In
mathematics, a closed manifold is a
manifold without boundary that is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
one-dimensional example is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
.
The
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
,
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
, and the
Klein bottle are all closed two-dimensional manifolds.
A
line is not closed because it is not compact.
A
closed disk
In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not.
For a radius, r, an open disk is usu ...
is a compact two-dimensional manifold, but it is not closed because it has a boundary.
Open manifolds
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.
Abuse of language
Most books generally define a manifold as a space that is, locally,
homeomorphic to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
(along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a
closed disk
In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not.
For a radius, r, an open disk is usu ...
, so authors sometimes define a
manifold with boundary
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
and abusively say ''manifold'' without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.
The notion of a closed manifold is unrelated to that of a
closed set. A line is a closed subset of the plane, and a manifold, but not a closed manifold.
Use in physics
The notion of a "
closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
.
See also
*
References
*
Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-P ...
: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, .
{{Manifolds
Differential geometry
Manifolds
Geometric topology