Fundamental theorem of Riemannian geometry

Summary

In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric. Because it is canonically defined by such properties, often this connection is automatically used when given a metric.

Statement of the theorem

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Fundamental theorem of Riemannian Geometry.[1] Let (M, g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection which satisfies the following conditions:

  • for any vector fields X, Y, and Z we have
     
    where X(g(Y, Z)) denotes the derivative of the function g(Y, Z) along vector field X.
  • for any vector fields X, Y,
     
    where [X, Y] denotes the Lie bracket of X and Y.

The first condition is called metric-compatibility of .[2] It may be equivalently expressed by saying that, given any curve in M, the inner 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 by parallel 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 a principal bundle connection on the orthonormal frame bundle.[5]

The second condition is sometimes called symmetry of .[6] It expresses the condition that the torsion of is zero, and as such is also called torsion-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 the metric 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 the contorsion tensor.

The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the Levi-Civita connection or (pseudo-)Riemannian connection. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either the second Christoffel identity or Koszul 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 is k-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 the Palatini variation of the Einstein–Hilbert action.

Proof of the theorem

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The proof of the theorem can be presented in various ways.[10] Here the proof is first given in the language of coordinates and Christoffel symbols, and then in the coordinate-free language of covariant 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.

Local coordinates

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Here the Einstein summation convention will be used, which is to say that an index repeated as both subscript and superscript is being summed over all values. Let m denote the dimension of M. Recall that, relative to a local chart, a connection is given by m3 smooth functions

 
with
 
for any vector fields X and Y.[11] Torsion-freeness of the connection refers to the condition that XY − ∇Y X = [X, Y] for arbitrary X and Y. Written in terms of local coordinates, this is equivalent to
 
which by arbitrariness of X and Y is equivalent to the condition Γi
jk
= Γi
kj
.[12] Similarly, the condition of metric-compatibility is equivalent to the condition[13]
 
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 the first Christoffel identity.[15] It can be contracted with the inverse of the metric, gkl, to find the second Christoffel identity:[16]
 
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.

Invariant formulation

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The above proof can also be expressed in terms of vector fields.[17] Torsion-freeness refers to the condition that

 
and metric-compatibility refers to the condition that
 
where X, Y, and Z are arbitrary vector fields. The computation previously done in local coordinates can be written as
 
This reduces immediately to the first Christoffel identity in the case that X, Y, and Z are coordinate vector fields. The equations displayed above can be rearranged to produce the Koszul formula or identity
 
This proves the uniqueness of a torsion-free and metric-compatible condition, since if g(W, Z) is equal to g(U, Z) for arbitrary Z, then W must equal U. This is a consequence of the non-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 of gkl. Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field XY when given X and Y, and it is routine to check that this defines a connection that is torsion-free and metric-compatible.[18]

Notes

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  1. ^ do Carmo 1992, Theorem 2.3.6; Helgason 2001, Theorem I.9.1; Jost 2017, Theorem 4.3.1; Kobayashi & Nomizu 1963, Theorem IV.2.2; Milnor 1963, Lemma 8.6; O'Neill 1983, Theorem 3.11; Petersen 2016, Theorem 2.2.2; Wald 1984, Theorem 3.1.1.
  2. ^ Jost 2017, Definition 4.2.1.
  3. ^ do Carmo 1992, pp.53-54; Milnor 1963, pp.47-48.
  4. ^ Petersen 2016, Proposition 2.2.5; Wald 1984, p.35.
  5. ^ Kobayashi & Nomizu 1963, Proposition IV.2.1.
  6. ^ do Carmo 1992, p.54; Milnor 1963, Definition 8.5.
  7. ^ Hawking & Ellis 1973, p.34; Helgason 2001, p.43; Jost 2017, Definition 4.1.7.
  8. ^ Wald 1984, section 3.1.
  9. ^ Hawking & Ellis 1973, p.41.
  10. ^ See for instance pages 54-55 of Petersen (2016) or pages 158-159 of Kobayashi & Nomizu (1963) for presentations differing from those given here.
  11. ^ Petersen 2016, p.66.
  12. ^ Jost 2017, Lemma 4.1.1; Kobayashi & Nomizu 1963, Proposition III.7.6; Milnor 1963, p.48.
  13. ^ Milnor 1963, p.48.
  14. ^ Wald 1984, p.35.
  15. ^ Milnor 1963, p.49.
  16. ^ Milnor 1963, p.49; Wald 1984, p.36.
  17. ^ do Carmo 1992, p.55; Hawking & Ellis 1973, p.40; Helgason 2001, p.48; Jost 2017, p.194; Kobayashi & Nomizu 1963, p.160; O'Neill 1983, p.61.
  18. ^ Jost 2017, p.194; O'Neill 1983, p.61.

References

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  • 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.