basic introduction to quotient geometry

Basic Introduction to Quotient Geometry

Raviteja Vemulapalli

University of Maryland, College Park

April 17, 2013

Raviteja Vemulapalli Slide number 1/28

Outline

• Quotient space, Horizontal space, Vertical space.

• Horizontal lift and Horizontal curves.

• Quotient space Riemannian metrics and Geodesics.


Quotient space

M1/G =M2

M2 is the quotient space ofM1 under a specified action by thegroup G.

What does this mean?


Quotient space (Ex 1: Space of concentric circles)

G = O2 ={R ∈ R2×2 | R>R = I2

}M1 = R2

Action: Left multiplication

Let x ∈M1 and g ∈ G. The action of g on x produces gx ∈M1.(because the action here is left multiplication)

The action of G on x ∈M1 produces the set[x] = Gx = {gx | g ∈ G}. Note that gx is inM1 for all g ∈ G.

Every element in Gx is said to be equivalent to x under thisgroup action. Gx is referred to as equivalence class of x (alsoreferred to as fiber).



The action of G dividesM1 into multiple disjoint equivalenceclasses (which happen to be circles centered at the origin inthis example).

Quotient spaceM2 is nothing but the set of all equivalenceclasses. In this example, quotient space is the space of circlescentered at the origin in R2.


Quotient space (Ex 2: Real Projective Space)

G = R− {0}M1 = R2 − {0}Action: Scalar multiplication

R2 − {0}/R− {0} = RP1

For a given x ∈M1, its equivalence class is given byGx = {gx | g ∈ R− {0}}, which can be associated with aunique line(except the origin) in R2.

Real projective space RP1 is nothing but the space of linespassing through the origin in R2.


Quotient space (Ex 2: Real Projective Space)


Quotient space (Exercise 1)

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3


What isM1/G ?

M1/G is the space of spheres centered at the origin in R3.


Quotient space (Exercise 2)


What isM1/G ?

M1/G is the real projective space RP2, i.e., the space of linespassing through the origin in R3.


Quotient geometry

We have seen how some manifolds can be interpreted asquotient spaces of others manifolds.

What is the use of this?

Given a manifold with closed form expressions for geodesics,parallel transport, distances, etc., we can easily (relatively)derive closed form expressions for these quantities on quotientspaces of that manifold.


Quotient geometry: Tangent space

Given a Riemannian manifoldM, at every point X ∈M, wehave a tangent space TXM.

M = R2

TXM = R2


Quotient geometry: Equivalence classes / Fibers

Given a manifoldM, group G and an action, we get a set ofequivalence classes / fibers onM by the action of G.

M1 = R2 M2 = R2 − {0}G =

{R ∈ R2×2 | R>R = I2

}G = R− {0}

Action: Left multiplication Action: Scalar multiplication


Quotient geometry: Vertical space

Vertical space VXM at X ∈M is the set of all tangent vectorsin TXM that are tangent to the equivalence class / fiber of X.

Vertical space can also be seen as the set of tangent vectors,movements along which keep X in its equivalence class / fiber.


Quotient geometry: Horizontal space

Horizontal space HXM at X ∈M is the orthogonalcomplement of vertical space VXM.

Horizontal space can also be seen as the set of tangentvectors, movements along which move X across equivalenceclasses / fibers.


Quotient geometry: Horizontal space

Horizontal vectors actually correspond to movements in thequotient space.

Horizontal vectors at a point X ∈M provide a representationfor the tangents vectors to the quotient spaceM/G at GX.

Dimensionality of quotient space is equal to the dimensionalityof horizontal space.

Dimensionality of tangent space is equal to the sum ofdimensionalities of Horizontal and Vertical spaces.


Quotient geometry: Horizontal and Vertical spaces

Space of circles in R2 centered at the origin is a onedimensional manifold.

Real projective space RP1 is a one dimensional manifold.



Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3


Fibers:

Spheres centered at the origin in R3

Tangent space at X ∈M1: R3 (3-dimensional)

Vertical space at X ∈M1: Tangent plane to the sphere passingthrough X (2-dimensional).

Horizontal space at X ∈M1: Normal to the sphere passingthrough X (1-dimensional).

Space of spheres in R2 centered at the origin is a onedimensional manifold.



Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3


Fibers: Spheres centered at the origin in R3







Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3



Tangent space at X ∈M1:

R3 (3-dimensional)






Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3









Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3




Vertical space at X ∈M1:

Tangent plane to the sphere passingthrough X (2-dimensional).





Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3









Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3





Horizontal space at X ∈M1:

Normal to the sphere passingthrough X (1-dimensional).




Exercise 1:

G = O3 ={R ∈ R3×3 | R>R = I3

}M1 = R3









Exercise 2:


Fibers:

Lines passing through the origin in R3.


Vertical space at X ∈M1: Line passing through origin and X(1-dimensional).

Horizontal space at X ∈M1: Plane orthogonal to the linepassing through origin and X (2-dimensional).

Space of lines passing through the origin in R3 is a twodimensional manifold.



Exercise 2:


Fibers: Lines passing through the origin in R3.







Exercise 2:



Tangent space at X ∈M1:

R3 (3-dimensional)






Exercise 2:









Exercise 2:




Vertical space at X ∈M1:

Line passing through origin and X(1-dimensional).





Exercise 2:









Exercise 2:





Horizontal space at X ∈M1:

Plane orthogonal to the linepassing through origin and X (2-dimensional).




Exercise 2:








Quotient geometry: Horizontal lift

Let ζ be a tangent vector to the quotient spaceM/G at GX.Then, the corresponding horizontal vector at X ∈M, denotedby ζ�X ∈ TXM is referred to as horizontal lift of ζ at X.


Quotient geometry: Riemannian metric

We need to equip the quotient spaceM/G with a Riemannianmetric to make it a Riemannian manifold.

We can derive a Riemannian metric for the quotient spaceM/G using the Riemannian metric ofM.

The idea is to define inner product between two tangent vectorsζ and η at GX ∈M/G in terms of inner product between theirhorizontal lifts ζ�X and η�X at X ∈M.

Since a single point GX ∈M/G corresponds to multiple pointsinM, we need to make sure that the value of the inner productbetween ζ and η at GX ∈M/G does not depend on theX ∈M used for lifting.


Quotient geometry: Riemannian metric

Space of circles: < ζ, η >GX= ζ>�Xη�X

Space of spheres: < ζ, η >GX= ζ>�Xη�X

Real projective space RP1: < ζ, η >GX=ζ>�Xη�XX>X

Real projective space RP2: < ζ, η >GX=ζ>�Xη�XX>X


Quotient geometry: Horizontal curveA horizontal curve c(t) ∈M is a curve such that the tangentvector c

′(t) is a horizontal vector for all t.

Horizontal curves onM correspond to curves in the quotientspaceM/G.


Quotient geometry: Geodesics in quotient space

Tangent vectors onM/G −→ Horizontal vectors onM.

Curves onM/G −→ Horizontal curves onM.

Geodesics onM/G −→

Horizontal geodesics onM ???



Tangent vectors onM/G −→ Horizontal vectors onM.

Curves onM/G −→ Horizontal curves onM.

Geodesics onM/G −→ Horizontal geodesics onM ???



It is not guaranteed that we will always have horizontalgeodesics onM.

In the above figure, horizontal curves are circles, which are notgeodesics onM(R2 − {0}).



What if we have horizontal geodesic curves onM ? Canwe represent the geodesic curves onM/G using thehorizontal geodesics onM?

• Stiefel and Grassmann manifolds – Yes.• General case – Not sure.


Quotient geometry: Summary

• Quotient space• Horizontal space• Vertical space• Horizontal lift• Quotient space Riemannian metric• Horizontal curves• Horizontal Geodesic curves


Quotient geometry: Take home message

If we know a manifoldM1 very well, i.e., we have closed formexpressions for geodesics, distances, parallel transport, etc.,then we can derive closed form expressions for these quantitieson quotient spaces ofM.


basic introduction to quotient geometry

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