1 Introduction
Preliminary
Course home page: http://math.kaist.ac.kr/~schoi/dgorb.htm
Helpful preliminary knowledge:
– Hatcher’s "Algebraic topology" Chapters 0,1. (better with Chapter 2...)
http://www.math.cornell.edu/~hatcher/AT/ATpage.html
– "Introduction to differentiable manifolds" by Munkres
– "Foundations of differentiable manifolds and Lie groups," by F. Warner.
– "Riemannian manfolds" by Do Carmo.
– S. Kobayashi and Nomizu, Foundations of differential geometry, Springer.
– R. Bishop and R. Crittendon, Geometry of manifolds.
Section 1: Manifolds and differentiable structures (Intuitive account)
– Manifolds
– Simplicial manifolds
– Pseudo-groups and G-structures.
– Differential geometry and G-structures.
– Principal bundles and connections, flat connections
Section 2: Lie groups and geometry
– Projective geometry and conformally flat geometry
– Euclidean geometry
– Spherical geometry
– Hyperbolic geometry and three models
– Discrete groups: examples
Part II. Topology of 2-orbifolds Subtitles are optional.
Section 3: Compact group actions and smooth topology
Section 4: Topology of 2-orbifold
– Topology and differentiable structures
– Covering orbifolds
– Euler characteristic.
Section 5: The universal covers and the fundamental group.
Section 6: Topological construction of 2-orbifolds: cut, paste, silvering, and
clarifying.
Part III. Geometry of 2-orbifolds Subtitles are optional.
Section 7: Geometric structures on orbifolds.
– Using atlas of charts
– Using sections.
– Covering maps of geometric orbifolds are good.
Section 8: Constructions of geometric orbifolds: spherical, Euclidean, hyperbolic,
conformally flat, projectively flat ones.
Section 9: Deformation spaces of geometric structures on orbifolds
Section 10: Deformation spaces of hyperbolic structures on 2-orbifolds
Section 11: Deformation spaces of real projective structures on 2-orbifolds.
Some advanced references for the course
W. Thurston, Lecture notes...: A chapter on orbifolds, 1977. (This is the principal
source)
W. Thurston, Three-dimensional geometry and topolgy, PUP, 1997
R.W. Sharp, Differential geometry: Cartan’s generalization of Klein’s Erlangen
program.
T. Ivey and J.M. Landsberg, Cartan For Beginners: Differential geometry via
moving frames and exterior differential systems, GSM, AMS
G. Bredon, Introduction to compact transformation groups, Academic Press,
1972.
M. Berger, Geometry I, Springer
S. Kobayashi and Nomizu, Foundations of differential geometry, Springer.
2 Manifolds and differentiable structures (Intuitive account)
2.1 Aim
The following theories for manifolds will be transfered to the orbifolds. We will
briefly mention them here as a "review" and will develop them for orbifolds later
(mostly for 2-dimensional orbifolds).
We follow coordinate-free approach to differential geometry. We do not need to
actually compute curvatures and so on.
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– G-structures
– Covering spaces
– Riemanian manifolds and constant curvature manifolds
– Lie groups and group actions
– Principal bundles and connections, flat connections
2.2 Manifolds
Topological spaces.
Quotient topology
We will mostly use cell-complexes: Hatcher’s AT P. 5-7 (Consider finite ones for
now.)
Operations: products, quotients, suspension, joins; AT P.8-10
Manifolds.
A topological n-dimensional manifold (n-manifold) is a Hausdorff space with
countable basis and charts to Euclidean spaces En; e.g curves, surfaces, 3-
manifolds.
The charts could also go to a positive half-space Hn. Then the set of points
mapping to Rn1 under charts is well-defined is said to be the boundary of the
manifold. (By the invariance of domain theorem)
Rn, Hn themselves or open subsets of Rn or Hn.
Sn the unit sphere in Rn+1. (use http://en.wikipedia.org/wiki/
Stereographic_projection)
RPn the real projective space. (use affine patches)
Manifolds.
An n-ball is a manifold with boundary. The boundary is the unit sphere Sn1.
Given two manifolds M1 and M2 of dimensions m and n respectively. The
product space M1 M2 is a manifold of dimension m + n.
An annulus is a disk removed with the interior of a smaller disk. It is also homeomorphic
to a circle times a closed interval.
The n-dimensional torus Tn is homeomorphic to the product of n circles S1.
2-torus: http://en.wikipedia.org/wiki/Torus
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More examples
Let Tn be a group of translations generated by Ti : x 7! x + ei for each i =
1; 2; ::; n. Then Rn=Tn is homeomorphic to Tn.
A connected sum of two n-manifolds M1 and M2. Remove the interiors of
two closed balls from Mi for each i. Then each Mi has a boundary component
homeomorphic to Sn1. We identify the spheres.
Take many 2-dimensional tori or projective plane and do connected sums. Also
remove the interiors of some disks. We can obtain all compact surfaces in this
way. http://en.wikipedia.org/wiki/Surface
2.3 Discrete group actions
Some homotopy theory (from Hatchers AT)
X; Y topological spaes. A homotopy is a f : X I ! Y .
Maps f and g : X ! Y are homotopic if f(x) = F(x; 0) and g(x) = F(x; 1)
for all x. The homotopic property is an equivalence relation.
Homotopy equivalences of two spaces X and Y is a map f : X ! Y with a map
g : Y ! X so that f g and g f are homotopic to IX and IY respectively.
Fundamental group (from Hatchers AT)
A path is a map f : I ! X.
A linear homotopy in Rn for any two paths.
A homotopy class is an equivalence class of homotopic map relative to endpoints.
The fundamental group (X; x0) is the set of homotopy class of path with endpoints
x0.
The product exists by joining. The product gives us a group.
A change of base-points gives us an isomorphism (not canonical)
The fundamental group of a circle is Z. Brouwer fixed point theorem
Induced homomorphisms. f : X ! Y with f(x0) = y0 induces f : (X; x0) ! (Y; y0).
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Van Kampen Theorem (AT page 43–50)
Given a space X covered by open subsets Ai such that any two or three of them
meet at a path-connected set, (X; ) is a quotient group of the free product
(Ai; ).
The kernel is generated by ij (a)ik(a) for any a in (Ai \ Aj ; ).
For cell-complexes, these are useful for computing the fundamental group.
If a space Y is obtained from X by attaching the boundary of 2-cells. Then
(Y; ) = (X; )=N where N is the normal subgroup generated by "boundary
curves" of the attaching maps.
Bouquet of circles, surfaces,...
Covering spaces and discrete group actions
Given a manifold M, a covering map p : ~M ! M from another manifold ~M is
an onto map such that each point of M has a neighborhood O s.t. pjp1(O) :
p1(O) ! O is a homeomorphism for each component of p1(O).
The coverings of a circle.
Consider a disk with interiors of disjoint smaller disks removed. Cut remove
edges and consider...
The join of two circles example: See Hatcher AT P.56–58
Homotopy lifting: Given two homotopic maps to M, if one lifts to ~M and so
does the other.
Given a map f : Y ! M with f(y0) = x0, f lifts to ~ f : Y ! ~M so that
~ f(y0) = ~x0 if f((Y; y0)) p(( ~M ; ~x0)).
Covering spaces and discrete group actions
The automorphism group of a covering map p : M0 ! M is a group of homeomorphisms
f : M0 ! M0 so that p f = f. (also called deck transformation
group.)
1(M) acts on ~M on the right by path-liftings.
A covering is regular if the covering map p : M0 ! M is a quotient map under
the action of a discrete group acting properly discontinuously and freely. Here
M is homeomorphic to M0=.
One can classify covering spaces of M by the subgroups of (M; x0). That is,
two coverings of M are equal iff the subgroups are the same.
Covering spaces are ordered by subgroup inclusion relations.
If the subgroup is normal, the corresponding covering is regular.
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A manifold has a universal covering, i.e., a covering whose space has a trivial
fundamental group. A universal cover covers every other coverings of a given
manifold.
~M has the covering automorphism group isomorphic to 1(M). A manifold
M equals ~M = for its universal cover ~M . is a subgroup of the deck transformation
group.
– Let ~M be R2 and T2 be a torus. Then there is a map p : R2 ! T2 sending
(x; y) to ([x]; [y]) where [x] = x mod 2 and [y] = y mod 2.
– Let M be a surface of genus 2. ~M is homeomorphic to a disk. The deck
transformation group can be realized as isometries of a hyperbolic plane.
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2.4 Simplicial manifolds
Simplicial manifolds
An n-simplex is a convex hull of n + 1-points (affinely independent). An n-
simplex is homeomorphic to Bn.
A simplicial complex is a locally finite collection S of simplices so that any face
of a simplex is a simplex in S and the intersection of two elements of S is an
element of S. The union is a topological set, a polyhedron.
We can define barycentric subdivisions and so on.
A link of a simplex is the simplicial complex made up of simplicies opposite
in a simplex containing .
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An n-manifoldX can be constructed by gluing n-simplices by face-identifications.
Suppose X is an n-dimensional triangulated space. If the link of every p-simplex
is homeomorphic to a sphere of (np1)-dimension, then X is an n-manifold.
If X is a simplicial n-manifold, we say X is orientable if we can give an orientations
on each simplex so that over the common faces they extend each other.
2.5 Surfaces
Surfaces
Canonical construction
Given a polygon with even number of sides, we assign identification by labeling by
alphabets a1; a2; ::; a1
1 ; a1
2 ; ; ; ; so that ai means an edge labelled by ai oriented
counter-clockwise and a1
i means an edge labelled by ai oriented clockwise. If a
pair ai and ai or a1
i occur, then we identify them respecting the orientations.
A bigon: We divide the boundary into two edges and identify by labels a; a1.
A bigon: We divide the boundary into two edges and identify by labels a; a.
A square: We identify the top segment with the bottom one and the right side
with the left side. The result is a 2-torus.
Any closed surface can be represented in this manner.
A 4n-gon. We label edges
a1; b1; a1
1 ; b1
1 ; a2; b2; a1
2 ; b1
2 ; :::an; bn; a1
n ; b1
n :
The result is a connected sum of n tori and is orientable. The genus of such a
surface is n.
A 2n-gon. We label edges a1a1a2a2::::anbn. The result is a connected sum of n
projective planes and is not orientable. The genus of such a surface is n.
The results are topological surfaces and a 2-dimensional simplicial manifold.
We can remove the interiors of disjoint closed balls from the surfaces. The results
are surfaces with boundary.
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The fundamental group of a surface can now be computed. A surface is a cell
complex starting from a 1-complex which is a bouquet of circles and attached
with a cell. (See AT P.51)
(S) = fa1; b1; :::; ag; bgj[a1; b1][a2; b2]:::[ag; bg]g
for orientable S of genus g.
An Euler characteristic of a simplicial complex is given by E F +V . This is a
topological invariant. We can show that the Euler characteristic of an orientable
compact surface of genus g with n boundary components is 2 2g n.
In fact, a closed orientable surface contains 3g 3 disjoint simple closed curves
so that the complement of its union is a disjoint union of pairs of pants, i.e.,
spheres with three holes. Thus, a pair of pants is an "elementary" surface.
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3 Pseudo-group and G-structures
Pseudo-groups
In this section, we introduce pseudo-groups.
However, we are mainly interested in classical geometries, Clifford-Klein geometries.
We will be concerned with Lie group G acting on a manifold M.
Most obvious ones are euclidean geometry where G is the group of rigid motions
acting on the euclidean space Rn. The spherical geometry is one where G is the
group O(n + 1) of orthogonal transformations acting on the unit sphere Sn.
Pseudo-groups
Topological manifolds form too large category to handle.
To restrict the local property more, we introduce pseudo-groups. A pseudogroup
G on a topological space X is the set of homeomorphisms between open
sets of X so that
– The domains of g 2 G cover X.
– The restriction of g 2 G to an open subset of its domain is also in G.
– The composition of two elements of G when defined is in G.
– The inverse of an element of G is in G.
– If g : U ! V is a homeomorphism for U; V open subsets of X. If U is a
union of open sets U for 2 I for some index set I such that gjU is in
G for each , then g is in G.
The trivial pseudo-group is one where every element is a restriction of the identity
X ! X.
Any pseudo-group contains a trivial pseudo-group.
The maximal pseudo-group of Rn is TOP, the set of all homeomorphisms between
open subsets of Rn.
The pseudo-group Cr, r 1, of the set of Cr-diffeomorphisms between open
subsets of Rn.
The pseudo-group PL of piecewise linear homeomorphisms between open subsets
of Rn.
(G;X)-pseudo group. Let G be a Lie group acting on a manifold X. Then we
define the pseudo-group as the set of all pairs (gjU;U) where U is the set of all
open subsets of X.
The group isom(Rn) of rigid motions acting on Rn or orthogonal group O(n +
1;R) acting on Sn give examples.
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3.1 G-manifold
G-manifold
G-manifold is obtained as a manifold glued with special type of gluings only in G.
Let G be a pseudo-group on Rn. A G-manifold is a n-manifold M with a maximal
G-atlas.
A G-atlas is a collection of charts (imbeddings) : U ! Rn where U is an
open subset of M such that whose domains cover M and any two charts are
G-compatible.
– Two charts (U; ); (V; ) are G-compatible if the transition map
= 1 : (U \ V ) ! (U \ V ) 2 G:
One can choose a locally finite G-atlas from a given maximal one and conversely.
A G-map f : M ! N for two G-manifolds is a local homeomorphism so that if
f sends a domain of a chart into a domain of a chart , then
f 1 2 G:
That is, f is an element of G locally up to charts.
3.2 Examples
Examples
Rn is a G-manifold if G is a pseudo-group on Rn.
f : M ! N be a local homeomorphism. If N has a G-structure, then so doesM
so that the map in a G-map. (A class of examples such as -annuli and -annuli.)
Let be a discrete group of G-homeomorphisms ofM acting properly and freely.
ThenM= has a G-structure. The charts will be the charts of the lifted open sets.
Tn has a Cr-structure and a PL-structure.
Given (G;X) as above, a (G;X)-manifold is a G-manifold where G is the restricted
pseudo-group.
A euclidean manifold is a (isom(Rn);Rn)-manifold.
A spherical manifold is a (O(n + 1); Sn)-manifold.
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4 Differential geometry and G-structures
Differential geometry and G-structures
We wish to understand geometric structures in terms of differential geometric
setting; i.e., using bundles, connections, and so on.
Such an understanding help us to see the issues in different ways.
Actually, this is not central to the lectures. However, we should try to relate to
the traditional fields where our subject can be studied in another way.
We will say more details later on.
4.1 Riemannian manifolds
Riemanian manifolds and constant curvature manifolds.
A differentiable manifold has a Riemannian metric, i.e., inner-product at each
tangent space smooth with respect coordinate charts. Such a manifold is said to
be a Riemannian manifold.
An isometric immersion (imbedding) of a Riemannian manifold to another one
is a (one-to-one) map that preserve the Riemannian metric.
We will be concerned with isometric imbedding of M into itself usually.
A length of an arc is the value of an integral of the norm of tangent vectors to the
arc. This gives us a metric on a manifold. An isometric imbedding of M into
itself is an isometry always.
A geodesic is an arc minimizing length locally.
A sectional curvature of a Riemannian metric along a 2-plane is given as the rate
of area growth of a triangle (An exact formula exists.)
A constant curvature manifold is one where the sectional curvature is identical
to a constant in every planar direction at every point.
Examples:
– A euclidean space En with the standard norm metric has a constant curvature
= 0.
– A sphere Sn with the standard induced metric from Rn+1 has a constant
curature = 1.
– Find a discrete torsion-free subgroup of the isometry group of En (resp.
Sn). Then En= (resp. Sn=) has constant curvature = 0 (resp. 1).
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4.2 Lie groups and group actions
Lie groups and group actions.
A Lie group is a smooth manifold G with an associative smooth product map
G G ! G with identity and a smooth inverse map : G ! G. (A Lie group
is often the set of symmetries of certain types of mathematical objects.)
For example, the set of isometries of Sn form a Lie group O(n + 1), which is a
classical group called an orthogonal group.
The set of isometries of the euclidean space Rn form a Lie group Rn O(n),
i.e., an extension of O(n) by a translation group in Rn.
Simple Lie groups are classified. Examples GL(n;R), SL(n;R), O(n;R), O(n;m),
GL(n;C), U(n), SU(n), SP(2n;R), Spin(n),....
An action of a Lie group G on a spaceX is a map GX ! X so that (gh)(x) =
g(h(x)).
For each g 2 G, g gives us a map g : X ! X where the identity element
correspond to the identity map of X.
Examples: Rn O(n) on Rn and O(n) on Sn.
4.3 Principal bundles and connections, flat connections
Principal bundles and connections, flat connections
Let M be a manifold and G a Lie group. A principlal fiber bundle P over M
with a group G:
– P is a manifold.
– G acts freely on P on the right. P G ! P.
– M = P=G. : P ! M is differentiable.
– P is locally trivial. : 1(U) ! U G.
Example 1: L(M) the set of frames of T(M). GL(n;R) acts freely on L(M).
: L(M) ! M is a principal bundle.
P a bundle space, M the base space. 1(x) a fiber.
1(x) = fugjg 2 Gg.
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A bundle can be constructed by mappings
f; : U \ U ! GjU;U"trivial" open subsets ofMg
so that
; =
; ;
for any triple U;U;U
.
G0;G Lie groups. f : G0 ! G a monomorphism. P(G0;M) ! P(G;M)
inducing identity M ! M is called a reduction of the structure group G to G0.
There maybe many reductions for given G0 and G.
P(G;M) is reducible to P(G0;M) if and only if ; can be taken to be in G0.
(See Kobayashi-Nomizu, Bishop-Crittendon for details.)
Associated bundles
Associated bundle: Let F be a manifold with a left-action of G.
G acts on P F on the right by
g : (u; x) ! (ug; g1(x)); g 2 G; u 2 M; x 2 F:
The quotient space E = P G F.
E is induced and 1
E (U) = U F. The structure group is the same.
Example: Tangent bundle T(M). GL(n;R) acts on Rn. Let F = Rn. Obtain
L(M) GL(n;R) Rn.
Example: Tensor bundltes Tr
s (M). GL(n;R) acts on Tr
s (R). Let F = Tr
s (R).
Connections
P(M;G) a principal bundle.
A connection decomposes each Tu(P) for each u 2 P into
– Tu(P) = Gu Qu where Gu is a subspace tangent to the fiber. (Gu the
vertical space, Qu the horizontal space.)
– Qug = (Rg)Qu for each g 2 G and u 2 P.
– Qu depend smoothly on u.
A horizontal lift of a piecewise-smooth path onM is a piecewise-smooth path
0 lifting so that the tangent vectors are all horizontal.
A horizontal lift is determined once the initial point is given.
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Given a curve onM with two endpoints, the lifts defines a parallel displacement
between fibers above the two endpoints. (commuting with G-actions).
Fixing a point x0 on M, this defines a holonomy group.
The curvature of a connection is a measure of how much a horizontal lift of small
loop in M is a loop in P.
The flat connection: In this case, we can lift homotopically trivial loops in Mn
to loops in P. Thus, the flatness is equivalent to local lifting of coordinate chart
of M to horizontal sections in P.
A flat connection on P gives us a smooth foliation of dimension n transversal to
the fibers.
The associated bundle E also inherits a connection and hence horizontal lifings.
The flatness is also equivalent to the local lifting property.
The flat connection on E gives us a smooth foliation of dimension n transversal
to the fibers.
Summary: A connection gives us a way to identify fibers given paths on X-
bundles over M. The flatness gives us locally consistent identifications.
The principal bundles and G-structures.
Given a manifold M of dimension n, a Lie group G acting on a manifold X of
dimension n.
We form a principal bundle P and then the associated bundle E fibered by X
with a flat connection.
A section f : M ! E which is transverse everywhere to the foliation given by
the flat connection.
This gives us a (G;X)-structure and coversely a (G;X)-structure gives us P;E; f
and the flat connection.
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