constant_sectional_curvature

[sectional_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

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” …

note-math