The [Levi-Civita connection](https://www.wikiwand.com/en/Levi-Civita_connection) is a fundamental concept in Riemannian geometry, providing a way to differentiate vector fields along curves on a Riemannian manifold. It is the unique connection on a Riemannian manifold that is compatible with the metric and is torsion-free. Here’s a detailed explanation:
### Definition
A **Levi-Civita connection** on a Riemannian manifold $(M, g)$ is a connection (or covariant derivative) ∇\nabla∇ that satisfies two key properties:
1. **Metric Compatibility**: The connection $\nabla$ preserves the Riemannian metric $g$. This means that for any vector fields $X, Y, Z$ on $M$, $X(g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$ This property ensures that the length of vectors and the angle between them are preserved under parallel transport.
2. **Torsion-Free**: The connection $\nabla$ has no torsion, meaning that for any vector fields $X$ and $Y$, $\nabla_X Y - \nabla_Y X = [X, Y]$ where $[X, Y]$ denotes the Lie bracket of $X$ and $Y$. This property ensures that the connection is symmetric.
### Construction
The Levi-Civita connection can be explicitly constructed using the Christoffel symbols. Given a Riemannian manifold $(M, g)$ with local coordinates $(x^1, x^2, \ldots, x^n)$, the Christoffel symbols $\Gamma^k_{ij}$ of the Levi-Civita connection are defined by: $\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \frac{\partial g_{il}}{\partial x^j} + \frac{\partial g_{jl}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^l} \right)$ where $g_{ij}$ are the components of the metric tensor and $g^{kl}$ are the components of the inverse metric tensor.
### Covariant Derivative
The covariant derivative of a vector field $Y$ in the direction of a vector field $X$, denoted $\nabla_X Y$, is given in local coordinates by: $(\nabla_X Y)^k = X^i \left( \frac{\partial Y^k}{\partial x^i} + \Gamma^k_{ij} Y^j \right)$ where $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$.
### Properties
- **Parallel Transport**: The Levi-Civita connection defines parallel transport along curves, preserving the length and angle of vectors.
- **Geodesics**: Geodesics are curves that parallel transport their own tangent vectors. They satisfy the geodesic equation: $\frac{d^2 x^k}{dt^2} + \Gamma^k_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} = 0$ where $x(t)$ represents the coordinates of the geodesic as a function of the parameter $t$.
- **Curvature**: The Riemann curvature tensor, Ricci curvature, and scalar curvature can all be defined in terms of the Levi-Civita connection. The curvature tensor RRR measures the failure of the covariant derivative to commute: $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$
### Applications
- **General Relativity**: In the theory of general relativity, the Levi-Civita connection is used to describe how spacetime curves in the presence of matter and energy, forming the basis for Einstein's field equations.
- **Riemannian Geometry**: The Levi-Civita connection is essential in defining and studying various geometric and topological properties of Riemannian manifolds, such as geodesics, curvature, and the behavior of tensor fields.
### Summary
The Levi-Civita connection is a unique connection on a Riemannian manifold that preserves the metric and is torsion-free. It allows for the definition of covariant derivatives, parallel transport, and geodesics, and plays a crucial role in understanding the geometry and curvature of the manifold.
# References
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