What is Riemannian Manifold intuitively?

Recently I was studying dimensionality reduction. When I come to a state-of-the-art dimensionality reduction algorithm -- UMAP, I couldn't understand their mathematics part. I think the first obstacle to understand it is -- I do not understand what is a Riemannian manifold. I have watched multiple videos and tutorials on the Riemannian manifold, but I still cannot catch the idea easily. Seems like the first thing they explain already requires complicated maths knowledge.

I would like to ask if anyone can really explain it in one simple sentence, that a normal computer science student or year one undergraduate STEM student will understand.

Thank you very much!

Solution 1:

Here is a rough description:

A topological manifold $M$ is an ambient space. Take a point $p$ in $M$, and consider two continuous parameterized curves $c_1(t), c_2(t)\in M$ defined for $-T<t<T$ such that $c_1(0)=p= c_2(0)$. Thus both $c_1$ and $c_2$ are curves that start at the point $p$ and for some time you can go forward and backward along either one of them. So far you can only talk about whether or not a curve passes through a point, and whether two curves intersect.

If $M$ additionally has a differentiable structure, so that $M$ is now a differentiable manifold, one can choose $c_1$ and $c_2$ to be differentiable and talk about their velocities $c_1'(0)$ and $c_2'(0)$ at $p$. Thus one can compare which one is faster, if they go in the same direction or not etc.

If $M$ additionally has a Riemannian structure, so that $M$ is a Riemannian manifold, now one can be even more specific about the directions of the curves $c_1$ and $c_2$ and even give the angle between their directions. Moreover if a curve $c(t)$ is defined for $a\leq t\leq b$, one can even compute its length (as a curve segment in $M$). Being able to compute angles and lengths are two of the fundamental features of Riemannian manifolds.

As an example consider the Euclidean plane $\mathbb{R}^2$. One can compute angles between vectors and lengths by using the dot product, which is the Riemannian structure of $\mathbb{R}^2$. Without the dot product one can not talk about angles between vectors. It would be a nice exercise to draw some pictures for this.

Thus in one sentence: a Riemannian manifold is an ambient space endowed with a gadget called a Riemannian metric that allows one to compute angles and lengths in a coherent fashion.

Solution 2:

Here's one longgggg not actually very simple sentence, that a good second year undergraduate STEM student who knows multivariable calculus might possibly maybe be able to understand:

While one ordinarily computes area in $\mathbb R^2$ by a double integral $$\int\!\!\int dx \, dy $$ or computes path length by a path integral $$\int \sqrt{\frac{dx}{dt}^2 + \frac{dy}{dt}^2} \,\,\,dt $$ on a 2-dimensional Riemannian manifold possessed with a metric described in coordinates by $$A(x,y) \, dx \otimes dx + B(x,y) \, dx \otimes dy + C(x,y) \, dy \otimes dy \quad\text{where $AC-B^2>0$} $$ one instead computes area by a double integral $$\int\!\!\int (AC-B^2) \, dx \, dy $$ and one computes path length by a path integral $$\int \sqrt{A \, \left(\frac{dx}{dt}\right)^2 + B \, \left(\frac{dx}{dt}\right) \cdot \left(\frac{dy}{dt}\right) + C \, \left(\frac{dy}{dt}\right)^2} \,\,\,\, dt $$