Fundamental theorem of Riemannian geometry

Last updated November 22, 2024

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, this connection is often automatically used when given a metric.

Statement

The theorem can be stated as follows:

Fundamental theorem of Riemannian Geometry.^[1] Let $(M, g)$ be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection $\nabla$ which satisfies the following conditions:
for any vector fields $X$ , $Y$ , and $Z$ we have $X{\big (}g(Y,Z){\big )}=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z),$ where $X (g (Y, Z))$ denotes the derivative of the function $g (Y, Z)$ along vector field $X$ .
for any vector fields $X$ , $Y$ , $\nabla _{X}Y-\nabla _{Y}X=[X,Y],$ where $[X, Y]$ denotes the Lie bracket of $X$ and $Y$ .

The first condition is called metric-compatibility of $\nabla$ .^[2] It may be equivalently expressed by saying that, given any curve in $M$ , the inner product of any two $\nabla$ –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 $\nabla$ to act on (0,2)-tensor fields: $\nabla g = 0$ .^[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 $\nabla$ .^[6] It expresses the condition that the torsion of $\nabla$ 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

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

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 $m 3$ smooth functions $\left\{\Gamma _{ij}^{l}\right\},$ with $(\nabla _{X}Y)^{i}=X^{j}\partial _{j}Y^{i}+X^{j}Y^{k}\Gamma _{jk}^{i}$ for any vector fields $X$ and $Y$ .^[11] Torsion-freeness of the connection refers to the condition that $\nabla X Y - \nabla Y X = [X, Y]$ for arbitrary $X$ and $Y$ . Written in terms of local coordinates, this is equivalent to $0=X^{j}Y^{k}(\Gamma _{jk}^{i}-\Gamma _{kj}^{i}),$ which by arbitrariness of $X$ and $Y$ is equivalent to the condition $Γ i jk = Γ i kj$ .^[12] Similarly, the condition of metric-compatibility is equivalent to the condition^[13] $\partial _{k}g_{ij}=\Gamma _{ki}^{l}g_{lj}+\Gamma _{kj}^{l}g_{il}.$ 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] ${\begin{aligned}\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij}&=\left(\Gamma _{ij}^{p}g_{pl}+\Gamma _{il}^{p}g_{jp}\right)+\left(\Gamma _{ji}^{p}g_{pl}+\Gamma _{jl}^{p}g_{ip}\right)-\left(\Gamma _{li}^{p}g_{pj}+\Gamma _{lj}^{p}g_{ip}\right)\\&=2\Gamma _{ij}^{p}g_{pl}\end{aligned}}$ 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, $g kl$ , to find the second Christoffel identity:^[16] $\Gamma _{ij}^{k}={\tfrac {1}{2}}g^{kl}\left(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij}\right).$ 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.

Invariant formulation

The above proof can also be expressed in terms of vector fields.^[17] Torsion-freeness refers to the condition that $\nabla _{X}Y-\nabla _{Y}X=[X,Y],$ and metric-compatibility refers to the condition that $X\left(g(Y,Z)\right)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z),$ where $X$ , $Y$ , and $Z$ are arbitrary vector fields. The computation previously done in local coordinates can be written as ${\begin{aligned}X\left(g(Y,Z)\right)&+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)\\&={\Big (}g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z){\Big )}+{\Big (}g(\nabla _{Y}X,Z)+g(X,\nabla _{Y}Z){\Big )}-{\Big (}g(\nabla _{Z}X,Y)+g(X,\nabla _{Z}Y){\Big )}\\&=g(\nabla _{X}Y+\nabla _{Y}X,Z)+g(\nabla _{X}Z-\nabla _{Z}X,Y)+g(\nabla _{Y}Z-\nabla _{Z}Y,X)\\&=g(2\nabla _{X}Y+[Y,X],Z)+g([X,Z],Y)+g([Y,Z],X).\end{aligned}}$ 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 $2g(\nabla _{X}Y,Z)=X\left(g(Y,Z)\right)+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)+g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X).$ This proves the uniqueness of a torsion-free and metric-compatible condition, since 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 $g kl$ . Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field $\nabla X Y$ 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

↑ 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.
↑ Jost 2017, Definition 4.2.1.
↑ do Carmo 1992, pp.53-54; Milnor 1963, pp.47-48.
↑ Petersen 2016, Proposition 2.2.5; Wald 1984, p.35.
↑ Kobayashi & Nomizu 1963, Proposition IV.2.1.
↑ do Carmo 1992, p.54; Milnor 1963, Definition 8.5.
↑ Hawking & Ellis 1973, p.34; Helgason 2001, p.43; Jost 2017, Definition 4.1.7.
↑ Wald 1984, section 3.1.
↑ Hawking & Ellis 1973, p.41.
↑ See for instance pages 54-55 of Petersen (2016) or pages 158-159 of Kobayashi & Nomizu (1963) for presentations differing from those given here.
↑ Petersen 2016, p.66.
↑ Jost 2017, Lemma 4.1.1; Kobayashi & Nomizu 1963, Proposition III.7.6; Milnor 1963, p.48.
↑ Milnor 1963, p.48.
↑ Wald 1984, p.35.
↑ Milnor 1963, p.49.
↑ Milnor 1963, p.49; Wald 1984, p.36.
↑ 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.
↑ Jost 2017, p.194; O'Neill 1983, p.61.

Related Research Articles

In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line".

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the $-sphere$ , hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.

In Riemannian or pseudo-Riemannian geometry, 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.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In differential geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection, then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.

In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.

In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles.

<span class="mw-page-title-main">Holonomy</span> Concept in differential geometry

In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence of the curvature of the connection. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group $O(p, q)$ . As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors $, that produces an output vector representing the displacement within a tangent space when the tangent space is developed along an infinitesimal parallelogram whose sides are . It is skew symmetric in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how a screw moves in opposite ways when it is twisted in two directions.$

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.

In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to:

In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of a Riemannian or pseudo-Riemannian manifold.

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.

In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite.

References

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.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[FOOTNOTEdo_Carmo1992Theorem_2.3.6Helgason2001Theorem_I.9.1Jost2017Theorem_4.3.1KobayashiNomizu1963Theorem_IV.2.2Milnor1963Lemma_8.6O'Neill1983Theorem_3.11Petersen2016Theorem_2.2.2Wald1984Theorem_3.1.1-1] 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.

[FOOTNOTEJost2017Definition_4.2.1-2] Jost 2017, Definition 4.2.1.

[FOOTNOTEdo_Carmo1992pp.53-54Milnor1963pp.47-48-3] Carmo 1992, pp.53-54; Milnor 1963, pp.47-48.

[FOOTNOTEPetersen2016Proposition_2.2.5Wald1984p.35-4] Petersen 2016, Proposition 2.2.5; Wald 1984, p.35.

[FOOTNOTEKobayashiNomizu1963Proposition_IV.2.1-5] Kobayashi & Nomizu 1963, Proposition IV.2.1.

[FOOTNOTEdo_Carmo1992p.54Milnor1963Definition_8.5-6] Carmo 1992, p.54; Milnor 1963, Definition 8.5.

[FOOTNOTEHawkingEllis1973p.34Helgason2001p.43Jost2017Definition_4.1.7-7] Hawking & Ellis 1973, p.34; Helgason 2001, p.43; Jost 2017, Definition 4.1.7.

[FOOTNOTEWald1984section_3.1-8] Wald 1984, section 3.1.

[FOOTNOTEHawkingEllis1973p.41-9] 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.

[FOOTNOTEPetersen2016p.66-11] Petersen 2016, p.66.

[FOOTNOTEJost2017Lemma_4.1.1KobayashiNomizu1963Proposition_III.7.6Milnor1963p.48-12] Jost 2017, Lemma 4.1.1; Kobayashi & Nomizu 1963, Proposition III.7.6; Milnor 1963, p.48.

[FOOTNOTEMilnor1963p.48-13] Milnor 1963, p.48.

[FOOTNOTEWald1984p.35-14] Wald 1984, p.35.

[FOOTNOTEMilnor1963p.49-15] Milnor 1963, p.49.

[FOOTNOTEMilnor1963p.49Wald1984p.36-16] Milnor 1963, p.49; Wald 1984, p.36.

[FOOTNOTEdo_Carmo1992p.55HawkingEllis1973p.40Helgason2001p.48Jost2017p.194KobayashiNomizu1963p.160O'Neill1983p.61-17] 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.

[FOOTNOTEJost2017p.194O'Neill1983p.61-18] Jost 2017, p.194; O'Neill 1983, p.61.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

v t e Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem