This may be a philosophical or physical question, so feel free to report me.
However, I was reading about how scientists (in the spirit of the motivation of Poincare's conjecture) tried to measure curvature of our three dimensional space to see if it is curved for a possibility that we live in a curved space, to see if it is possible for us to travel in a geodesic/straight line in the universe and be able to get back to where we started.
But my question is, how does a being on a manifold measure its intrinsic properties like curvature, as in, how would we actually know if we are moving in a curved path or a straight bath given that we live on the Riemannian manifold?
For instance, living on earth, assume for the sake of argument earth is a perfect sphere, one way to know that we are walking in a straight line would be to hold up a straight ruler, project the ruler on earth orthogonally and follow that path which will be a segment of a great circle, hence it is a geodesic. However, this already employs the fact that the surface of the earth is an embedded manifold in $\mathbb{R}^3$ and the metric is the induced metric from the ambient manifold. If we were truly a 2-dimensional being living on the surface of the earth, this would not be a viable option. In fact, the way we define covariant derivative rigorously is by the tangential Levi-Civita connection which requires that we first computing the covariant derivative of a curve in $\mathbb{R}^3$.
So now we are in a much more similar situation as 3-dimensional being living in 3 dimensional space (I'm aware that we actually live in spacetime that is a 4-dimensional pseudo-Riemannian manifold, yes). How exactly would we measure curvature in space - living in space and having no awareness of an ambient manifold (As far as I know the Whitney's embedding theorem is non-constructive)?
Thank you
You mention 3-dimensional and 4-dimensional manifolds but what you should review is the notion of Gaussian curvature of a 2-dimensional manifold (i.e., a surface). Gaussian curvature is an intrinsic invariant, in that to calculate it, one does not need an isometric embedding of the surface into Euclidean space. The simplest definition of Gaussian curvature does make reference to an embedding in 3-space (it is the product of the principal curvatures, or the determinant of the Weingarten map). However, Gauss proved that it is independent of the embedding!