Thefundamental theorem ofRiemannian geometry states that on anyRiemannian manifold (orpseudo-Riemannian manifold) there is a uniqueaffine connection that istorsion-free and metric-compatible, called theLevi-Civita connection or(pseudo-)Riemannian connection of the given metric. Because it is canonically defined by such properties, this connection is often automatically used whengiven a metric.

The theorem can be stated as follows:

Fundamental theorem of Riemannian Geometry.^[1] Let(M,g) be aRiemannian manifold (orpseudo-Riemannian manifold). Then there is a unique connection∇ that satisfies the following conditions:

• for anyvector fieldsX,Y, andZ we have$${\displaystyle X(g(Y,Z))=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z),}$$ whereX(g(Y,Z)) denotes the derivative of the functiong(Y,Z) along vector fieldX.
• for any vector fieldsX,Y,$${\displaystyle {\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y],}$$ where[X,Y] denotes theLie bracket ofX andY.

The first condition is calledmetric-compatibility of∇.^[2] It may be equivalently expressed by saying that, given any curve inM, theinner product of any two∇–parallel vector fields along the curve is constant.^[3] It may also be equivalently phrased as saying that the metric tensor is preserved byparallel transport, which is to say that the metric is parallel when considering the natural extension of∇ to act on (0,2)-tensor fields:∇g = 0.^[4] It is further equivalent to require that the connection is induced by aprincipal bundle connection on theorthonormal frame bundle.^[5]

The second condition is sometimes calledsymmetry of∇.^[6] It expresses the condition that thetorsion of∇ is zero, and as such is also calledtorsion-freeness.^[7] There are alternative characterizations.^[8]

An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving themetric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is thecontorsion tensor.

The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called theLevi-Civita connection or(pseudo-)Riemannian connection. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either thesecond Christoffel identity orKoszul formula as obtained in the proofs below. This explicit definition expresses the Levi-Civita connection in terms of the metric and its first derivatives. As such, if the metric isk-times continuously differentiable, then the Levi-Civita connection is(k − 1)-times continuously differentiable.^[9]

The Levi-Civita connection can also be characterized in other ways, for instance via thePalatini variation of theEinstein–Hilbert action.

The proof of the theorem can be presented in various ways.^[10] Here the proof is first given in the language of coordinates andChristoffel symbols, and then in the coordinate-free language ofcovariant derivatives. Regardless of the presentation, the idea is to use the metric-compatibility and torsion-freeness conditions to obtain a direct formula for any connection that is both metric-compatible and torsion-free. This establishes the uniqueness claim in the fundamental theorem. To establish the existence claim, it must be directly checked that the formula obtained does define a connection as desired.

Here theEinstein summation convention will be used, which is to say that an index repeated as bothsubscript and superscript is being summed over all values. Letm denote the dimension ofM. Recall that, relative to a local chart, aconnection is given bym³ smooth functions$${\displaystyle \left\{{\mathrm{\Gamma }}^l{}_{ij}\right\},}$$with$${\displaystyle ({\mathrm{\nabla }}_{X}Y{)}^i=X^j{\mathrm{\partial }}_j{Y}^i+X^j{Y}^k{\mathrm{\Gamma }}^i{}_{jk}}$$for any vector fieldsX andY.^[11] Torsion-freeness of the connection refers to the condition that∇_XY − ∇_YX = [X,Y] for arbitraryX andY. Written in terms of local coordinates, this is equivalent to$${\displaystyle 0=X^j{Y}^k({\mathrm{\Gamma }}^i{}_{jk}-{\mathrm{\Gamma }}^i{}_{kj}),}$$which by arbitrariness ofX andY is equivalent to the conditionΓⁱ_jk = Γⁱ_kj.^[12] Similarly, the condition of metric-compatibility is equivalent to the condition^[13]$${\displaystyle {\mathrm{\partial }}_k{g}_{ij}={\mathrm{\Gamma }}^l{}_{ki}{g}_{lj}+{\mathrm{\Gamma }}^l{}_{kj}{g}_{il}.}$$In this way, it is seen that the conditions of torsion-freeness and metric-compatibility can be viewed as a linear system of equations for the connection, in which the coefficients and 'right-hand side' of the system are given by the metric and its first derivative. The fundamental theorem of Riemannian geometry can be viewed as saying that this linear system has a unique solution. This is seen via the following computation:^[14]in which the metric-compatibility condition is used three times for the first equality and the torsion-free condition is used three times for the second equality. The resulting formula is sometimes known as thefirst Christoffel identity.^[15] It can be contracted with the inverse of the metric,g^kl, to find thesecond Christoffel identity:^[16]$${\displaystyle {\mathrm{\Gamma }}^k{}_{ij}=\frac{1}{2}{g}^{kl}\left({\mathrm{\partial }}_i{g}_{jl}+{\mathrm{\partial }}_j{g}_{il}-{\mathrm{\partial }}_l{g}_{ij}\right).}$$This proves the uniqueness of a torsion-free and metric-compatible condition; that is, any such connection must be given by the above formula. To prove the existence, it must be checked that the above formula defines a connection that is torsion-free and metric-compatible. This can be done directly.

The above proof can also be expressed in terms of vector fields.^[17] Torsion-freeness refers to the condition that$${\displaystyle {\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y],}$$and metric-compatibility refers to the condition that$${\displaystyle X\left(g(Y,Z)\right)=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z),}$$whereX,Y, andZ are arbitrary vector fields. The computation previously done in local coordinates can be written asThis reduces immediately to the first Christoffel identity in the case thatX,Y, andZ are coordinate vector fields. The equations displayed above can be rearranged to produce theKoszul formula oridentity$${\displaystyle 2g({\mathrm{\nabla }}_{X}Y,Z)=X\left(g(Y,Z)\right)+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)+g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X).}$$This proves the uniqueness of a torsion-free and metric-compatible condition, since ifg(W,Z) is equal tog(U,Z) for arbitraryZ, thenW must equalU. This is a consequence of thenon-degeneracy of the metric. In the local formulation above, this key property of the metric was implicitly used, in the same way, via the existence ofg^kl. Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field∇_XY when givenX andY, and it is routine to check that this defines a connection that is torsion-free and metric-compatible.^[18]

• do Carmo, Manfredo Perdigão (1992).Riemannian geometry. Mathematics: Theory & Applications. Translated from the second Portuguese edition by Francis Flaherty. Boston, MA:Birkhäuser Boston, Inc.ISBN 0-8176-3490-8.MR 1138207.Zbl 0752.53001.
• Hawking, S. W.;Ellis, G. F. R. (1973).The large scale structure of space-time. Cambridge Monographs on Mathematical Physics. Vol. 1. London−New York:Cambridge University Press.doi:10.1017/CBO9780511524646.ISBN 9780521099066.MR 0424186.Zbl 0265.53054.
• Helgason, Sigurdur (2001).Differential geometry, Lie groups, and symmetric spaces.Graduate Studies in Mathematics. Vol. 34 (Corrected reprint of the 1978 original ed.). Providence, RI:American Mathematical Society.doi:10.1090/gsm/034.ISBN 0-8218-2848-7.MR 1834454.Zbl 0993.53002.
• Jost, Jürgen (2017).Riemannian geometry and geometric analysis. Universitext (Seventh edition of 1995 original ed.).Springer, Cham.doi:10.1007/978-3-319-61860-9.ISBN 978-3-319-61859-3.MR 3726907.Zbl 1380.53001.
• Kobayashi, Shoshichi;Nomizu, Katsumi (1963).Foundations of differential geometry. Vol I. New York–London:John Wiley & Sons, Inc.MR 0152974.Zbl 0119.37502.
• Milnor, J. (1963).Morse theory. Annals of Mathematics Studies. Vol. 51. Princeton, N.J.:Princeton University Press.MR 0163331.Zbl 0108.10401.
• O'Neill, Barrett (1983).Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. Vol. 103. New York:Academic Press, Inc.doi:10.1016/s0079-8169(08)x6002-7.ISBN 0-12-526740-1.MR 0719023.Zbl 0531.53051.
• Petersen, Peter (2016).Riemannian geometry.Graduate Texts in Mathematics. Vol. 171 (Third edition of 1998 original ed.).Springer, Cham.doi:10.1007/978-3-319-26654-1.ISBN 978-3-319-26652-7.MR 3469435.Zbl 1417.53001.
• Wald, Robert M. (1984).General relativity. Chicago, IL:University of Chicago Press.ISBN 0-226-87032-4.MR 0757180.Zbl 0549.53001.

• Connection (mathematics)
• Riemannian geometry
• Riemannian manifolds
• Theorems in Riemannian geometry

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