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!
 A: 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.
A: The idea of Riemannian Geometry is motivated by the 'normal' differential geometry sitting in $\mathbb{R}^n$. You probably learned something like the first fundamental form and the second fundamental in studying the hypersurface. The first fundamental form $g$ in the Euclidean space is a positive definite 2-tensor in term of local coordinate. $$g=g_{ij}\, du^i\otimes du^j$$ where the $g_{ij}=\langle \frac{\partial F}{\partial u_i},\frac{\partial F}{\partial u_j}\rangle$ where $\langle \cdot,\cdot\rangle$ is the standard dot product.
Then , we can use the first fundamental form to define angle and length s.t it encodes the geometric property of the hypersurface for example Volume, length of the curve, etc. After that, by Theorema Egregium, the Gaussian curvature depends on the first fundamental form only. It means that the curvature of the hypersurface can be measured intrinsically.
Under this consequence, we assume the manifold does not sit inside a Euclidean Space. But now we can measure the curvature of the manifold intrinsically, if we can define a positive definite 2 tensor on the manifold which is called Riemannian metric (The first fundamental form manifold version)
For example, we can parameterize a round sphere $\mathbb{S}^2$ by spherical coordinate with radius $R$ its $F(\theta,\varphi)=(R\cos\theta\sin\varphi,R\sin\theta\sin\varphi,R\sin\varphi)$ Then consider their partial derivative and dot it together we will get the first fundamental form explicitly $$g=g_{\theta\theta}d\theta\otimes d\theta+g_{\theta\varphi}d\theta\otimes d\varphi+g_{\varphi\theta}d\varphi\otimes d\theta+g_{\varphi\varphi}d\varphi\otimes d\varphi=R^2d\varphi\otimes d\varphi+R^2\sin^2\varphi d\theta\otimes d\theta$$
Under the Riemannian metric, we can define length, volume, curvature intrinsically.
Therefore, if a smooth manifold $M$ endow a positive definite inner product (Riemannian metric) on the manifold $g$. We name this pair $(M,g)$ is a Riemannian Manifold.
A: 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
$$\int \sqrt{A \, \left(\frac{dx}{dt}\right)^2 + B \, \left(\frac{dx}{dt}\right) \cdot \left(\frac{dy}{dt}\right) + C \, \left(\frac{dy}{dt}\right)^2} \,\,\,\,  dt 
$$

