An invariant differential operator is a differential operator that commutes with the action of a group of transformations, meaning it behaves nicely under the transformations specified by the group.

An isotropic manifold is a mathematical concept primarily found in the field of differential geometry. More specifically, isotropic manifolds often relate to the study of Riemannian manifolds or pseudo-Riemannian manifolds with special properties regarding distances and angles. In general, a manifold is considered to be isotropic if its geometry is invariant under transformations that preserve angles and distances in some sense, meaning that the curvature properties of the manifold do not depend on the direction.

The Kenmotsu manifold is a specific type of Riemannian manifold known in the context of differential geometry. It is characterized by having certain curvature properties and is considered in the study of submanifolds and their embeddings. To be more precise, a Kenmotsu manifold is a type of 3-dimensional (or higher-dimensional) contact metric manifold that satisfies certain conditions relating to its contact structure and the metric.

The Kulkarni–Nomizu product is a mathematical operation used in the context of differential geometry, particularly for constructing new geometric structures on manifolds. Specifically, it is a way to combine two Riemannian manifolds using their cotangent bundles to create a new manifold, often involving the introduction of a new metric.

The Lichnerowicz formula is a result in differential geometry, specifically in the study of Riemannian manifolds. It is an important tool in the context of the study of the spectrum of the Laplace operator on Riemannian manifolds and has applications in the theory of harmonic functions, heat equations, and more. The Lichnerowicz formula gives a relationship between the Laplacian of a spinor field and the geometric properties of the manifold.

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

Mean curvature flow is a mathematical concept used in differential geometry and geometric analysis. It describes the evolution of a surface in space as it flows in the direction of its mean curvature. The mean curvature of a surface at a point is intuitively understood as a measure of how the surface curves at that point; it is essentially the average of the curvatures in all directions.

The Nadirashvili surface is a notable example of a minimal surface, which is a surface that locally minimizes area. More specifically, it is a type of mathematical surface that is defined in terms of its geometric properties and is studied in differential geometry. The Nadirashvili surface is particularly interesting due to its unique characteristics: it is a complete minimal surface that has finitely many singular points, yet it is not embedded, meaning that it intersects itself.

A nonlinear partial differential equation (PDE) is a type of equation that relates a function of multiple variables to its partial derivatives, where the relationship involves nonlinear terms. In contrast to linear PDEs, where the solution can be combined using superposition due to linearity, nonlinear PDEs can exhibit more complex behavior and often require different analytical and numerical methods for their solution.

Parabolic geometry is a branch of differential geometry that studies geometric structures that are modeled on a special class of homogeneous spaces known as parabolic geometries. These structures relate to the study of certain types of manifolds and their associated symmetries, particularly those that arise from a specific class of Lie groups and their actions. ### Key Features of Parabolic Geometry: 1. **Parabolic Structures**: Parabolic geometries are associated with parabolic subalgebras of Lie algebras.

A projective vector field is a concept that arises in the context of differential geometry and dynamical systems, particularly in relation to the study of vector fields defined on manifolds. In the simplest terms, a vector field on a manifold assigns a vector to each point on the manifold. A projective vector field is a special type of vector field that is defined up to a certain equivalence relation.

The term "tangent" can have multiple meanings depending on the context. Here are a few common interpretations: 1. **Mathematics**: In trigonometry, the tangent (often abbreviated as "tan") is a function that relates the angle of a right triangle to the ratio of the opposite side to the adjacent side.

Teichmüller space is a fundamental concept in the field of complex analysis and algebraic geometry, specifically in the study of Riemann surfaces. It is named after the mathematician Oswald Teichmüller.

The Schouten–Nijenhuis bracket is an important tool in differential geometry and algebraic topology, particularly in the study of multivector fields and their relations to differential forms and Lie algebras. It generalizes the Lie bracket of vector fields to multivector fields, which are generalized objects that can be thought of as skew-symmetric tensors of higher degree. ### Definition 1. **Multivector Fields**: Let \( V \) be a smooth manifold.

Spectral shape analysis refers to a method used to characterize and interpret the spectral content of signals, sounds, or images based on their shape in the frequency domain. This technique is particularly useful in fields such as audio signal processing, speech analysis, music information retrieval, and various applications in physics and engineering. ### Key Components of Spectral Shape Analysis: 1. **Spectral Representation**: The process often starts with transforming a time-domain signal into the frequency domain using techniques like the Fourier transform.

In mathematics, the term "twist" can refer to several different concepts depending on the context. Here are a few interpretations: 1. **Topological Twist**: In topology, a twist can refer to a kind of transformation or modification to a surface or shape. For example, the Möbius strip is considered a "twisted" form of a cylinder where one end is turned half a turn before being attached to the other end.

Warped geometry refers to a concept in geometry and theoretical physics where the structure of space is not uniform but instead distorted or "warped" in a way that can affect the behavior of objects within that space. This idea often arises in contexts involving general relativity, string theory, and higher-dimensional theories. In general relativity, gravity is interpreted as the curvature of spacetime caused by mass and energy.

In differential geometry, the concept of a "web" is related to a specific arrangement of curves or surfaces. More formally, a web can be defined as a collection of curves in a manifold that satisfy certain intersection properties and can be used to study geometric structures.

Symplectization is a concept from the field of differential geometry and symplectic geometry, which is the study of geometric structures that arise in classical mechanics and Hamiltonian systems. The process of symplectization involves turning a given manifold into a symplectic manifold by introducing an additional dimension.

The winding number is a concept from topology, particularly in the context of complex analysis and algebraic topology. It measures the total number of times a curve wraps around a point in the plane.