The Frölicher–Nijenhuis bracket is a mathematical construct that comes from the field of differential geometry and differential algebra. It is a generalization of the Lie bracket, which is typically defined for vector fields. The Frölicher–Nijenhuis bracket allows us to define a bracket operation for arbitrary differential forms and multilinear maps.

Equivalent latitude is a concept used in atmospheric science and meteorology to describe the latitude corresponding to a particular atmospheric condition or property that is typically associated with a certain latitude in the atmosphere. It is often used in the context of phenomena such as the stratosphere, tropopause, or specific atmospheric trace gases. One common application of equivalent latitude is in the study of the ozone layer and the polar vortex.

The Filling Area Conjecture is a concept from the field of geometric topology, particularly in the study of three-dimensional manifolds. It concerns the relationship between the topological properties of a surface and its geometric properties, specifically focusing on the area of certain types of surfaces. The conjecture originates from the study of isotopy classes of simple curves on surfaces.

A **Finsler manifold** is a generalization of a Riemannian manifold that allows for the length of tangent vectors to be defined in a more flexible way. While Riemannian geometry is based on a positive-definite inner product that varies smoothly from point to point, Finsler geometry introduces a more general function, referred to as the **Finsler metric**, which defines the length of tangent vectors.

"Evolute" can refer to different concepts depending on the context. Here are a few possibilities: 1. **Mathematics**: In mathematics, particularly in differential geometry, an evolute is the locus of the centers of curvature of a given curve. It captures the idea of how the curvature of the original curve behaves and represents the "envelope" of the normals to that curve. 2. **Business/Technology**: Evolute may refer to companies or products that carry the name.

The hyperkähler quotient is a concept from the field of differential geometry and mathematical physics, particularly in the study of hyperkähler manifolds and symplectic geometry. It generalizes the notion of a symplectic quotient (or Marsden-Weinstein quotient) to the context of hyperkähler manifolds, which possess a rich geometric structure.

"Discoveries" by Josep Comas Solà is not a widely known work or publication, so it may not have significant recognition in mainstream literature or academic contexts.

In programming, particularly in the context of functional programming and data processing, a "flat map" is a higher-order function that applies a map function to each element of a data structure (like a list or an array), and then flattens the result into a single structure.

The Frankel conjecture is a hypothesis in differential geometry, specifically related to the topology of certain kinds of manifolds. It was proposed by Theodore Frankel in the 1950s and pertains to Kähler manifolds, which are complex manifolds that have a hermitian metric whose imaginary part is a closed differential form. The conjecture states that if a Kähler manifold has a Kähler class that is ample, then any morphism from the manifold to a projective space is surjective.

The Frenet–Serret formulas are a set of differential equations that describe the intrinsic geometry of a space curve in three-dimensional space. They provide a way to relate the curvature and torsion of a curve to the behavior of its tangent vector, normal vector, and binormal vector. The formulas are fundamental in the study of curves in differential geometry and are named after the mathematicians Jean Frédéric Frenet and Joseph Alain Serret.

A Haken manifold is a specific type of 3-manifold in the field of topology, particularly in the study of 3-manifolds and their properties. Named after the mathematician Wolfgang Haken, a Haken manifold is characterized by several important properties that contribute to its structure and classification.

A **G-fibration** is a concept in the field of algebraic topology, particularly in relation to homotopy theory and the study of fiber spaces. It is a generalization of the notion of a fibration, and it is typically associated with certain kinds of structured spaces and diagrams. In a broad sense, a G-fibration is a fibration where the fibers are not just sets but are equipped with a group action, typically from a topological group \( G \).

In differential geometry, a \( G \)-structure on a manifold is a mathematical framework that generalizes the structure of a manifold by introducing additional geometric or algebraic properties. More specifically, a \( G \)-structure allows you to define a way to "view" or "furnish" the manifold with additional structure that can be treated similarly to how one treats vector spaces or tangent spaces.

A G2-structure is a mathematical concept within the field of differential geometry, particularly in the study of special types of manifolds. More specifically, G2-structures are related to the notion of "exceptional" symmetries and are associated with the G2 group, which is one of the five exceptional Lie groups.

The gauge covariant derivative is a fundamental concept in the framework of gauge theories, which are essential for describing fundamental interactions in particle physics, most notably in the Standard Model. It is a modification of the ordinary derivative that accounts for the presence of gauge symmetry and the associated gauge fields. ### Definition and Purpose In a gauge theory, the fundamental fields are often associated with certain symmetry groups, such as U(1) for electromagnetism or SU(2) for weak interactions.

Gauss curvature flow is a geometric evolution equation that describes the behavior of a surface in terms of its curvature. Specifically, it is a variation of curvature flow that involves the Gaussian curvature of the surface. In mathematical terms, given a surface \( S \) in \( \mathbb{R}^3 \), the Gauss curvature \( K \) is a measure of how the surface bends at each point.

Gaussian curvature is a measure of the intrinsic curvature of a surface at a given point. It is defined as the product of two principal curvatures at that point, which are the maximum and minimum curvatures of the surface in two perpendicular directions.

General covariance is a principle from the field of theoretical physics and mathematics, particularly in the context of general relativity and differential geometry. It refers to the idea that the laws of physics should take the same form regardless of the coordinate system used to describe them. In other words, the equations that govern physical phenomena should be invariant under arbitrary smooth transformations of the coordinates.

A **generalized complex structure** is a mathematical concept that arises in the study of differential geometry, particularly in the context of **generalized complex geometry**. This notion generalizes the classical notions of complex and symplectic structures on smooth manifolds. ### Definition: A **generalized complex structure** on a smooth manifold \(M\) is defined in terms of the tangent bundle of \(M\).

A geodesic is the shortest path between two points on a curved surface or in a curved space. In mathematics and physics, this concept is often applied in differential geometry and general relativity. - **In Geometry**: On a sphere, for example, geodesics are represented by great circles (like the equator or the lines of longitude).