Levi-Civita Connection
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 ► ► {displaystyle nabla } is a Levi-Civita connection if it meets the following conditions
- Preserve the metricfor any vector fields X, And, Z We've got Xg(And,Z)=g(► ► XAnd,Z)+g(And,► ► XZ){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 ► ► XAnd− − ► ► AndX=[chuckles]X,And]{displaystyle nabla _{X}Y-nabla _{Y}X=[X,Y]}Where [chuckles]X,And]{displaystyle [X,Y]} is the Lie corchete of the vector fields X and And.
![{displaystyle nabla _{X}Y-nabla _{Y}X=[X,Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6e5703b32d7cbf681d8c5c7304eb681147024d)
![{displaystyle [X,Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94470b44d283fde62130212956058ca6b727da37)
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
- DdtV=► ► γ γ ! ! (t)V{displaystyle {frac {D}{dt}}V=nabla _{{{dot {gamma}}(t)}V}.
Standard connection of R. no {displaystyle mathbb {R} ^{n}}
For two vector fields X,And{displaystyle X,Y} in the n-dimensional eucliding space, this is given by the rule
- DXAnd=(JAnd)X{displaystyle D_{X}Y=(JY)X,}
where JAnd{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) R3{displaystyle mathbb {R} ^{3}) can induce a covariant derivative by calculation
- ► ► XAnd=DXAnd− − n,DXAnd n{displaystyle nabla _{X}Y=D_{X}Y-langle n,D_{X}Yrangle n}
relationship known as Gauss equation. It's easy to prove ► ► XAnd{displaystyle nabla _{X}Y} satisfy the same properties as D.
