Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.

Showing that a conformal structure determines a complex structure is more difficult.

One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.

In mathematical terms, this defines a conformal structure.

He found examples (non-Euclidean nilmanifolds and solvmanifolds) of closed 3-manifolds which fail to admit flat conformal structures.

In this sense, Weyl was not just referring to one metric but to the conformal structure defined by .

This allows one to define conformal curvature and other invariants of the conformal structure.

The operator is especially important in conformal geometry, because in a suitable sense it depends only on the conformal structure.

The unit ball B with its conformal structure is the Poincaré model of hyperbolic 3-space.

In an appropriate sense, they depend only on the conformal structure of the manifold.