Mathematics > Differential Geometry > Manifolds
Differential Geometry is a field of mathematics that studies the geometry of curves and surfaces through the use of calculus and linear algebra. Within this field, the concept of a manifold is fundamental. A manifold is a topological space that, on a small enough scale, resembles Euclidean space and allows for the application of calculus. In other words, a manifold is a mathematical space that is locally similar to Euclidean spaces of a certain dimension, but can have a more complicated global structure.
Definition and Examples:
A \( n \)-dimensional manifold, or \( n \)-manifold, is a topological space \( M \) such that each point \( p \in M \) has a neighborhood homeomorphic to an open subset of \( \mathbb{R}^n \). This essentially means that around every point, the space looks like \( \mathbb{R}^n \).
Formally, for each point \( p \in M \), there exists an open set \( U \) containing \( p \) and a homeomorphism \( \phi : U \to V \subset \mathbb{R}^n \), where \( V \) is open in \( \mathbb{R}^n \). The pair \( (U, \phi) \) is called a chart, and a collection of such charts that cover \( M \) is known as an atlas.
Examples of Manifolds:
Euclidean Space \( \mathbb{R}^n \):
Euclidean space itself is an example of an n-manifold. It is trivially a manifold as each point has a neighborhood homeomorphic to itself.The Circle \( S^1 \):
The 1-dimensional circle \( S^1 \) is a 1-manifold. Each point on the circle has a neighborhood that can be mapped to an open interval in \( \mathbb{R} \).The Sphere \( S^2 \):
The 2-dimensional surface of a sphere is a 2-manifold. Around each point on the sphere, there exists a neighborhood homeomorphic to an open disk in \( \mathbb{R}^2 \).
Functions and Differentiability:
A crucial aspect of manifolds is the ability to perform calculus. A function \( f : M \to \mathbb{R} \) or between manifolds \( f : M \to N \) is said to be smooth (or differentiable) if, when expressed in terms of local charts, it behaves like a differentiable function in Euclidean space. That is, for charts \( (U, \phi) \) around \( p \in M \) and \( (V, \psi) \) around \( f(p) \in N \), the composition
\[ \psi \circ f \circ \phi^{-1} : \phi(U) \subset \mathbb{R}^n \to \psi(V) \subset \mathbb{R}^m \]
is a smooth function in the usual sense on \( \mathbb{R} \).
Tensors and Tangent Spaces:
A fundamental structure associated with a manifold is its tangent space. At each point \( p \) on a manifold \( M \), the tangent space \( T_p M \) is a vector space that intuitively represents the directions in which one can tangentially pass through \( p \). This allows the use of multidimensional calculus directly on manifolds.
Given local coordinates \( (x^1, x^2, \ldots, x^n) \), the tangent vectors at \( p \) can be expressed as
\[ \frac{\partial}{\partial x^i} \bigg|_p \]
and any tangent vector can be written as a linear combination of these basis vectors.
Applications:
The theory of manifolds is foundational in many areas of mathematics and physics. In theoretical physics, manifolds serve as the setting for the general theory of relativity, where spacetime itself is modeled as a 4-dimensional manifold. In pure mathematics, manifolds are studied within topology, geometry, and analysis, providing a rich structure that connects various domains.
In summary, manifolds are an essential construct within differential geometry, enabling the application of differential and integral calculus in diverse and complex topological spaces. These abstract mathematical surfaces have profound implications in both theoretical research and practical applications across numerous scientific disciplines.