Mathematics > Differential Geometry > Connection and Curvature
Differential Geometry is a field of mathematics that employs the techniques of calculus and linear algebra to study problems in geometry. Within this vast field, the concepts of connection and curvature are fundamental for understanding the intrinsic properties of differentiable manifolds and the behavior of geometric structures on these manifolds.
A connection is a tool that allows us to compare geometric objects at different points on a manifold. More formally, it enables the differentiation of tensor fields on a manifold. One common example of a connection is the Levi-Civita connection, which is unique for each Riemannian manifold and is defined by the properties of being torsion-free and metric-compatible.
Mathematically, a connection on a smooth manifold \( M \) can be expressed in terms of a connection 1-form \( \omega \), which is a type of differential form. For a vector bundle \( E \) over \( M \), a connection \( \nabla \) allows us to differentiate sections of \( E \). If \( X \) and \( Y \) are vector fields on \( M \), the Levi-Civita connection \( \nabla_X Y \) tells us how to transport \( Y \) along \( X \).
The concept of curvature describes how a manifold deviates from being flat. For a given connection, the curvature is measured by the curvature tensor \( R \). In the context of a Riemannian manifold with a Levi-Civita connection, the Riemann curvature tensor \( R \) is defined as:
\[ R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z, \]
where \( X, Y, \) and \( Z \) are vector fields on the manifold, and \( [X,Y] \) is the Lie bracket of \( X \) and \( Y \).
The curvature tensor has several important symmetries and can be used to define other curvature-related concepts such as the Ricci curvature and the scalar curvature. The Ricci curvature \( \text{Ric}(Y, Z) \) is obtained by taking a trace of the Riemann curvature tensor, and the scalar curvature \( R \) is the trace of the Ricci curvature, encapsulating how the volume of small geodesic balls in the manifold differs from those in the Euclidean space.
These notions of connection and curvature are pivotal in many areas of geometry and theoretical physics, including the theory of General Relativity where the curvature of spacetime is directly related to the distribution of matter and energy.
Connections also generalize to other types of geometric structures like principal bundles and vector bundles. In each case, understanding the connection and curvature provides deep insights into the geometric and topological properties of the manifold in question.