Differential geometry of singular spaces

  • #1
Korybut
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TL;DR Summary: Reference request

Hello!

Reading the book "Differential geometry of Singular Spaces and Reduction of symmetry" by J. Sniatycki https://www.cambridge.org/core/book...-of-symmetry/7D73498C35A5975594605428DA8F9267
I found that not every definition and statement is clear to me and alternative source of information is highly desirable. Can someone provide additional reference to this subject? Many thanks in advance
 

FAQ: Differential geometry of singular spaces

What is differential geometry of singular spaces?

Differential geometry of singular spaces is a branch of mathematics that extends the methods of differential geometry to spaces that are not smooth, meaning they may have singularities or points where the usual rules of calculus do not apply. This involves studying the properties and structures of these spaces using techniques from algebraic geometry, topology, and analysis.

Why is the study of singular spaces important?

The study of singular spaces is important because many natural and mathematical phenomena are modeled by spaces that are not smooth. Singularities occur in various contexts such as in the solutions to differential equations, in algebraic varieties, and in physical theories like general relativity. Understanding these spaces can lead to insights in both pure mathematics and applied fields.

What are some common examples of singular spaces?

Common examples of singular spaces include algebraic varieties with singular points, spaces with conical singularities, and orbifolds. Specific examples are the cusp of a curve defined by \(y^2 = x^3\), the cone \(x^2 + y^2 = z^2\), and quotient spaces formed by group actions that have fixed points.

How do mathematicians handle singularities in differential geometry?

Mathematicians handle singularities by using various tools and techniques such as resolution of singularities, which involves replacing a singular space with a smooth one that has the same essential features, and by developing theories like intersection homology and perverse sheaves that can cope with the lack of smoothness. They also use metric techniques to study the geometry of spaces with singularities.

What are the main challenges in the differential geometry of singular spaces?

The main challenges in the differential geometry of singular spaces include defining and understanding curvature and other geometric invariants in the presence of singularities, developing tools to analyze the local and global structure of these spaces, and finding ways to generalize classical theorems of differential geometry to the singular setting. Additionally, computational challenges arise when trying to model and study these spaces practically.

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