Is there any way to uniquely characterise manifolds ( their geometrical and topological properties) without refering to charts or to a particular hyperspace containing the manifold. For example could the 2 sphere be defined as a set of points on which all geodesics are its circles of equal lenght ,which in turn are just sets which are the codomain of the exponential map. Could a metric tensor be defined on such a manifold in a coordinate free way. This is just an example of an attempt , but the idea is to use intrinsic curvatures and topological properties to define manifolds.

Is there any way to uniquely characterise manifolds ( their geometrical and topological properties) without refering to charts or to a particular hyperspace containing the manifold. For example could the 2 sphere be defined as a set of points on which all geodesics are its circles of equal lenght ,which in turn are just sets which are the codomain of the exponential map. Could a metric tensor be defined on such a manifold in a coordinate free way. This is just an example of an attempt , but the idea is to use intrinsic curvatures and topological properties to define manifolds. The point is , how does one characterise the geometry of a manifold ( all the intrinsic curvatures of differential geometry , gaussian curvature most importantly) without refering to a basis or an embedding space and how that would work on some concrete exaples, like spheres or ellipsoids.