Riemannian Geometry Basics
Riemannian geometry is a branch of differential geometry that studies smooth manifolds with a Riemannian metric. This metric allows for the definition of various geometric concepts such as angles, distances, and curvature. It is fundamental in the formulation of General Relativity, where spacetime is modeled as a curved manifold.
1. Manifolds
A manifold is a topological space that locally resembles Euclidean space. More formally, an n-dimensional manifold is a space where every point has a neighborhood that is homeomorphic to an open subset of R^n.
Example
Consider the surface of a sphere. Locally, small patches of the sphere can be flattened to resemble R^2, but globally, the surface has a different topology.
2. Riemannian Metrics
A Riemannian metric is a smoothly varying positive-definite inner product on the tangent space of a manifold. This allows us to measure lengths of curves and angles between vectors.
Definition
If- M is a smooth manifold, and - g is a Riemannian metric on M,
then for any two tangent vectors X, Y in the tangent space T_pM at a point p, the inner product is given by:
$$ g(X, Y) > 0 $$
Practical Example
In 2D Euclidean space, the Riemannian metric can be represented by the standard inner product:
$$ g(X, Y) = X^T Y $$
where X and Y are vectors in R^2.
3. Geodesics
Geodesics are curves that represent the shortest path between two points on a manifold. They generalize the concept of straight lines in Euclidean space.
Example: The Great Circle
On a sphere, the shortest path between two points is along a great circle. For instance, the path taken by airplanes often follows great circles, which minimizes travel distance.
4. Curvature
Curvature is a measure of how a geometric object deviates from being flat. In Riemannian geometry, curvature can be described in several ways:
- Gaussian Curvature: Measures intrinsic curvature. For example, the sphere has positive Gaussian curvature, while a flat plane has zero curvature. - Sectional Curvature: Associated with two-dimensional planes in the tangent space.
Example
The Gaussian curvature of a sphere of radius R is given by:
$$ K = rac{1}{R^2} $$
5. Applications in General Relativity
In General Relativity, the Riemannian geometry framework is used to describe the curvature of spacetime caused by mass and energy. The Einstein field equations relate the curvature of spacetime to the energy-momentum tensor, thus linking geometry and physics directly.
Summary
Riemannian geometry provides the mathematical language to describe curvature and geometric properties essential for understanding the structure of spacetime in General Relativity. Mastery of its basics is crucial for exploring more advanced concepts in the field.