Riemannian Geometry

Riemannian Geometry is a branch of Differential Geometry that focuses on the study of smooth manifolds equipped with an inner product on the tangent space at each point. These inner products are chosen smoothly from point to point, and their presence imparts the manifold with a rich geometric structure allowing for the generalization of many classical results from Euclidean geometry to more abstract surfaces. The central object of study in Riemannian Geometry is the Riemannian manifold.

Riemannian Manifolds

A Riemannian manifold (M, g) consists of a smooth manifold \( M \) equipped with a Riemannian metric \( g \), which is essentially a smoothly varying positive-definite inner product on the tangent space \( T_pM \) at each point \( p \) of the manifold. The Riemannian metric allows for the measurement of geometrical concepts such as distances, angles, volumes, and curvatures.

Geodesics and Distance

Geodesics play a crucial role in Riemannian Geometry; these are curves that represent the shortest path between points locally. Mathematically, a geodesic \( \gamma(t) \) in a Riemannian manifold satisfies the geodesic equation:
\[ \frac{D}{dt} \frac{d \gamma}{dt} = 0, \]
where \( \frac{D}{dt} \) denotes the covariant derivative along the curve. The distance between two points \( p \) and \( q \) in \( M \) is then defined as the infimum of the lengths of all piecewise smooth curves joining \( p \) and \( q \), where the length of a curve \( \gamma \) is given by:
\[ L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)} \left( \frac{d\gamma}{dt}, \frac{d\gamma}{dt} \right)} \, dt. \]

Curvature

Another fundamental concept in Riemannian Geometry is curvature, which measures how the manifold deviates from being flat. The Riemann curvature tensor \( R \) is an essential tool for this purpose. For a given vector field \( X, Y, Z \), the curvature tensor \( R \) is defined by:
\[ R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]}Z, \]
where \( \nabla \) denotes the Levi-Civita connection. Two important derived quantities are the Ricci curvature and the scalar curvature, which are contractions of the Riemann curvature tensor.

Ricci Curvature and Scalar Curvature

The Ricci curvature tensor \( \text{Ric} \) is obtained by taking the trace of the Riemann curvature tensor over the first and third indices:
\[ \text{Ric}(X, Y) = \sum_i R(e_i, X, Y, e_i), \]
where \( \{e_i\} \) is an orthonormal basis for the tangent space. The scalar curvature \( S \) is then the trace of the Ricci curvature:
\[ S = \sum_i \text{Ric}(e_i, e_i). \]

Applications

Riemannian Geometry has profound implications in both pure and applied mathematics. It forms the mathematical backbone of Einstein’s General Theory of Relativity, where spacetime is modeled as a four-dimensional Riemannian manifold with a Lorentzian metric. In pure mathematics, it plays a crucial role in topology, particularly in the study of the global properties of manifolds using techniques such as the study of critical points of smooth functions.

In summary, Riemannian Geometry extends concepts from classical geometry to the broader context of smooth manifolds, providing the tools and frameworks necessary to explore both the local and global properties of these spaces. Its interplay with physics, particularly in the theory of General Relativity, underscores its significance and utility in comprehending the universe.