- For oriented compact surfaces, the Euler characteristic tells you the topology of the surface. The Euler characteristic can be calculated from any Riemannian metric by integrating its Gauss curvature with respect to the volume form given by the metric.
In higher dimensions Euler characteristic does not determine the topology of the manifold even though it can still be calculated from the curvature. For instance the Euler characteristic of any 3 dimensional manifold is zero.
In higher dimensions one has a curvature tensor, rather than just the Gauss curvature. Knowing this tensor still does not tell you the topology. For instance in three dimensions there are 30 something different compact oriented Riemannian manifolds whose curvature tensor is identically zero.
- In this part I am not completely sure if this is right but .....While requiring that coordinate transformations be analytic is the actual definition of a complex manifold, on a surface, a complex manifold may be thought of as a real manifold with a multiplication by i on each tangent plane. Such a multiplication is a linear map on each tangent plane whose square is multiplication by -1.
A surface together with a complex structure is a Riemann surface. This idea does not require the idea of a metric. It is really a different idea. A given topological surface has many inequivalent complex structures. So it can be many different complex manifolds.
However, a Riemannian metric does determine a complex structure on a surface. To multiply a tangent vector by i, just rotate it so that the ordered pair v,iv is positively oriented. One can show that the surface has a coordinate system in which the metric looks like multiplication by a scalar - so called isothermal coordinates - and that the coordinate transformations between isothermal coordinates are analytic. Thus the metric determines a conformal structure on the surface. However, many metrics determine the same conformal structure.
- The Gauss curvature does not tell you the metric. On a torus for instance there are many conformal structures whose metric has Gauss curvature zero.
One can see from this that there is more to Riemann surfaces than geometry and topology. There is also its possible conformal(complex) structures. Study of conformal structures is a whole field of mathematics.