manifold

a topological space that locally resembles Euclidean space near each point.

an $n$ -dimensional manifold is a topological space with the property that each point has a neighbourhood that is homeomorphic to an open subset of $n$ -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 $\mathbb{R}^n$ for some non-negative integer $n$

Implies that either the point is an isolated point $n=0$ , or it has a neighborhood homeomorphic to the open ball:

\mathbf{B}^n = \{(x_{1},x_{2},\ldots, x_n) \in \mathbb{R}^n : x_1^2 + x_2^2 + \ldots x_n^2 <1\}

differentiable manifold

a topological manifold with a globally defined differential structure.

Pseudo-Riemannian manifold

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 $n$ -dimensional differentiable manifold $M$ associated with each point $p$ .

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 $g$

$g: T_p M \times T_p M \to \mathbb{R}$

The map is symmetric and bilinear, so if $X, Y, Z \in T_p M$ are tangent vectors at point $p$ to the manifold $M$ then we have:

\begin{aligned} g(X,Y) &= g(Y,X) \\ g(aX + Y, Z) &= ag(X,Z) + g(Y,Z) \end{aligned}

for any real number $a \in \mathbb{R}$

$g$ is non-degenerate means there is no non-zero $X \in T_p M$ such that $g(X,Y)=0 \forall \space Y \in T_p M$

neighborhood

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 $X$ is a topological space and $p$ is a point in $X$ , then a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$ :
$p \in U \subseteq V \subseteq X$
This is equivalent to the point $p \in X$ belonging to the topological interior of $V$ in $X$ .

properties

the neighbourhood $V$ need not be an open subset of $X$ .

manifold

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differentiable manifold

Pseudo-Riemannian manifold

metric tensors

neighborhood

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