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Isothermal coordinates refer to a specific type of coordinate system used in differential geometry, particularly in the study of surfaces and Riemannian manifolds. These coordinates are characterized by their property that the metric induced on the surface can be expressed in a particularly simple form.

K-noid is a term that may refer to specific concepts or topics depending on the context, but it is not widely recognized in mainstream discourse or academic literature. However, it is possible that "K-noid" could pertain to a niche subject such as blockchain technology, programming, a concept in a game, or something else entirely.

The Lyusternik–Fet theorem, also known as the Lyusternik–Fet homotopy theorem, is a result in the field of algebraic topology. It primarily deals with the properties of topological spaces in terms of their homotopy type.

The Mabuchi functional is an important concept in differential geometry, particularly in the study of Kähler manifolds and the geometric analysis of the space of Kähler metrics. It was introduced by the mathematician Toshiki Mabuchi in the context of Kähler geometry. The Mabuchi functional is a functional defined on the space of Kähler metrics in a fixed Kähler class and is closely related to the notion of Kähler-Einstein metrics.

A nonholonomic system refers to a type of dynamical system that is subject to constraints which are not integrable, meaning that the constraints cannot be expressed purely in terms of the coordinates and time. These constraints typically involve the velocities of the system, leading to a situation where the motion cannot be fully described by a potential function alone.

In differential geometry, the **normal bundle** is a specific construction associated with an embedded submanifold of a differentiable manifold. It provides a way to understand how the submanifold sits inside the ambient manifold by considering directions that are orthogonal (normal) to the submanifold. ### Definition Let \( M \) be a smooth manifold, and let \( N \subset M \) be a smooth embedded submanifold.

An osculating circle is a circle that best approximates a curve at a given point. It is defined as the circle that has the same tangent and curvature as the curve at that point. In other words, the osculating circle touches the curve at that point and shares the same slope and curvature in a local neighborhood around that point.

The Paneitz operator is a mathematical object that arises in the context of differential geometry, particularly in the study of Riemannian manifolds and the analysis of conformal geometry. Named after the mathematician S. Paneitz, the operator is a fourth-order differential operator defined on a Riemannian manifold.

A parallel curve is a concept used in geometry and differential geometry. It involves the creation of a new curve that maintains a constant distance from a given original curve at all points. This new curve can be thought of as being "offset" from the original curve by a specific distance, which can be positive (creating a curve that is outward from the original) or negative (creating a curve that is inward).

The Fourier integral operator is a mathematical operator used in the context of Fourier analysis and signal processing. It is designed to generalize the concept of the Fourier transform and is particularly useful for analyzing functions in terms of their frequency components. The Fourier integral operator transforms a function defined in one domain (often time or space) into its representation in the frequency domain. ### Definition Let \( f(x) \) be a function defined on the real line.

Hasok Chang is a philosopher of science, particularly known for his work in the philosophy of physics and the history of science. He is a professor at the University of Cambridge and has written extensively on topics such as scientific realism, the nature of scientific knowledge, and the interactions between science and society. His research also often emphasizes the importance of historical context in understanding scientific concepts and practices.

In equivariant cohomology, the localization theorem relates the equivariant cohomology of a space to the data at fixed points under a group action.

Plücker embedding is a mathematical construction that embeds a projective space into a higher-dimensional projective space. Specifically, it is most commonly associated with the embedding of the projective space \( \mathbb{P}^n \) into \( \mathbb{P}^{\binom{n+1}{2} - 1} \) using the concept of the lines in \( \mathbb{P}^n \).

A Poisson manifold is a particular type of differentiable manifold equipped with a Poisson bracket, which is a bilinear operation that satisfies certain algebraic properties.

Regular homotopy is a concept from algebraic topology, specifically in the field of differential topology. It relates to the study of two smooth maps from one manifold to another and the idea of deforming one map into another through smooth transformations. In a more precise sense, let \( M \) and \( N \) be smooth manifolds.

The Seifert conjecture is a conjecture in the field of topology, specifically dealing with the properties of certain types of manifolds known as Seifert fibered spaces. It was proposed by the mathematician Herbert Seifert in the late 1950s. The conjecture posits that: **Every Seifert fibered manifold (which is a type of 3-manifold) has an incompressible surface.

The Serre–Swan theorem is a fundamental result in algebraic topology and differential geometry that establishes a profound connection between vector bundles and sheaves of modules.

Stunted projective space is a type of topological space that can be defined in the context of algebraic topology. More specifically, it involves modifying the standard projective space in a way that truncates it or "stunts" its structure.

In mathematics, particularly in the field of differential geometry, a **submanifold** is a subset of a manifold that itself has the structure of a manifold, often with respect to the topology and differential structure induced from the larger manifold.

In differential geometry, the tangent bundle is a fundamental construction that enables the study of the properties of differentiable manifolds. It provides a way to associate a vector space (the tangent space) to each point of a manifold, facilitating the analytical treatment of curves, vector fields, and differential equations. ### Definition: For a differentiable manifold \( M \), the tangent bundle \( TM \) is defined as the collection of all tangent spaces at each point of \( M \).