Intrinsic low-dimensional manifold

Imagine you have a really big piece of paper. On that paper, draw a bunch of dots all over it. Now, imagine you have a special pen that can draw lines between those dots. But there's a catch - you can only draw in two dimensions. That means you can't draw up or down, only side to side and back and forth.

Even though you can only draw in two dimensions, you might notice that some of the dots on the paper are actually close to each other in a third, hidden dimension. Maybe they're all on the surface of a curved object, like a ball, and you can't see that curve because you're stuck drawing in two dimensions.

This is kind of like how an intrinsic low-dimensional manifold works. A manifold is a kind of abstract object that mathematicians study, and an intrinsic manifold is one where the object's properties are independent of how it's embedded in space. So, just like how the dots on our paper might actually be part of a three-dimensional curve, an intrinsic manifold might have properties that aren't immediately obvious because they're hidden in extra dimensions.

In other words, a manifold might be like a paper with dots, but the dots are actually points in a higher-dimensional space. But, even though the points might be in a higher-dimensional space, their properties and relationships can still be studied in the lower-dimensional space of the manifold itself.