Please answer the following question: Information: - In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. - In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in C, such that the transition maps are holomorphic. - Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. - In complex geometry , a Hopf manifold ( Hopf 1948 ) is obtained as a quotient of the complex vector space ( with zero deleted ) ( ( \ Bbb C ) ^ n \ backslash 0 ) by a free action of the group \ Gamma \ cong ( \ Bbb Z ) of integers , with the generator \ gamma of \ Gamma acting by holomorphic contractions . Here , a holomorphic contraction is a map \ gamma : \ ; ( \ Bbb C ) ^ n \ mapsto ( \ Bbb C ) ^ n such that a sufficiently big iteration \ ; \ gamma ^ N puts any given compact subset ( \ Bbb C ) ^ n onto an arbitrarily small neighbourhood of 0 . Two dimensional Hopf manifolds are called Hopf surfaces . - The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions Given the paragraphs above, decide what entity has the relation 'subclass of' with 'complex manifold'.
Answer:
hopf manifold