Recent Advances in Differentiable Manifolds

Dear Colleagues,

In various branches of mathematics, one can find spaces that can be described locally by n-tuples of real numbers (x₁, x₂, ..., x_n). Such objects are called manifolds, i.e., a manifold is a topological space that is locally homeomorphic to the Euclidean space Rⁿ. Thus, we can regard a manifold as being made of pieces of Rⁿ attached by homeomorphisms. If these attached homeomorphisms are taken to be differentiable, we obtain the notion of a differentiable manifold. In general, manifolds look similar locally but different globally. Therefore, our main purpose will be to discover the way of describing the difference between manifolds from a global point of view.

When manifolds occur naturally in a branch of mathematics (here, we mean, e.g., in algebraic geometry, differential geometry and differential topology), there always appears to be some extra structure: a Riemannian metric, a Kählerian metric, a conformal structure, a smooth structure, etc. This structure is often the main object of interest; the manifold itself is merely the setting. However, many mathematicians in these areas mainly study the manifold itself; hence, the extra structures are used only as tools.

In the 21st century, manifold theory has made much progress in various context. We are now setting up a new Special Issue of Axioms entitled “Recent Advances in Differentiable Manifolds”. We welcome original research and review papers on this topic.

Prof. Dr. Kazuhiro Sakuma
Guest Editor

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• algebraic varieties
• solutions of system of polynomial equations
• Riemannian geometry
• symplectic geometry
• conformal geometry
• gauge theory
• differential topology
• algebraic topology
• singularity theory of differentiable maps