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In geometry, a "hedgehog" refers to a specific topological structure that can be visualized as a shape resembling the spiny animal after which it is named. More formally, in the context of topology and geometric topology, a hedge-hog is often defined as a higher-dimensional generalization used in various mathematical contexts.

The Henneberg surface is a mathematical construct in the field of topology and geometric analysis. It is a type of non-orientable surface that can be described as a specific sort of 2-dimensional manifold. The surface is named after the mathematician Heinz Henneberg. One of the significant characteristics of the Henneberg surface is its unique structure.

A Hermitian Yang–Mills connection is a mathematical concept that arises in the field of differential geometry and gauge theory, particularly in the study of Yang–Mills theories and the geometry of complex manifolds. It is an important tool in areas such as algebraic geometry, gauge theory, and mathematical physics. ### Key Components: 1. **Hermitian Manifolds**: A Hermitian manifold is a complex manifold equipped with a Hermitian metric.

A Hermitian manifold is a type of complex manifold equipped with a Riemannian metric that is compatible with the complex structure. More formally, a Hermitian manifold consists of the following components: 1. **Complex Manifold**: A manifold \( M \) that is equipped with an atlas of charts where the transition functions are holomorphic mappings. This means that the local coordinates can be expressed in terms of complex variables.

A Hermitian symmetric space is a type of Riemannian manifold that possesses a certain symmetric structure along with a compatible complex structure. More specifically, a Hermitian symmetric space is defined as a homogeneous space \( G/K \) where: 1. **Complex Structure**: The space has a complex manifold structure, meaning it can be described using complex coordinates, and it possesses a compatible Hermitian metric \( g \).

Hilbert's lemma, specifically referring to a result concerning sequences or series, typically pertains to the field of functional analysis and has implications in various areas of mathematics, particularly in the study of series and convergence.

A Hilbert manifold is a specific type of manifold that is modeled on a Hilbert space, which is a complete inner product space. To understand the concept of a Hilbert manifold, it's helpful to break down the terms involved: 1. **Manifold**: A manifold is a topological space that locally resembles Euclidean space. Formally, it is a topological space where every point has a neighborhood that is homeomorphic to an open subset of Euclidean space.

The Hilbert scheme is an important concept in algebraic geometry that parametrizes subschemes of a given projective variety (or more generally, an algebraic scheme) in a systematic way. More precisely, for a projective variety \( X \), the Hilbert scheme \( \text{Hilb}^n(X) \) is a scheme that parametrizes all closed subschemes of \( X \) with a fixed length \( n \).

Hitchin's equations are a set of differential equations that arise in the context of mathematical physics, particularly in the study of stable connections and Higgs bundles on Riemann surfaces. They were introduced by Nigel Hitchin in the early 1990s and have connections to gauge theory, algebraic geometry, and string theory, among other fields.

A Hitchin system is a mathematical structure that arises in the study of integrable systems, particularly in the context of differential geometry and algebraic geometry. It is named after Nigel Hitchin, who introduced these systems in the context of the theory of stable bundles and the geometry of moduli spaces. More specifically, a Hitchin system is typically defined on a compact Riemann surface and can be understood as a certain type of symplectic manifold.

The Holmes–Thompson volume is a concept in differential geometry, particularly in the study of manifolds and their geometric structures. It is associated with the geometric measure theory and is a specific volume measure defined for certain types of Riemannian manifolds. More specifically, the Holmes–Thompson volume is used to generalize the notion of volume in the context of certain spaces where traditional notions of volume may not apply directly.

Holonomy is a concept from differential geometry and mathematical physics that describes the behavior of parallel transport around closed loops in a manifold. It provides insight into the geometric properties of the space, including curvature and how certain geometric structures behave under parallel transport.

Homological mirror symmetry (HMS) is a conjectural framework in mathematical physics and algebraic geometry that relates certain aspects of symplectic geometry and algebraic geometry. It emerges primarily from the work of Maxim Kontsevich in the late 1990s. The conjecture provides a deep relationship between the geometry of a space and the derived category of coherent sheaves on that space, particularly in the context of mirror symmetry—a phenomenon that originated in string theory.

The Hopf conjecture is a statement in differential geometry and topology that concerns the curvature of Riemannian manifolds. More specifically, it was proposed by Heinz Hopf in 1938. The conjecture states that if a manifold is a compact, oriented, and simply connected Riemannian manifold of even dimension, then its total scalar curvature is non-negative.

The Hopf fibration is a mathematical construction that represents a particular way of decomposing certain spheres into circles. Named after Heinz Hopf, who introduced the concept in 1931, it provides a fascinating connection between topology, geometry, and algebra. Specifically, the Hopf fibration describes a fibration of the 3-sphere \( S^3 \) over the 2-sphere \( S^2 \) with the fibers being circles \( S^1 \).

Huisken's monotonicity formula is a key result in the study of geometric analysis, particularly in the context of the Ricci flow and mean curvature flow. It describes a property of the area of certain geometric objects as they evolve under a flow. This formula is particularly significant in the understanding of the behavior of these flows and the singularities that may arise within them.

A Hyperkähler manifold is a special type of Riemannian manifold that has a rich geometric structure. It is characterized by several key properties: 1. **Riemannian Manifold**: A Hyperkähler manifold is a Riemannian manifold, meaning it is equipped with a Riemannian metric that allows the measurement of distances and angles. 2. **Complex Structure**: It possesses a complex structure, which means that it can be viewed as a complex manifold.

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.

In mathematics, particularly in the field of differential topology, an **immersion** is a type of function between differentiable manifolds. Specifically, if we have two differentiable manifolds \(M\) and \(N\), a function \(f: M \to N\) is called an immersion if its differential \(df\) is injective at every point in \(M\).

In differential geometry, the **induced metric** (or **submanifold metric**) refers to the metric that a submanifold inherits from an ambient manifold.