### curvature

(unflatness of space)

**Curvature** of space (i.e., a **curved space**) is space
that deviates from the well-known rules of geometry.
Curved spaces can be consistently described mathematically,
and general relativity considers the space of the universe
(as well as 4D spacetime) as non-flat,
to help describe gravity,
and cosmological theories assume some curvature.
In both cases, the curvature is too small to show up in day-to-day
measurements.

A curvature of space can be detected by testing
the known rules of flat (Euclidean) geometry, in an analogous
manner to how one might measure the curvature of
the Earth's surface through measurement: for example,
laying out a large triangle on the surface consisting
of a given Earth altitude (e.g., sea level) will yield
a shape whose angles do not add up to 180°
or laying out a circle and measuring the ratio
of its circumference and diameter will not yield π.
Precisely these same tests can, in principle,
be done in space, and given enough curvature
and large enough shapes, the curvature could
be measured. Sensitive trials have been carried
out to see if any curvature is detectable.

A full description of curvature of space requires more than a
single scalar,
but one simple measure of curvature of space at a point within that space
is the **scalar curvature** (or **curvature scalar**), a number
representing the degree of curvature. With it, the volume
of spheres surrounding the point can be related to that
of spheres of identical radius but in flat space.
The scalar curvature can be positive (analogous to the surface
of a sphere) or negative (analogous to the surface of a
saddle shape, or the inner edge of a doughnut).

(*cosmology,mathematics,geometry*)
**Further reading:**

http://en.wikipedia.org/wiki/Curved_space

http://en.wikipedia.org/wiki/Scalar_curvature

**Referenced by pages:**

anti-de Sitter space (AdS)

conformal field theory (CFT)

distance modulus (μ)

geodesic

general relativity (GR)

gravitational wave (GW)

inflation

Lambda-CDM model (ΛCDM)

Lovelock gravity

luminosity distance (d_{L})

parameterized post-Newtonian formalism (PPN formalism)

radio source counts

redshift-angular size relation

spacetime

wormhole

Index