[sectional-curvature]
According to symmetry-of-curvature,
Sectional curvature is a quadratic form (possibly degenerate) restricted to the direction space i.e. restricted to unit length
Curvature can be recovered from sectional-curvature. Proof does not require non-degeneracy, symmetric bilinear forms can be recovered from quadratic forms
Prop
[constant-sectional-curvature] <==>
i.e. the curvature only has a scalar part and the scalar curvature is constant
Proof
constant-sectional-curvature <==>
<==> is a zero quadratic form
<==>
The orthogonal decomposition of gives with
[constant-sectional-curvature-imply-Einstein-metric]
Proof trace-free Ricci-curvature = 0
[constant-sectional-curvature-low-dimension]
-
==> constant-sectional-curvature = Einstein-metric = constant-scalar-curvature
-
==> constant-sectional-curvature = Einstein-metric Proof 3D + (Einstein <==> )
[quadratic-manifold] :=
where
[quadratic-manifold-is-constant-sectional-curvature] Quadratic manifold has constant-sectional-curvature
Proof
Using submanifold techniques. A point on the submanifold has tangent space and normal space in
Submanifold geodesic coordinates for the point + normal space as coordinates for the manifold
The coordinate-frame of at point in these coordinates is orthonormal
Separate tangent, normal,
The metric-dual of curvature
==>
The curvature of is zero
==>
Quadratic manifold co-dimension 1, normal space dimension 1, normal field with unit normal field
So
In ordinary coordinates at point , and
==>
==>
Cosmological constant
Lorentz manifolds in quadratic manifolds have "static coordinates", i.e. the metric will be in static form in static coordinates
- static coordinates :=
Decomposition into radius + hyperbola + sphere
Coordinates with
metric will be
- static coordinates :=
Decomposition into radius + sphere + sphere
Coordinates with
metric will be
The behavior of the time axis of is like. And there exists closed time-like geodesicm, hence not causal
The time axis behavior of the "single-sheeted hyperboloid" is like, and the space existence is like. There exists closed space-like geodesic
can be "time-sliced" into . is a diffeomorphism of
metric
Example of "visualization" of : in or , single-sheeted hyperboloid
Although time-like geodesics of are always closed, appearing as ellipses, time-like non-geodesics can have infinite length, for example, they can continuously approach light-like geodesics
Light-like geodesics appear as "parabolas" โฆ