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Irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the ...

A collection of identities which hold on a Kähler manifold, also called the Hodge identities. Let omega be a Kähler form, d=partial+partial^_ be the exterior derivative, ...

A Kähler structure on a complex manifold M combines a Riemannian metric on the underlying real manifold with the complex structure. Such a structure brings together geometry ...

An (m+1)-dimensional subspace W of an (n+1)-dimensional vector space V can be specified by an (m+1)×(n+1) matrix whose rows are the coordinates of a basis of W. The set of ...

Define the zeta function of a variety over a number field by taking the product over all prime ideals of the zeta functions of this variety reduced modulo the primes. Hasse ...

A map u:M->N, between two compact Riemannian manifolds, is a harmonic map if it is a critical point for the energy functional int_M|du|^2dmu_M. The norm of the differential ...

For omega a differential (k-1)-form with compact support on an oriented k-dimensional manifold with boundary M, int_Mdomega=int_(partialM)omega, (1) where domega is the ...

Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four ...

A differential k-form is a tensor of tensor rank k that is antisymmetric under exchange of any pair of indices. The number of algebraically independent components in n ...

Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal matrix--that shares the same fundamental ...