Are there spheres of non-constant curvature?

Question

I mean, are there any metrics on $n$-sphere of non-constant curvature? Maybe for a specific dimension or curvature (sectional, ricci, scalar)...

What changes in curvature when we use other metrics than the round one on spheres? What about conformal changes?

If you have any good references, please cite. It would be very nice to read them.

Answer 1

are there any metrics on $n$-sphere of non-constant curvature?

Let's assume it's about intrinsic curvature and $n\geqslant 2$. Then you can get such a metric quite easily, here an example in 2 dimensions:

Take a potato and map it's surface 1:1 and smoothly to $S_2 = \{x\in\mathbb R^2/\|x\|_2=1\}$. Then define the distance of two points on $S_2$ to be the distance of their preimages on the potato. This defines a metric, and the curvature is non-constant.