Tangent space
Introduction
In the field of differential geometry, a tangent space is a fundamental concept that plays a crucial role in understanding the geometry of differentiable manifolds. It is a vector space associated with each point of a differentiable manifold, intuitively capturing the idea of directions that can be followed within the manifold.
Definition
Given a differentiable manifold M of dimension n, the tangent space at a point p in M, denoted by T_pM, is a real vector space of dimension n. It is formed by considering all possible velocity vectors of curves passing through p.
Construction
There are several equivalent ways to define the tangent space, including via derivations, equivalence classes of curves, and velocities of curves.
Derivations
In this approach, a tangent vector at a point p is defined as a derivation at p, which is a linear map from the space of smooth functions defined near p to the real numbers that satisfies the Leibniz rule.
Equivalence Classes of Curves
In this approach, a tangent vector at a point p is defined as an equivalence class of curves through p, where two curves are considered equivalent if they have the same velocity at p.
Velocities of Curves
In this approach, a tangent vector at a point p is defined as the velocity of a curve passing through p at that point.
Properties
Tangent spaces have several important properties that make them useful in differential geometry.
Vector Space Structure
Each tangent space is a real vector space, which means it has operations of vector addition and scalar multiplication that satisfy the usual properties of a vector space.
Dimension
The dimension of the tangent space at any point of a differentiable manifold is equal to the dimension of the manifold itself.
Tangent Bundle
The collection of all tangent spaces at all points of a manifold forms a new differentiable manifold called the tangent bundle, denoted by TM. The tangent bundle is twice the dimension of the original manifold.
Applications
Tangent spaces are used in many areas of mathematics and physics.
Differential Geometry
In differential geometry, tangent spaces are used to define vectors and vector fields on manifolds, which are essential in the study of curves and surfaces.
Calculus of Variations
In the calculus of variations, tangent spaces are used to define variations of functions, which are important in the study of optimization problems.
General Relativity
In general relativity, tangent spaces are used to define vectors and tensors at each point of spacetime, which are crucial in the formulation of the laws of physics in curved spacetime.