Manifold geometry and Non - Euclidean geometry

Question

What is the difference between Manifold geometry and Non-Euclidean geometry; what connection is there between them?

Answer 1

Consider the following definitions:

A Lie group $G$ is a smooth manifold $G$ that is also a group, with the property that the multiplication map $m: G \times G \rightarrow G$ and inversion map $i: G \rightarrow G$, given by $m(g,h) = gh$, and $i(g) = g^{-1}$, are both smooth.

A model geometry is a simply connected smooth manifold $X$ together with a transitive action (only one group orbit, i.e., for all $x,y \in X$, there is a group element $g\in G$ so that $gx=y$) of a Lie group $G$ on $X$ with compact stabilizers (the set of all group elements $g$ such that $gx=x$, for $x \in X$, is compact).

A geometric structure on a manifold $\mathcal{M}$ is a diffeomorphism $\varphi : \mathcal{M} \rightarrow X \big/ \Gamma$ for some model geometry $X$, where $\Gamma$ is a discrete subgroup of $G$ acting freely on $X$.

The Geometrization conjecture ensures that the components $\mathcal{N}_{i}$ of a closed prime 3-manifold $\mathcal{N}$ each have a geometric structure with finite volume of one of the eight Thurston Geometries:

Euclidean Geometry $\mathbb{E}^3$, Spherical Geometry $\mathbb{S}^3$, Hyperbolic Geometry $\mathbb{H}^3$, the geometry of $\mathbb{S}^2 \times \mathbb{R}$, the geometry of $\mathbb{H}^2 \times \mathbb{R}$, the geometry of the universal cover of $\text{SL}_{2}(\mathbb{R})$, Nil geometry, and Sol geometry.

So, when you ask about "manifold geometry", the notion you are likely looking for is a geometric structure on a manifold, and for 3-manifolds we have that the Geometrization Conjecture tells us what these geometric structures look like. When you simply refer to non-euclidean geometry as is, you are likely referring to a metric space with a non-euclidean metric.