Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.