a topological space that locally resembles Euclidean space near each point.
an -dimensional manifold is a topological space with the property that each point has a neighbourhood that is homeomorphic to an open subset of -dimensional Euclidean space.
Formally, a topological manifold is a second countable Hausdorff space that is locally homeomorphic to a Euclidean space.
Locally homeomorphic to a Euclidean space
every point has a neighborhood homeomorphic to an open subset of the Euclidean space for some non-negative integer
Implies that either the point is an isolated point , or it has a neighborhood homeomorphic to the open ball:
a topological manifold with a globally defined differential structure.
abbrev: Lorentzian manifold
with a metric tensor that is everywhere non-degenerate
application used in general relativity is four-dimensional Lorentzian manifold for modeling space-time
metric tensors
A tangent space is a -dimensional differentiable manifold associated with each point .
a non-degenerate, smooth, symmetric bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold.
metric tensor
The map is symmetric and bilinear, so if are tangent vectors at point to the manifold then we have:
for any real number
Lien vers l'originalis non-degenerate means there is no non-zero such that
think of open set or interior.
intuition: a set of point containing that point where one can move some amount in any direction away from that point without leaving the set.
definition
if is a topological space and is a point in , then a neighbourhood of is a subset of that includes an open set containing :
This is equivalent to the point belonging to the topological interior of in .
properties
the neighbourhood need not be an open subset of .