Glossary

1.1 Definition and examples of smooth manifolds

3 min read•august 7, 2024

Smooth manifolds are spaces that locally resemble . They're the foundation for doing calculus on curved surfaces, combining topology and . This concept is crucial for understanding more advanced topics in Morse Theory.

Manifolds come in various shapes and sizes, from familiar Euclidean spaces to spheres and tori. We'll explore their definitions, structures, and examples, setting the stage for deeper dives into their properties and applications in later sections.

Topological Manifolds and Smooth Structures

Defining Topological Manifolds

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  • Topological manifold is a topological space that locally resembles Euclidean space near each point
  • Topological manifolds are Hausdorff, second countable
  • Every point of an n-dimensional topological manifold has a neighborhood that is homeomorphic to an open subset of the Euclidean space Rn\mathbb{R}^n
  • Examples of topological manifolds include the Euclidean space Rn\mathbb{R}^n, the SnS^n, and the TnT^n

Smooth Structures and Atlases

  • Smooth structure on a topological manifold is a collection of smoothly compatible coordinate covering the manifold
  • Atlas is a collection of charts that cover the entire manifold
    • Charts are homeomorphisms from open subsets of the manifold to open subsets of Euclidean space
    • Charts in an atlas must be smoothly compatible where they overlap
  • is a topological manifold equipped with a smooth structure
  • Smooth structures allow calculus to be performed on the manifold

Diffeomorphisms and Equivalence of Smooth Structures

  • is a smooth bijective map between smooth manifolds with a smooth inverse
    • Diffeomorphisms preserve the smooth structure of the manifolds
  • Two smooth structures on a topological manifold are considered equivalent if there exists a diffeomorphism between the resulting smooth manifolds
  • Smooth structures on a given topological manifold may not be unique (exotic smooth structures)

Euclidean and Non-Euclidean Spaces

Euclidean Spaces

  • Euclidean space is the fundamental space of classical geometry
  • Euclidean n-space, denoted Rn\mathbb{R}^n, is the set of all n-tuples of real numbers (x1,…,xn)(x_1, \ldots, x_n)
  • Euclidean space is equipped with the standard Euclidean metric and topology
  • Euclidean spaces are the simplest examples of smooth manifolds

Spheres and Tori

  • n-sphere, denoted SnS^n, is the set of points in Rn+1\mathbb{R}^{n+1} that are a unit distance from the origin
    • 1-sphere S1S^1 is the circle
    • 2-sphere S2S^2 is the ordinary sphere in 3-dimensional space
  • n-torus, denoted TnT^n, is the product of n circles S1×…×S1S^1 \times \ldots \times S^1
    • 1-torus T1T^1 is homeomorphic to the circle S1S^1
    • 2-torus T2T^2 is the surface of a donut-shaped object
  • Spheres and tori are examples of compact, connected smooth manifolds

Projective Spaces

  • , denoted RPn\mathbb{RP}^n, is the space of lines through the origin in Rn+1\mathbb{R}^{n+1}
    • Points in RPn\mathbb{RP}^n correspond to pairs of antipodal points in SnS^n
  • , denoted CPn\mathbb{CP}^n, is the space of complex lines through the origin in Cn+1\mathbb{C}^{n+1}
  • Projective spaces are examples of non-Euclidean smooth manifolds with rich geometric and topological properties

Key Terms to Review (22)

Charts: In the context of smooth manifolds, a chart is a mathematical tool that provides a way to describe the local structure of a manifold by associating an open set of the manifold with an open set in Euclidean space via a smooth map. Charts allow mathematicians to translate complex geometric properties into more manageable mathematical language, facilitating the study of properties like tangent spaces and cotangent spaces by providing local coordinates.
Complex Projective n-Space: Complex projective n-space, denoted as $$ ext{CP}^n$$, is a smooth manifold that represents the set of all complex lines through the origin in $$ ext{C}^{n+1}$$. Each point in $$ ext{CP}^n$$ corresponds to a line spanned by a non-zero vector in this space, effectively allowing the exploration of higher-dimensional geometric structures and topological properties. This space plays a crucial role in algebraic geometry, topology, and various mathematical fields due to its rich structure and properties.
Diffeomorphism: A diffeomorphism is a smooth, bijective mapping between smooth manifolds that has a smooth inverse. It preserves the structure of the manifolds, meaning that both the mapping and its inverse are smooth, allowing for a seamless transition between the two spaces without losing any geometric or topological information.
Differentiable Structure: A differentiable structure on a manifold is a collection of charts that allows for the definition of differentiability of functions between manifolds. It provides the necessary framework to apply calculus on manifolds, ensuring that transition maps between overlapping charts are smooth functions. This structure is crucial for understanding how smooth manifolds operate and interact with concepts like gradient vector fields, which rely on differentiability to define their properties.
Differential Geometry: Differential geometry is a branch of mathematics that uses the techniques of calculus and algebra to study the properties and structures of geometric objects. It focuses on smooth manifolds, which are spaces that locally resemble Euclidean space, allowing for the analysis of curves, surfaces, and higher-dimensional analogs. This area provides powerful tools for understanding the curvature and other intrinsic properties of these manifolds, forming a crucial connection to various applications in physics and engineering.
Embedding: Embedding is a mathematical concept that refers to a way of placing one mathematical object into another in such a way that the structure of the original object is preserved. This is particularly important in understanding how smooth manifolds relate to each other and how they can be represented in higher-dimensional spaces, allowing for complex interactions like the sphere eversion problem and the analysis of smooth functions.
Euclidean Space: Euclidean space is a fundamental concept in mathematics, referring to a space characterized by a flat geometry where the familiar notions of distance and angles apply. It serves as the standard framework for various mathematical concepts and is crucial for understanding both smooth manifolds and the behavior of gradient vector fields on these manifolds. This geometric foundation allows for the application of calculus and linear algebra, enabling the exploration of complex structures in higher dimensions.
General Relativity: General relativity is a theory of gravitation proposed by Albert Einstein that describes gravity not as a force, but as a curvature of spacetime caused by mass and energy. This groundbreaking idea connects the geometry of smooth manifolds with the behavior of objects in gravitational fields, showing how massive bodies like planets and stars influence the structure of the universe around them.
Henri Poincaré: Henri Poincaré was a French mathematician, theoretical physicist, and philosopher known for his foundational contributions to topology and dynamical systems. His work laid the groundwork for modern mathematics and significantly influenced the development of the theory of smooth manifolds, which are essential in understanding complex geometric structures in various fields such as physics and engineering.
Implicit Function Theorem: The Implicit Function Theorem states that under certain conditions, a relation defined by an equation can be expressed as a function of some of its variables. This theorem is crucial in understanding the structure of smooth manifolds and provides a bridge between algebraic relationships and the smooth functions that define them, ensuring that local behavior near a point can be described by smooth functions.
Inverse Function Theorem: The Inverse Function Theorem states that if a function between smooth manifolds is continuously differentiable and its derivative is invertible at a point, then there exists a neighborhood around that point where the function has a continuous inverse. This theorem is crucial because it helps establish when smooth maps can be locally inverted, connecting to the structure of smooth manifolds and their tangent spaces.
John Milnor: John Milnor is a prominent American mathematician known for his groundbreaking work in differential topology, particularly in the field of smooth manifolds and Morse theory. His contributions have significantly shaped modern mathematics, influencing various concepts related to manifold structures, Morse functions, and cobordism theory.
Lie Groups: Lie groups are mathematical structures that combine the concepts of group theory and differential geometry. They are smooth manifolds that also have a group structure, allowing for smooth operations of multiplication and inversion. This connection to smooth manifolds makes Lie groups essential in various fields such as physics and geometry, particularly in the study of symmetries and continuous transformations.
N-sphere: An n-sphere is a generalization of the concept of a sphere to n dimensions. In mathematical terms, it is defined as the set of points in (n+1)-dimensional Euclidean space that are at a fixed distance (the radius) from a central point. The n-sphere serves as a fundamental example of a smooth manifold and helps in understanding the properties and structures of higher-dimensional spaces.
N-torus: An n-torus is a product of n copies of the circle, denoted as $T^n = S^1 \times S^1 \times \cdots \times S^1$ (n times). It can be visualized as a generalization of the familiar 2-torus, which resembles the surface of a doughnut. The n-torus serves as a fundamental example of a smooth manifold, providing insights into topology and geometry.
Real Projective n-Space: Real projective n-space, denoted as $$\mathbb{RP}^n$$, is a topological space that represents the set of all lines through the origin in $$\mathbb{R}^{n+1}$$. It can be viewed as the quotient of the sphere $$S^n$$, where antipodal points are identified, making it a fundamental example of a smooth manifold with nontrivial topology. This identification leads to a space that is compact, connected, and has intriguing geometric properties.
Smooth atlas: A smooth atlas is a collection of charts that together cover a smooth manifold, allowing for the description of its differentiable structure. Each chart in the atlas maps an open subset of the manifold to an open subset of Euclidean space and ensures that the transition maps between overlapping charts are smooth. This concept is fundamental in understanding how smooth manifolds can be analyzed using calculus and differential geometry.
Smooth manifold: A smooth manifold is a topological space that locally resembles Euclidean space and is equipped with a differentiable structure, allowing for smooth transitions between coordinate charts. This concept is fundamental in many areas of mathematics and physics, particularly in understanding complex geometric and topological properties through calculus.
Submanifold: A submanifold is a subset of a manifold that itself has a manifold structure, meaning it satisfies the properties of being locally homeomorphic to Euclidean space. This concept is crucial because submanifolds can inherit the smooth structure of the ambient manifold and serve as important examples and applications in differential geometry, allowing for the study of lower-dimensional spaces within higher-dimensional ones.
Tangent Space: The tangent space at a point on a smooth manifold is a vector space that intuitively represents the possible directions in which one can tangentially pass through that point. This concept helps in understanding the geometry of manifolds, as it relates to the behavior of curves and surfaces locally around a point, forming a bridge to more advanced ideas such as differential forms and their applications in topology and geometry.
Topological Spaces: A topological space is a set of points, along with a collection of open sets that satisfy specific axioms, allowing the study of continuity, convergence, and compactness. This concept provides a framework for understanding the nature of spaces in both abstract and geometric contexts, and is fundamental in defining structures like smooth manifolds.
Transition Functions: Transition functions are smooth maps that relate different coordinate charts on a smooth manifold, ensuring the compatibility of the manifold's structure. They are essential for understanding how to smoothly 'move' between local representations of the manifold and play a crucial role in defining smooth functions on the manifold. These functions guarantee that the transition from one chart to another preserves the manifold's smooth structure, making them fundamental for working with manifolds and their properties.
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