In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection that satisfies these properties. The connection formalizes and generalizes the "rolling without slipping or twisting" method of transporting tangent planes of a smooth surface embedded in R 3 {\displaystyle \mathbb {R} ^{3}} (or generally, any Riemannian manifold, by the Nash embedding theorems). The covariant derivative is defined given any affine connection. In the theory of Riemannian and pseudo-Riemannian manifolds, the "covariant derivative" by default refers to the one defined using the Levi-Civita connection. The components (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols.
History The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature. In 1917, Tullio Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space. He interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space. The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding M n ⊂ R n ( n + 1 ) / 2 . {\displaystyle M^{n}\subset \mathbf {R} ^{n(n+1)/2}.}
In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results. In the same year, Hermann Weyl generalized Levi-Civita's results.
Notation (M, g) denotes a pseudo-Riemannian manifold. TM is the tangent bundle of M. g is the pseudo-Riemannian metric of M. X, Y, Z are smooth vector fields on M, i. e. smooth sections of TM. [X, Y] is the Lie bracket of X and Y. It is again a smooth vector field. The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. In the latter case, the inner product of Xp, Yp is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields act (by definition) as differential operators on smooth functions. In local coordinates ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} , the action reads
X ( f ) = X i ∂ ∂ x i f = X i ∂ i f {\displaystyle X(f)=X^{i}{\frac {\partial }{\partial x^{i}}}f=X^{i}\partial _{i}f}
where Einstein's summation convention is used.
Formal definition An affine connection ∇ {\displaystyle \nabla } is called a Levi-Civita connection if
it preserves the metric, i.e., ∇ g = 0 {\displaystyle \nabla g=0} . it is torsion-free, i.e., for any vector fields X {\displaystyle X} and Y {\displaystyle Y} we have ∇ X Y − ∇ Y X = [ X , Y ] {\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y]} , where [ X , Y ] {\displaystyle [X,Y]} is the Lie bracket of the vector fields X {\displaystyle X} and Y {\displaystyle Y} . Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text.
Fundamental theorem of (pseudo-)Riemannian geometry
Theorem Every pseudo-Riemannian manifold ( M , g ) {\displaystyle (M,g)} has a unique Levi-Civita connection ∇ {\displaystyle \nabla } . Proof: To prove uniqueness, unravel the definition of the action of a connection on tensors to find
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