section
right inverse of a fiber bundle map
quadratic differential
a section of the symmetric square of the holomorphic cotangent bundle; the vector space of holomorphic quadratic differentials on a Riemann surface is naturally isomorphic to the cotangent space to the Teichmüller space
density
section of the trivial line bundle associated to the determinant representation of the frame bundle
vector field
assignment of a vector to each point in a subset of Euclidean space
tensor density
generalization of tensor fields
tensor field
assignment of a tensor continuously varying across a mathematical space