diffeomorphisminvariant

A diffeomorphism invariant is a property or quantity associated with a geometric object, such as a manifold or a metric space, that remains unchanged under a diffeomorphism. A diffeomorphism is a smooth bijective map between two smooth manifolds whose inverse is also smooth. Essentially, diffeomorphisms are structure-preserving transformations that do not introduce or remove any "smoothness" or "geometric detail." Therefore, anything that is invariant under diffeomorphisms is considered a fundamental geometric feature, independent of the specific coordinate system or the way the object is embedded in a larger space.

Examples of diffeomorphism invariants include the fundamental group of a manifold, its Betti numbers (which measure the number of holes of different dimensions), and the Euler characteristic. For manifolds equipped with a Riemannian metric, diffeomorphism invariants are more restricted. For instance, the Ricci scalar is not a diffeomorphism invariant, as it can change under a change of metric. However, properties related to the topology of the manifold, like its Euler characteristic, remain invariant. In differential geometry, the search for and understanding of diffeomorphism invariants is crucial for classifying and understanding the intrinsic properties of manifolds. These invariants help in distinguishing between different geometric objects that might appear different in specific representations but are fundamentally the same from a geometric perspective.