The Blaschke
Conjecture
Ben McKay
1883:
Light and Sound on Surfaces
(1) On a thin sphere of
glass, sound from North Pole focuses at South Pole; same with light.
(2) Light and sound travel
along surface, not across empty space.
(3) Light rays travel outward
in all directions at same speed, along shortest paths (geodesics).
Blaschke Conjecture
Conjecture (Wilhelm Blaschke 1921)
Light pulses from any point
focus all at once at some other point only on spheres.
Proved (Leon
Green, 1961).
Riemannian Manifolds
Riemannian manifold: a metric
space locally isometric to a smooth object in Euclidean space.
Geodesic: locally shortest
path.
Example: equator (locally
shortest, not globally).
Blaschke Manifolds
Blaschke manifold: all geodesics from any point collide with
one another at the same time (i.e. same distance).
Maybe not
all at the same point.
Blaschke conjecture: Blaschke
manifolds are projective spaces (R,C,H or O) or spheres.
Geometric Astronomy
Imagine our Riemannian manifold
as the space in which we live.
A distant star looks like a
point of light if no geodesics collide:
Looks like 2 points (equally
bright) if two geodesics collide at the same distance.
Like a dim point and a bright
one if along geodesics of different distances.
Diameter
To make astronomy easy, say
that we can’t see a point (too dim) if light ray travels all the way across the
universe (beyond diameter).
Blaschke Astronomy
In a Blaschke
Universe, light rays from a point focus all at once. Closer than focusing
distance, a star looks like a point of light. At focusing distance, could have
multiple images.
Theorem (Blaschke):
Focusing distance=diameter
Proof: (1) After 2 geods collide, neither is shortest. (2) But all geods collide at focusing distance. So after focusing
distance, no path is shortest. (3) Shortest paths exist. Therefore, focusing
distance is maximum distance.
Example: Sphere
If universe is a sphere, and
you stand at the North Pole, then every star appears as a point of light,
unless it lies at the South Pole. A star at the South Pole looks like a circle
around you.
Example: Real Projective Plane
Real projective plane is sphere
with opposite points identified:
North Pole=South Pole.
After light from North Pole
hits Equator, it is now going back to North Pole.
Example: Real Projective Plane
A star appears as a point of
light, unless it is on the equator, where it appears as two equally bright
points of light in opposite directions.
Einstein Rings
A black hole can focus light
from a star behind it, forming an image called an Einstein ring.
Blaschke-Einstein Rings
We have no black holes, but
global spatial geometry.
Theorem (Omori): In a Blaschke universe, every star appears as
a single point of light, or as an Einstein ring surrounding you (a great
sphere in your horizon sphere).
Blaschke Topology
Corollary: Every Blaschke manifold is given by drawing great spheres (Einstein
rings) on the boundary of a solid ball (your horizon ball), and collapsing
these spheres to points.
How many different patterns
of Einstein rings are possible?
Drawing Spheres on Spheres
Theorem (Browder):
Every fibration of a sphere is one of:
|
0-spheres (two points)
drawn on n-sphere |
|
circles drawn on 2n+1
sphere |
|
3-spheres drawn on 4n+3
sphere |
|
7-spheres drawn on 15
sphere |
|
all of n-sphere drawn on
itself |
Corollary: Every Blaschke manifold has cohomology
of a unique projective space over R,C,H or O or sphere. Call that
projective space or sphere its model.
Blaschke Geometry
Topology is still unknown,
beyond cohomology.
Theorem (M-): Every great
circle foliation of a sphere is equivalent to every other, by a diffeomorphism.
Corollary: Every Blaschke manifold modeled on complex projective space is diffeomorphic to its model.
Still might not be the usual
shape.
Blaschke Conjecture: The Blaschke
manifolds are the spheres and projective spaces, with their usual metrics.
State of the Art
Theorem (Berger & Kazdan): Blaschke manifolds
modeled on spheres or real projective spaces are isometric to their models.
Theorem (Gluck,
Warner and Ziller): Blaschke
manifolds modeled on projective planes are homeomorphic
to their models.
First open case: complex
projective plane (dimension 4).
Circles
Theorem(M-): Great circle foliations of a sphere are all diffeomorphic to one another.
Ideas in proof:
·
Consider a circle
foliation as a geometric structure on a sphere.
·
Calculate
“curvature” following Cartan.
·
The “curvature”
is a map to complex projective space.