talk 20040160 poincare

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PoincaresConjecture

Kim Taeik

Shape of theUniverse


Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture

Poincares Conjecture

Proof

Geometry

Euclidean

Riemannean

Geometrization

Poincares ConjectureA picture for shape of Universe

Kim Taeik

KAIST Undergratuated

December 11, 2009
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture

Poincares ConjectureProof

Geometry

Euclidean

Riemannean

Geometrization

Question

What is the shape of the Universe?How can we know the shape of the Universe?
http://find/

7/30/2019 Talk 20040160 Poincare

3/76

PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Shape of Earth

We have the world map

If we sail to east we will return
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Shape of Earth


http://find/http://goback/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Shape of Earth



7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Shape of Earth

If we sail to north,we will return
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Shape of Earth

http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Shape of Earth

http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Problem

Which is the shape of Earth?
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Problem

Which is the shape of Earth?

Solution

Escape to the Universe! Earth is sphere!
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Question

http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Question

What is the shape of the Universe?How can we know the shape of the Universe?Solution

Escape from the universe!
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Question



Is it possible?

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Earth Problem

How can we determine the shape without escaping theEarth?
http://find/

7/30/2019 Talk 20040160 Poincare

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Property of Spherical shell

P i

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization


a rope

P i
http://find/

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PoincaresConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization


shrink

slide on surface

Poincares
http://find/

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Poincare sConjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization


Poincares
http://find/

7/30/2019 Talk 20040160 Poincare

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Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization


Poincares
http://find/

7/30/2019 Talk 20040160 Poincare

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Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization


Poincares
http://find/

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Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization


Rope can shrink to point

Poincares
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Torus Surface

Poincares
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Torus Surface

Poincares
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Torus Surface

Poincares
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Geometry

Euclidean

Riemannean

Geometrization

Torus Surface

PoincaresC j
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Torus Surface

PoincaresC j t

7/30/2019 Talk 20040160 Poincare

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Torus Surface

the rope cannot shrink

PoincaresConject re

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Simply Connected

Spherical Shell = simply connected

PoincaresConjecture
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Simply Connected

Spherical Shell = simply connected

Torus simply connected

PoincaresConjecture
http://find/

7/30/2019 Talk 20040160 Poincare

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

2D Simply Connected

simply connected 2-D spherical shell.

PoincaresConjecture
http://find/

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Conjecture

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization


closed(compact) 3-D : simply connected 3D sphere?

PoincaresConjecture
http://find/

7/30/2019 Talk 20040160 Poincare

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C j u

Kim Taeik



Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Question



Is it possible?

PoincaresConjecture
http://find/

7/30/2019 Talk 20040160 Poincare

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j

Kim Taeik


Shape of the

UniverseShape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Question


Solution

Check simply connected.

PoincaresConjecture

G C
http://find/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Geometization Conjecture

Too many, too complicated...

PoincaresConjecture

G i i C j
http://find/

7/30/2019 Talk 20040160 Poincare

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Geometization Conjecture

Too many, too complicated...

Are there some structures?

PoincaresConjecture

G

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Geometry

S3

E3

H

3

S2 R

H2 R

SL2

(R)Nil

Sol

PoincaresConjecture

G
http://find/

7/30/2019 Talk 20040160 Poincare

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Geometry

For exampleS3

E3

H

3

S2 R

H2 R

SL2

(R)Nil

Sol
http://find/

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PoincaresConjecture

L th f Li

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Length of Line

x

y

i

f

f(x)

PoincaresConjecture

L th f Li
http://find/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Length of Line

x

y

i

f

f(x)

To calculate the length

fi

ds

PoincaresConjecture

Length of Line

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Length of Line

x

y

i

f

f(x)

To calculate the length

fi

ds

What is ds?

PoincaresConjecture

Euclidean Geometry
http://find/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry

Axioms

PoincaresConjecture

Euclidean Geometry
http://find/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry

Axioms1. to draw a straight line from any point to any point.2. to produce a finite straight line3. To describe a circle with any center and distance4. That all right angles are equal to one another5. The parallel postulate

PoincaresConjecture

Euclidean Geometry
http://find/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry

5. The parallel postulate

PoincaresConjecture

Euclidean Geometry
http://find/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry


Pythagoras theorem (ds)2 = (dx)2 + (dy)2

PoincaresConjecture

Ki T ik Euclidean Geometry
http://goforward/http://find/http://goback/

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry



The sum of angles in triangle is 180

PoincaresConjecture

Ki T ik Euclidean Geometry

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry




Does the postulate need?

PoincaresConjecture

Kim Taeik Euclidean Geometry

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Euclidean Geometry




Does the postulate need?

If the postulate false, how change ds?

PoincaresConjecture

Kim Taeik Without Postulate

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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Without Postulate

PoincaresConjecture


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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Without Postulate

PoincaresConjecture


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Kim Taeik


Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

Without Postulate

PoincaresConjecture


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Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

the sum of angles larger than 180

PoincaresConjecture

http://find/

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Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

The postulate not always true!

PoincaresConjecture

http://find/

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Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

The postulate not always true!Space is characterized by curvature

PoincaresConjecture

http://find/

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Shape of the


Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

ds must have terms


PoincaresConjecture

http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

ds must have terms

How can we determine ?


PoincaresConjecture

Kim Taeik Metric
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

ds ds()

PoincaresConjecture

Kim Taeik Metric
http://find/

7/30/2019 Talk 20040160 Poincare

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

ds ds()(ds)

PoincaresConjecture

Kim Taeik Metric
http://find/

7/30/2019 Talk 20040160 Poincare

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

ds ds()(ds)

metric : a distance between elements of a set.

d(x,y)

PoincaresConjecture

Kim Taeik Metric
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

ds ds()(ds)

metric : a distance between elements of a set.

d(x,y)

Essential characteristic of space : metric

PoincaresConjecture

Kim Taeik Ricci Flow
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij

PoincaresConjecture

http://find/

7/30/2019 Talk 20040160 Poincare

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij

gij

PoincaresConjecture

http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij

gijRicci flow canonical form

PoincaresConjecture

http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij


Space

PoincaresConjecture

http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij


Space manifold with uniform metricmodel

PoincaresConjecture

http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij


Space manifold with uniform metricmodel

PoincaresConjecture

http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

tgij = 2Rij


Space manifold with uniform metricconnected

Ricci flow

grouping

The eight geometries

PoincaresConjecture

Kim Taeik Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

manifold with gij

PoincaresConjecture

Kim Taeik Geometry
http://find/

7/30/2019 Talk 20040160 Poincare

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

PoincaresConjecture

Kim Taeik Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

Ricci flow:flatiron

PoincaresConjecture

Kim Taeik

Sh f h

Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

PoincaresConjecture

Kim Taeik

Sh f th

Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

Conversely

PoincaresConjecture

Kim Taeik

Shape of the

Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

characterized by metric

PoincaresConjecture

Kim Taeik

Shape of the

Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

Ricci flow1:wrinkling

PoincaresConjecture

Kim Taeik

Shape of the

Geometry
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

S3

E3

H3

S2 R

H2 R

SL2(R)

Nil

Sol

Ricci flow1:wrinkling

PoincaresConjecture

Kim Taeik

Shape of the

The Poincare Conjecture

1 Sh f h U i
http://find/

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Shape of Earth

Simply Connected

Spherical Shell

Simply Connected

PoincaresConjecture


Proof

Geometry

Euclidean

Riemannean

Geometrization

1 Shape of the UniverseShape of the UniverseShape of Earth

2 Simply ConnectedSpherical Shell

Simply Connected3 Poincares Conjecture


4 GeometryEuclideanRiemanneanGeometrization
http://find/

talk 20040160 poincare

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