Hi,

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

A surface is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2.

We see that all definitions of surfaces and manifolds are connected with the notions of Euclidean planes and spaces which are metric in their nature. This means that the definition of a surface is also connected with the notion of metric.

Hope this information is useful for you. Good luck!

This article is about mathematics and related concepts in geometry. For other uses, see Curvature (disambiguation).

File:Cell-Shape-Dynamics-From-Waves-to-Migration-pcbi.1002392.s007.ogv

A migrating wild-type Dictyostelium discoideum cell whose boundary is colored by curvature. Scale bar: 5 µm.

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) – in a way that relates to the radius of curvature of circles that touch the object – and intrinsic curvature, which is defined in terms of the lengths of curves within a Riemannian manifold.

This article deals primarily with extrinsic curvature. Its canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.

Curvature is normally a scalar quantity, but one may also define a curvature vector that takes into account the direction of the bend in addition to its magnitude. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.

This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.

The definition is independent of the metric, in the sense that functions of the curvature can be shown to be  invariant under general coordinate transformations, which don't leave the metric invariant. 

The metric is the gauge field for gravity, the curvature is the field strength and, like all gauge theories, with non-abelian transformations, certain functions of the curvature define  the appropriate gauge invariant quantities.

(The difference, indeed, with gauge theories, with internal symmetry groups, is that the connection in the latter case is the gauge field. This does have consequences when coupling matter fields  with spin to gravity.)

There is also a notion of curvature in positive characteristic: the p-curvature (cf. the wikipedia page). It captures a certain aspect of the usual curvature without needing a metric.

https://en.wikipedia.org/wiki/P-curvature

That's misleading, in fact, because there is a connection, in this case, too-and it's the invariance of the ``p-curvature'' that matters, once more. What matters is what quantity plays the role of the connection, not whether a quantity plays the role of the metric. Of course neither is invariant under the ``interesting'' transformations, but they just are used to construct the invariants. One shouldn't focus too much on names.