1.1 Definition and examples of smooth manifolds

Smooth manifolds are spaces that locally resemble Euclidean space. They're the foundation for doing calculus on curved surfaces, combining topology and differential geometry. 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 topological spaces
• Every point of an n-dimensional topological manifold has a neighborhood that is homeomorphic to an open subset of the Euclidean space $\mathbb{R}^n$
• Examples of topological manifolds include the Euclidean space $\mathbb{R}^n$, the n-sphere $S^n$, and the n-torus $T^n$

Smooth Structures and Atlases

• Smooth structure on a topological manifold is a collection of smoothly compatible coordinate charts 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
• Smooth manifold 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

• Diffeomorphism 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 $\mathbb{R}^n$, is the set of all n-tuples of real numbers $(x_1, \ldots, x_n)$
• 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 $S^n$, is the set of points in $\mathbb{R}^{n+1}$ that are a unit distance from the origin
  • 1-sphere $S^1$ is the circle
  • 2-sphere $S^2$ is the ordinary sphere in 3-dimensional space
• n-torus, denoted $T^n$, is the product of n circles $S^1 \times \ldots \times S^1$
  • 1-torus $T^1$ is homeomorphic to the circle $S^1$
  • 2-torus $T^2$ is the surface of a donut-shaped object
• Spheres and tori are examples of compact, connected smooth manifolds

Projective Spaces

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