuscaloosa, AL, USA

SMILE VIGRE Program, July, 2013

Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 1 / 33

Objective

Parallel transport along a hyperbolic triangle Compare angle of initial and final vector Compute area of hyperbolic triangle Compare area and angles of parallel transports of hyperbolic triangles

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Albert Einstein and Hermann Minkowski

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Applications

Complex variables Topology of two and three dimensional manifolds Finitely presented infinite groups Physics Computer science

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Postulate 1

A straight line segment can be drawn joining any two points.

A B

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Postulate 2

Any straight line segment can be extended indefinitely in a straight line

A B C

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Postulate 3

Given any straight line segment, a circle can be drawn having the segment as a radius and one endpoint as center.

A B

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Postulate 4

All right angles are congruent.

A B C D

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Parallel Postulate

Through any given point not on a line there passes exactly one line that is parallel to that given line in the same plane.

A B C

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Parallel Transport

w v

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Parallel Transport About a Euclidean Triangle

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Upper Half-Plane

Definition

Upper half-plane: R2

+ = {(x, y) ∈ R2 : y > 0}.

Complex plane: H2 = {x + iy : x, y ∈ R, y > 0}.

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Geodesic

Definition

Geodesic is a line, curved or straight, between two points such that the acceleration of the line is 0. So given a curve γ defined on an open interval I it must satisfy D

dt dγ dt = 0 ∈ Tγ(t) for all t ∈ I.

Euclidean Geometry Hyperbolic Geometry

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Covariant Derivative

Definition

There is a unique correspondence that associates the vector field DV

dt

along the differentiable curve γ to the vector field V. The vector field V is referred to as the covariant derivative.

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Results from the Covariant Derivative

From the Covariant Derivative we obtain a set of two differential equations, using the curve γ = (γ1(t), γ2(t)) and the vector field V = (f(t), g(t)), that a parallel vector field must satisfy. f ′(t) = γ′

2(t)

γ2(t)f(t) + γ′

1(t)

γ2(t)g(t) g′(t) = γ′

1(t)

γ2(t)f(t) + γ′

2(t)

γ2(t)g(t)

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Original Vector: V0

π 2

1 2

γ1(t) = (t, 1) η π

2 (t) = ( π

2 , t)

γ2(t) = (t, 2) η0(t) = ( π

2 , t)

V0

π 8 Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 16 / 33

Transport Along γ1(t) to V1

π 2

1 2

γ1(t) = (t, 1) η π

2 (t) = ( π

2 , t)

γ2(t) = (t, 2) η0(t) = ( π

2 , t)

V0

π 8

V1

3π 8 Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 17 / 33

Transport Along the Geodesic η π

2 to V2

π 2

1 2

γ1(t) = (t, 1) η π

2 (t) = ( π

2 , t)

γ2(t) = (t, 2) η0(t) = ( π

2 , t)

V0

π 8

V1

3π 8

V2

7π 8 3π 8 Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 18 / 33

Transport Back Along γ2(t) to V3

π 2

1 2

γ1(t) = (t, 1) η π

2 (t) = ( π

2 , t)

γ2(t) = (t, 2) η0(t) = ( π

2 , t)

V0

π 8

V1

3π 8

V2

7π 8 3π 8

V3

3π 8 7π 8 Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 19 / 33

Transport Down the Geodesic η0 to Vf

π 2

1 2

γ1(t) = (t, 1) η π

2 (t) = ( π

2 , t)

γ2(t) = (t, 2) η0(t) = ( π

2 , t)

V0

π 8

V1

3π 8

V2

7π 8 3π 8

V3

3π 8 7π 8

Vf

3π 8 π 4 Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 20 / 33

Area Calculation

To find the area of the rectangle, we use the following integral: A =

• π

2

2

1

dydx y2 =

• π

2

(−1 y

• 2

1

) dx =

• π

2

1 2 dx = 1 2x

• π

2

= π 4

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General Equation

Given the curve c(t) = (t, mt + b), we used Mathematica to find the equation for the parallel vector field: V1(t) = e−

t b+mt

• e

t b+mt − 1

• ,

V2(t) = e−

t b+mt Burden, Glorioso, Landry, White (Southern University ,Southeastern Louisiana University, and University of Alabama) Reyes Project 1 Smile 2013 22 / 33

Line: y = mx + b

Parallel Transport about the line 2t + 1.

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General Parallel Transport

When a vector is being transported around a hyperbolic triangle it maintains the angle with the tangent vectors of the curve on which it is moving.

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Area

Area Equation Step 1

For a hyperbolic triangle with angles α and β and γ = 0, which is an ideal triangle, we get the equation for the area: A = cos β

− cos α

∞ √

1−x2

dydx y2 = π − (α + β)

.5 1 1.5 2 .5 1 1.5 2 2.5

A B AB C D α β

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Area Calculation

For the triangle AB∞ A = cos β

− cos α

∞ √

1−x2

dydx y2 = cos β

− cos α

−1 y

• ∞

√

1−x2 dx

= cos β

− cos α

1 √ 1 − x2 dx = arcsin(x)

• cos β

− cos α

= arcsin(sin(π 2 − β)) − arcsin(− sin(π 2 − α)) = ((π 2 − β) + (π 2 − α)) = π − (α + β)

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Area

Area equation

We can find the area of a general hyperbolic triangle with angles α, β, and γ by subtracting the areas of two ideal hyperbolic triangles: A = π − (α + β + γ)

.5 1 1.5 2 .5 1 1.5 2 2.5

A B X Y C D α β1 γ β2 180 − γ

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Area Calculation

For the triangle ABC A = cos β

− cos α

∞ √

1−x2

dydx y2 − cos β2

− cos(π−γ)

∞ √

1−x2

dydx y2 = cos β

− cos α

−1 y

• ∞

√

1−x2 dx −

cos β2

− cos(π−γ)

−1 y

• ∞

√

1−x2 dx

= cos β

− cos α

1 √ 1 − x2 dx − cos β2

− cos(π−γ)

1 √ 1 − x2 dx = arcsin(x)

• cos β

− cos α − arcsin(x)

• cos β2

− cos(π−γ)

= ((π 2 − β) + (π 2 − α)) − ((π 2 − β2) + (π 2 − γ)) = π − (α + β) − (π − (γ + β2)) = π − (α + β1 + β2) − (π − (γ + β2)) = π − (α + β + γ)

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Area vs. Angle of Initial to Transported Vector

Theorem

The area of any hyperbolic triangle in R2

+ is equal to the angle between

the initial and final transported vectors.

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Non-Normal Hyperbolic Triangle

1 2 3 4 5 6 1 2 3 4

A′ C′ B′ C B A

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What are Fractional Transformations?

A fractional transformation is a special representation of a matrix that is used in Möbius Transformations.

Fractional Transformation

Given matrix M = a b c d

• , where a, b, c, d ∈ R, and the point

p = (x, y). The fractional transformation of p in terms of M is written as: fM(p) = ap + b cp + d = a(x + yi) + b c(x + yi) + d

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Future Studies

Parallel Transport in Different Models of Hyperbolic Geometry

Poincaré Disk Hyperboloid

B A C

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Acknowledgments

We would like to give a big thanks to our graduate student Andrew Holmes for helping us with this project, in addition to Dr. Edgar Reyes for providing the information needed for this project. Also, we would like to thank the SMILE program for allowing us to have this great

• pportunity.

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