STEIN MANIFOLD Meaning and
Definition
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A Stein manifold is a concept in mathematics that arises in the field of complex analysis. It refers to a specific type of complex manifold that possesses certain desirable analytical properties.
Formally, a Stein manifold is a complex manifold equipped with a holomorphic coordinate system that allows for a complete analytic description of the manifold. This means that every holomorphic function defined on an open subset of the manifold can be extended to a function defined on the entire manifold. In other words, the holomorphic functions on a Stein manifold have no obstructions.
The name "Stein" is derived from the mathematician Karl Stein who provided groundbreaking research on these types of manifolds. Stein manifolds have become an important area of study in complex analysis due to their numerous applications in various branches of mathematics, including algebraic geometry, differential geometry, and mathematical physics.
One of the key features of Stein manifolds is the existence of a so-called Stein domain, which is a special type of domain where the manifold is isomorphic to the Euclidean space. This property allows for the application of Euclidean tools and techniques, facilitating the study and understanding of complex analytic phenomena.
Overall, a Stein manifold is a complex manifold that possesses a rich analytic structure, allowing for a deep investigation of its holomorphic functions and connections to other areas of mathematics.
Etymology of STEIN MANIFOLD
The word "Stein manifold" is derived from the name of the mathematician Karl-Heinz Stein.
Karl-Heinz Stein was a German mathematician who made significant contributions to complex analysis and the theory of manifolds. In the 1950s, he introduced a class of manifolds in complex analysis now known as "Stein manifolds".
Stein manifolds are complex manifolds with certain analytical properties. They are characterized by the existence of a "Stein domain", which is an open subset of the manifold on which every holomorphic function can be holomorphically extended to the whole manifold.
The term "Stein manifold" was coined by other mathematicians in recognition of Karl-Heinz Stein's pioneering work in this area. It has since become a standard term in complex analysis and manifold theory to describe this class of manifolds.
