Riemannian manifold

A Riemannian manifold is a mathematical space that allows for the measurement of geometric properties. These properties include distance, angles, length, volume, and curvature. In a standard smooth manifold, you can talk about points and paths, but you cannot inherently measure them. A Riemannian manifold solves this by adding a specific structure called a Riemannian metric. This metric acts as a tool to define how much space exists at every single point.

To understand how this works, we must look at the tangent space. At every point on a smooth manifold, there is an associated vector space called a tangent space. You can visualize these as tiny, flat planes touching the surface at that specific spot.

While a manifold provides these tangent spaces, it does not automatically provide a way to measure the vectors within them. A Riemannian metric provides a smoothly varying inner product for each tangent space. This inner product acts like a measuring stick, allowing us to determine the length of vectors and the angles between them. By using the tools of calculus, we can extract more complex data from this metric. For example, integration of the metric allows us to calculate the Riemannian distance function between points. Differentiation of the metric allows us to define curvature and the concept of parallel transport.

Riemannian manifolds can be categorized by how their metrics are defined. Some are submanifolds, which are shapes that exist inside a larger Euclidean space. For instance, a sphere or an ellipsoid sitting in three-dimensional space is a Riemannian manifold. The metric on these shapes is simply the restriction of the standard Euclidean metric to the surface. However, many manifolds are defined using an intrinsic point of view. This means the geometry is defined directly on the abstract space itself without referencing any outside space. This is often necessary for complex shapes like hyperbolic space or projective space. Some metrics are also constructed using group actions to move an inner product across the entire manifold. Others are built using tools from partial differential equations to create special metrics, such as Kähler–Einstein metrics.

The history of this field began with important discoveries in the 19th century. In 1827, Carl Friedrich Gauss discovered the Theorema Egregium, or the "remarkable theorem." This theorem proved that the Gaussian curvature of a surface is an intrinsic property. This means curvature can be determined by local measurements made entirely within the surface. Bernhard Riemann expanded on these ideas in 1854 when he first conceptualized Riemannian manifolds. While Riemann introduced the ideas, they were not fully formalized until much later. The concept of a smooth manifold was explicitly defined in 1913 by Hermann Weyl. Later, mathematicians like Élie Cartan and Levi-Civita contributed essential concepts like the Cartan connection and the Levi-Civita connection.

One of the most significant applications of this math is in the study of the universe. Albert Einstein used a generalization called pseudo-Riemannian manifolds to develop his theory of general relativity. In this theory, spacetime is modeled as a 4-dimensional pseudo-Riemannian manifold. The Einstein field equations act as constraints on the curvature of this spacetime. This connection shows how the geometry of a manifold can dictate the physical laws of gravity. Beyond physics, these manifolds are vital in computer graphics, machine learning, and cartography. They also have deep links to geometric topology, complex geometry, and algebraic geometry.

There are several ways to view the relationship between different manifolds. An isometry is a function between two Riemannian manifolds that preserves all geometric structure. If an isometry exists between two manifolds, they are considered to be the same for the purposes of Riemannian geometry. We can also create new manifolds by combining existing ones. For example, if you take the product of two manifolds, you can create a product manifold with a new metric. A common example is the flat torus, which is created from the product of two circles.

Finally, mathematicians study many variations of these structures. Generalizations include Finsler manifolds, sub-Riemannian manifolds, and pseudo-Riemannian manifolds. A fundamental theorem states that every smooth manifold admits a Riemannian metric. This was proven using a mathematical tool called a partition of unity. Another important result comes from John Nash, who proved that every Riemannian manifold can be embedded as a submanifold of a Euclidean space. This connects the abstract, intrinsic view of Riemann to the visual, extrinsic view of shapes sitting in space.