Affine differential geometry is a branch of mathematics that studies the properties and structures of affine manifolds, which are manifolds equipped with an affine connection. Unlike Riemannian geometry, which relies on the notion of a metric to define geometric properties like lengths and angles, affine differential geometry primarily focuses on the properties that are invariant under affine transformations, such as parallel transport and affine curvature.

Analytic torsion is a concept in mathematical analysis, particularly in the fields of differential geometry and topology, relating to the behavior of certain types of Riemannian manifolds. It arises in the context of studying the spectral properties of differential operators, especially the Laplace operator.

A developable surface is a type of surface in geometry that can be flattened into a two-dimensional plane without distortion. This means that the surface can be "unfolded" or "rolled out" in such a way that there is no stretching, tearing, or compressing involved. Developable surfaces include shapes like: 1. **Planes**: Flat surfaces are obviously developable as they are already two-dimensional.

Cartan's equivalence method is a powerful mathematical framework developed by the French mathematician Henri Cartan in the early 20th century. It is primarily used in the field of differential geometry and the theory of differential equations, particularly for understanding the equivalence of geometric structures and their associated systems of differential equations.

Clairaut's relation, also known as Clairaut's theorem, is a fundamental result in differential geometry that relates the curvature of a surface to the derivatives of the surface's height function. Specifically, it applies to surfaces of revolution, which are surfaces generated by rotating a curve about an axis.

Clifford analysis is a branch of mathematical analysis that extends classical complex analysis to higher-dimensional spaces using the framework of Clifford algebras. It focuses on functions that operate in spaces equipped with a geometric structure defined by Clifford algebras, which generalize the concept of complex numbers to higher dimensions. In Clifford analysis, the primary objects of interest are functions that are defined on domains in Euclidean spaces and take values in a Clifford algebra.

A **closed geodesic** is a type of curve on a manifold that has several important properties in differential geometry and topology. Here are the key characteristics: 1. **Geodesic**: A geodesic is a curve that locally minimizes distance and is a generalization of the concept of a "straight line" to curved spaces. It can be defined as a curve whose tangent vector is parallel transported along the curve itself.

A **closed manifold** is a type of manifold that is both compact and without boundary. More specifically, a manifold \( M \) is called closed if it satisfies the following conditions: 1. **Compact**: This means that the manifold is a bounded space that is also complete, meaning that every open cover of the manifold has a finite subcover. In simple terms, a compact manifold is one that is "finite" in a sense and can be covered by a finite number of open sets.

A **differential invariant** is a property or quantity in differential geometry that remains unchanged under particular types of transformations, usually involving differentiable functions or mappings. These invariants play a crucial role in studying geometric objects and their properties without being affected by the coordinate system or parameterization used to describe them.

Dynamic fluid film equations are mathematical formulations that describe the behavior of thin films of fluid that flow under the influence of various forces, such as gravity, surface tension, and viscous forces. These equations are crucial in understanding phenomena in various fields, including materials science, engineering, and fluid dynamics. In general, a fluid film can be considered a thin layer of fluid with a small thickness compared to its lateral dimensions.

In differential geometry, the concept of the Laplace operator, often denoted as \(\Delta\) or \(\nabla^2\), is a generalization of the Laplacian from classical analysis to manifolds. It plays a significant role in understanding the geometric and analytical properties of functions defined on a manifold.

"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.

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.

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.

The Gibbons–Hawking ansatz is a concept in theoretical physics, particularly in the study of gravitational instantons, which are solutions to the classical equations of general relativity. Named after the physicists Gary Gibbons and Stephen Hawking, the ansatz constructs a specific form of metric that is useful for exploring the properties of four-dimensional manifolds, especially in the context of quantum gravity and the study of black hole thermodynamics.

In differential geometry and algebraic geometry, the concept of a **stable normal bundle** primarily arises in the context of vector bundles over a variety or a manifold. A normal bundle is associated with a submanifold embedded in a manifold.

A radio beacon is a device that transmits specific radio signals to provide information about its location or to assist in navigation. These signals can be used by ships, aircraft, and other vehicles to determine their position.

The "Glossary of Riemannian and Metric Geometry" typically refers to a collection of terms and definitions commonly used in the fields of Riemannian geometry and metric geometry. These fields study the properties of spaces that are equipped with a notion of distance and curvature.

A glossary of differential geometry and topology typically includes key terms and concepts that are fundamental to these fields of mathematics. Here are some important terms that you might find in such a glossary: ### Differential Geometry 1. **Differentiable Manifold**: A topological manifold with a structure that allows for the differentiation of functions. 2. **Tangent Space**: The vector space consisting of the tangent vectors at a point on a manifold.