Euler classes are topological invariants associated with vector bundles, which are mathematical structures that generalize the concept of vector fields. They provide important information about the geometry and topology of the underlying space. Specifically, the Euler class can be used to determine whether a vector bundle has a non-zero section, which relates to the existence of solutions to certain equations. In the context of differential geometry, the Euler class is particularly significant in the study of manifolds and characteristic classes. It is defined using the Chern classes of complex vector bundles and plays a crucial role in index theory, which connects analysis, topology, and geometry.