\chapter{Overview of Geometry}
\section{Distinction between geometry \& topology}
geometery is when there's *local* structure
(notably curvature / torsion / a connection);
topologic is global structure
EG, orientability is a global (topological) feature,
curvature is a local (geometric) feature.
A boundary is symplectic structure:
Sp *sounds* geometric (by analogy with Orthogonal),
but there is no local symplectic structure (Darboux's theorem).
Hence it's properly called "symplectic topology" (though the term is
less common)
[I don't know if there's a different line that people draw]
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\section{What do you mean, ``a geometry''?}
Lie Group -> Klein geometry (global) -> Cartan geometry (local)
Lie Group -> Homogeneous Space -> Manifold with G-structure
[Kleinian geometry was just "have a notion of congruence/rigid motion";
it was very natural step from Euclidean geometry]
(symmetric space, locally symmetric space ?)
[*model space*, transitive symmetry groups]
(constant curvature, geometry; eg of 8 geometries in 3D)
topologically,
Diff / PL / Top
structure
geometrically,
G-structure
(inc. orientation, spin)
more precisely,
special holonomy
(inc. K\"ahler, Calabi-Yau)
[is this just a special G-structure?]
these are *very* special (other than Kahler?)
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\section{Euclidean/Kleinian/Cartan geometry: comparison}
classical Euclidean geometry was very axiomatic
Klein geometry (homogeneous spaces) is v.algebraic:
it's lattices in Lie groups
(Mostow rigidity, Margulis super-rigidity)
Cartan geometry (local structure, differential geometry) is
very computational.
(you don't need to work in coordinates,
but the point is that there is a huge tensor hanging around)
Why was Euclidean geometry not enough?
- 4+ dimensions
- projective geometry
- non-Euclidean geometry
Euclidean geometry was "about" lines and angles and circles;
today we'd talk about what structures are preserved under a group of
symmetries (Kleinian POV)
...or what local structures are preserved (Cartan POV)
Kleinian geometry not enough b/c often don't have global symmetries:
curvature can change from point to point
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\section{By dimension}
0, 1 trivial
2, 3 geometric
4 weird
5+ algebraic
(geom fails in dim >= 4; still get a class of manifolds)
indeed, in 4, surgery works top but not smooth,
so the weirdness is in diff structures
embed / immerse / submersion: codim > 2 is surgery, codim 2 is weird,
codim 1 & 0 pretty easy
(separate / covering space; codim < 0 is also fibre bundle, like codim
0:
think of codim 0 as spec. case of submerse (codim, rel. dim))
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Big theme:
connection between different notions
different notions (polyhedral, smooth, etc.) are different;
notably, S^7 has 28 differential structures, and R^4 has uncountably
infinite
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some things get harder in more dimensions, as expected
(notably classification of algebraic varieties:
curves are managable, but surfaces are hard, and 3-folds are VERY hard!)
...but surprisingly, some things get easier!
(due to surgery theory)
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Weaker equivalences
(algebraic geometry)
other equivalences, and breaking into pieces:
birational like cobordant: weaker equivalence;
cobordisms built up of handles (Morse theory); birational hopefully
built of blowups/blowdowns
[there's also simple homotopy]
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\section{Axiomatic geometry}
Euclidean geometry not just flat,
but also very axiomatic.
Curved items have always been around.
Original generalizations of Euclidean geometry still satisfied
the other axioms (2 points determine a line, etc.)
[not a coincidence, right -- this is b/c Kleinian / homogeneous space?]
...but more general (Cartan) not globally axiomatic
(still have lines, now called geodesics,
but 2 points don't determine line -- think of cylinder!)
Can do axiomatic geometry, but not a central subject
(inc. combinatorial affine / projective spaces!)
[over reals, turns out every projective space in dim > 2 comes from KP^n,
but there are exotic projective planes]
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Big geometric areas:
- differential geometry
curvature etc.
- algebraic geometry
"classical" is geometry; "modern" is algebraic
- geometric topology / geometry \& topology (of manifolds)
- PDEs / microlocal analysis
symplectic topology, etc.
algebraic topology is an important tool,
and can be geometric, but it's pretty algebraic
more broadly, can also include all symmetries: group theory,
esp. representation theory
(and even some kinds of combinatorics!)
...but these are also very algebraic / discrete
(Thurston: geometry as "our visual perception brain module")
Geometry has moduli; topology doesn't (this is a way of distinguishing
them; another is "local structure")
In topology, nearby objects are the same.
[A certain duality]
Moduli related to moduli spaces and represented functors.
Can apply same philosophy in algebra: have deformations of a given
structure.
Lie etc. algebras rigid...but have non-commutative Hopf algebra
deformations,
called "quantum groups".