Levi-Civita Connection

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In Riemannian geometry, the Levi-Civita connection (named after Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a Riemann metric (or pseudo-Riemannian metric) Dadaist. The fundamental theorem of Riemann geometry states that there is a unique connection that these properties satisfy.

In the theory of a Riemannian manifold or a pseudo-Riemannian manifold the term covariant derivative is often used for the Levi-Civita connection. The expression in spatial coordinates of the connection is called the Christoffel symbols.

Formal definition

Sea (M, g) a variety of Riemann (or a pseudoriemannian variety) then an affin connection $nabla$ is a Levi-Civita connection if it meets the following conditions

Preserve the metricfor any vector fields X, And, Z We've got ${displaystyle Xg(Y,Z)=g(nabla _{X}Y,Z)+g(Y,nabla _{X}Z)}$ Where X g(And, Z) denotes the derivative of the function g(And, Z) along the vector field X.

He's free torsion.for any vector fields X and And We've got ${displaystyle nabla _{X}Y-nabla _{Y}X=[X,Y]}$ Where ${displaystyle [X,Y]}$ is the Lie corchete of the vector fields X and And.

Derivative along a curve

The Levi-Civita connection also defines a derivative along a curve, usually denoted by D.

Given a differentiable curve γ over (M, g) and a vector field V in γ its derivative is defined as

{displaystyle {frac {D}{dt}}V=nabla _{{dot {gamma }}(t)}V}

Standard connection of R. no {displaystyle mathbb {R} ^{n}}

For two vector fields ${displaystyle X,Y}$ in the n-dimensional eucliding space, this is given by the rule

{displaystyle D_{X}Y=(JY)X,}

where ${displaystyle JY}$ It's the Jacobin of Y.

Induced connection on surfaces of R. 3 {displaystyle mathbb {R} ^{3}}

For a pair of tangent vector fields to a surface (codimension 1 variety) ${mathbb {R}}^{3}$ ) can induce a covariant derivative by calculation

{displaystyle nabla _{X}Y=D_{X}Y-langle n,D_{X}Yrangle n}

relationship known as Gauss equation. It's easy to prove ${displaystyle nabla _{X}Y}$ satisfy the same properties as D.

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