# Metric tensor

A **metric tensor** is a smooth choice of inner product on the tangent space (flat vector space of directions at each point of a curved manifold) at every point of a manifold, written as a symmetric (0, 2) tensor field \( g \). It supplies the machinery for measuring lengths of tangent vectors, angles between them, and, by integration along curves, distances between points, in exactly the way the dot product does on [Euclidean space](https://www.edgepedia.org/euclidean-space).<sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/MetricTensor.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A symmetric, positive-definite tensor field of type (0, 2) on a smooth manifold; equivalently, a smoothly varying inner product on each tangent space \( T_pM \)<sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup> |
| Riemannian manifold | A smooth manifold endowed with such a metric; every smooth manifold admits one<sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup><sup> • </sup><sup>[3](https://math.berkeley.edu/~ltomczak/notes/Mich2022/DG_Notes.pdf)</sup>  |
| Line element | In a coordinate basis, \( ds^2 = g_{\mu\nu}\, dx^{\mu} dx^{\nu} \); in Euclidean space \( g_{ij} = \delta_{ij} \), reproducing the Pythagorean theorem<sup>[4](https://web.mit.edu/edbert/GR/gr1.pdf)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/MetricTensor.html)</sup>  |
| Index gymnastics | The inverse \( g^{ij} \), defined by \( g^{ij} g_{jk} = \delta^i_k \), raises and lowers indices: \( A_i = g_{ik} A^k \)<sup>[5](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)</sup> |
| Levi-Civita connection | There is a unique torsion-free connection compatible with \( g \); its components are the Christoffel symbols<sup>[3](https://math.berkeley.edu/~ltomczak/notes/Mich2022/DG_Notes.pdf)</sup> |
| Relativistic form | In general relativity \( g_{\mu\nu} \) generalizes the Minkowski metric \( \eta_{\mu\nu} = \mathrm{diag}(1,-1,-1,-1) \) and is the dynamical field of Einstein's equations \( G_{\mu\nu} = 8\pi G T_{\mu\nu} \)<sup>[6](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/Einstein%27s%20Field%20Equations.pdf)</sup> |
| Origin | Bernhard Riemann, then 28, proposed the generalization of Gauss's surface theory to spaces of any order in his 1854 Göttingen Habilitation lecture<sup>[7](https://onlinelibrary.wiley.com/doi/abs/10.1002/phbl.19540100702)</sup><sup> • </sup><sup>[8](https://www.nature.com/articles/119117c0)</sup>  |

## Definition and basic properties

Let \( M \) be a smooth manifold. A metric tensor \( g \) is a symmetric tensor field of type (0, 2): a map that takes two vector fields and returns a smooth function, with \( g(X, Y) = g(Y, X) \) and linearity in each argument. It is **Riemannian** when it is positive definite, meaning \( g(X, X) \ge 0 \) with equality only for the zero vector; then \( g \) determines an inner product \( g_p \) on each tangent space \( T_pM \), varying smoothly from point to point.<sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup> A manifold with such a metric is a [Riemannian manifold](https://www.edgepedia.org/riemannian-manifold).<sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup>

A common terminological trap is worth stating plainly: a [Riemannian metric](https://www.edgepedia.org/riemannian-metric) is not a distance function in the sense of metric-space theory. It is a generalization of an inner product, and the distance function on the manifold is derived from it by taking the infimum of lengths of curves joining two points.<sup>[9](https://www.wim.uni-mannheim.de/media/Lehrstuehle/wim/schmidt/FSS2024/Riemannian_Geometry/Web/RGch3.html)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/MetricTensor.html)</sup>

Existence is not a restriction. Since every vector bundle admits an inner product, every smooth manifold admits a Riemannian metric.<sup>[3](https://math.berkeley.edu/~ltomczak/notes/Mich2022/DG_Notes.pdf)</sup><sup> • </sup><sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup>

## Lengths, angles, and the line element

At a point \( p \), the metric gives the length of a tangent vector \( X \) as \( \|X\|_g = \sqrt{g(X, X)} \) and the angle \( \theta \) between vectors \( X \) and \( Y \) through \( \cos\theta = g(X, Y) / (\|X\|\, \|Y\|) \).<sup>[9](https://www.wim.uni-mannheim.de/media/Lehrstuehle/wim/schmidt/FSS2024/Riemannian_Geometry/Web/RGch3.html)</sup>

In a coordinate basis, the squared infinitesimal distance, the **line element**, takes the form

\[ ds^2 = g_{\mu\nu}\, dx^{\mu} dx^{\nu}. \]

This equation is a scalar statement, valid in any basis, and it can be taken as the definition of the metric tensor: given the requirement that infinitesimal squared distances be measurable, only one (0, 2) tensor can produce them.<sup>[4](https://web.mit.edu/edbert/GR/gr1.pdf)</sup> In Euclidean \( \mathbb{R}^n \) with standard coordinates, \( g_{ij} = \delta_{ij} \), the [Kronecker delta](https://www.edgepedia.org/kronecker-delta), and the line element reduces to \( ds^2 = dx_1^2 + dx_2^2 + \cdots \), the [Pythagorean theorem](https://www.edgepedia.org/pythagorean-theorem).<sup>[2](https://mathworld.wolfram.com/MetricTensor.html)</sup><sup> • </sup><sup>[10](https://archive.lib.msu.edu/crcmath/math/math/m/m217.htm)</sup>

**Coordinates change the components, not the geometry.** In polar coordinates \( (\rho, \theta) \) on the Euclidean plane, the line element becomes

\[ ds^2 = d\rho^2 + \rho^2\, d\theta^2, \]

so \( g_{\rho\rho} = 1 \) and \( g_{\theta\theta} = \rho^2 \). The coordinate basis vector \( \vec{e}_{\theta} \) is not a unit vector; its squared length is \( \rho^2 \). All information about lengths is carried by the metric components rather than by normalized basis vectors.<sup>[4](https://web.mit.edu/edbert/GR/gr1.pdf)</sup> Under a coordinate change \( x^{\mu} \to x^{\mu'} \), the components transform as

\[ g'_{\mu\nu} = \frac{\partial x^{\alpha}}{\partial x^{\mu'}} \frac{\partial x^{\beta}}{\partial x^{\nu'}}\, g_{\alpha\beta}, \]

which is exactly the (0, 2) tensor transformation law, forced by the invariance of \( ds^2 \).<sup>[5](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)</sup>

## Raising and lowering indices

Because \( g_{ij} \) is nonsingular, it has an inverse \( g^{ij} \), defined by \( g^{ij} g_{jk} = \delta^i_k \), which itself transforms as a (2, 0) tensor.<sup>[5](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)</sup> The metric and its inverse convert vectors into one-forms and back: lowering an index means \( A_i = g_{ik} A^k \), and raising means \( B^i = g^{ij} B_j \). The operation extends to tensors of any rank, for example \( D_{ij}{}^{\;\;\;pq} = g_{lp} g_{mq} D_{ij}{}^{lm} \), with care needed over which indices are contracted.<sup>[5](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)</sup> The scalar product of two vectors is the contraction \( g_{ij} A^i B^j \), and the squared magnitude of \( A \) is \( g_{ij} A^i A^j \), both invariant under coordinate change.<sup>[5](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)</sup>

## The Levi-Civita connection

A metric unlocks the central constructions of [Riemannian geometry](https://www.edgepedia.org/riemannian-geometry). The fundamental theorem states that on a Riemannian manifold there exists a unique torsion-free connection that is orthogonal, that is, compatible with \( g \) in the sense that the metric is covariantly constant.<sup>[3](https://math.berkeley.edu/~ltomczak/notes/Mich2022/DG_Notes.pdf)</sup> This is the **Levi-Civita connection**, and its coordinate components \( \Gamma^i_{jk} \) are the [Christoffel symbols](https://www.edgepedia.org/christoffel-symbols).<sup>[3](https://math.berkeley.edu/~ltomczak/notes/Mich2022/DG_Notes.pdf)</sup>

## Pseudo-Riemannian metrics and relativity

If the positive-definiteness requirement is relaxed to nondegeneracy alone, the result is a **pseudo-Riemannian** (or semi-Riemannian) metric, and these manifolds are the framework for relativity theory.<sup>[1](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)</sup> In special relativity the invariant interval is \( d\tau^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta} \) with the Minkowski metric \( \eta_{\alpha\beta} = \mathrm{diag}(1, -1, -1, -1) \); in general relativity \( g_{\mu\nu} \) generalizes \( \eta_{\mu\nu} \).<sup>[6](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/Einstein%27s%20Field%20Equations.pdf)</sup> In flat [Minkowski space](https://www.edgepedia.org/minkowski-space) in spherical coordinates \( (ct, r, \theta, \varphi) \), the metric already acquires position-dependent components, \( g_{\theta\theta} = r^2 \) and \( g_{\varphi\varphi} = r^2 \sin^2\theta \).<sup>[11](https://hepweb.ucsd.edu/ph110b/110b_notes/node74.html)</sup>

In general relativity the metric tensor is the dynamical field. Einstein's field equations,

\[ G_{\mu\nu} = 8\pi G T_{\mu\nu}, \qquad G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2} R\, g_{\mu\nu}, \]

determine \( g_{\mu\nu} \) from the energy-momentum tensor \( T_{\mu\nu} \), with \( R_{\mu\nu} \) the Ricci tensor and \( R \) the curvature scalar.<sup>[6](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/Einstein%27s%20Field%20Equations.pdf)</sup> Large masses can change the metric significantly and can introduce off-diagonal components.<sup>[11](https://hepweb.ucsd.edu/ph110b/110b_notes/node74.html)</sup> [Spacetime](https://www.edgepedia.org/spacetime) is locally Minkowskian: at any event there is a coordinate transformation making the metric Minkowskian at that event, though generally not throughout a neighborhood unless spacetime is flat there.<sup>[5](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)</sup><sup> • </sup><sup>[13](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S3.html)</sup> Solving the vacuum field equations \( T_{ij} = 0 \) around a spherically symmetric mass yields the classical tests of the theory, including the perihelion rotation of Mercury at 42.9 arc-seconds per century and the 1.75 arc-second deflection of starlight grazing the Sun.<sup>[12](https://www.grc.nasa.gov/www/k-12/Numbers/Math/Mathematical_Thinking/field_equations.htm)</sup>

## Historical note

The concept descends from [Bernhard Riemann](https://www.edgepedia.org/bernhard-riemann)'s Habilitation lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen", delivered on 11 June 1854 before the Philosophical Faculty at [Göttingen](https://www.edgepedia.org/gottingen), in which he generalized Gauss's theory of surfaces to spaces of any order.<sup>[7](https://onlinelibrary.wiley.com/doi/abs/10.1002/phbl.19540100702)</sup> Bianchi, Beltrami, and others developed the subject, and Ricci extended it with tensor analysis and his absolute calculus; the connection between Riemannian geometry and physics became explicit only decades later in Einstein's general relativity.<sup>[8](https://www.nature.com/articles/119117c0)</sup><sup> • </sup><sup>[7](https://onlinelibrary.wiley.com/doi/abs/10.1002/phbl.19540100702)</sup>

## References

1. [Riemannian Manifolds, lecture notes (Riemannian metrics), Texas Tech University](https://www.math.ttu.edu/~apampano/Riemannian_Geometry.pdf)
2. [Metric Tensor, Wolfram MathWorld](https://mathworld.wolfram.com/MetricTensor.html)
3. [Differential Geometry lecture notes, L. Tomczak-Jaegermann seminar notes, UC Berkeley (Section 8, Riemannian geometry)](https://math.berkeley.edu/~ltomczak/notes/Mich2022/DG_Notes.pdf)
4. [E. Bertschinger, Introduction to Tensor Calculus for General Relativity, MIT notes](https://web.mit.edu/edbert/GR/gr1.pdf)
5. [M. Hendry, Space Time and Gravity lecture notes, University of Glasgow](https://www.astro.gla.ac.uk/users/martin/teaching/gr2/oldnotes/grnotes2_complete.pdf)
6. [The Einstein Field Equations, JPL MPG source document](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/Einstein%27s%20Field%20Equations.pdf)
7. [B. Riemann (1954), "Über die Hypothesen, welche der Geometrie zu Grunde liegen", Physikalische Blätter 10, 296–306](https://onlinelibrary.wiley.com/doi/abs/10.1002/phbl.19540100702)
8. [Review of L. P. Eisenhart, Riemannian Geometry, Nature (1926)](https://www.nature.com/articles/119117c0)
9. [Metrics and Connections, Riemannian Geometry course notes, University of Mannheim](https://www.wim.uni-mannheim.de/media/Lehrstuehle/wim/schmidt/FSS2024/Riemannian_Geometry/Web/RGch3.html)
10. [Metric Tensor, CRC Concise Encyclopedia of Mathematics archive](https://archive.lib.msu.edu/crcmath/math/math/m/m217.htm)
11. [The Metric Tensor, UCSD Physics 110B notes](https://hepweb.ucsd.edu/ph110b/110b_notes/node74.html)
12. [Field Equations and Equations of Motion, NASA Glenn Research Center](https://www.grc.nasa.gov/www/k-12/Numbers/Math/Mathematical_Thinking/field_equations.htm)
13. [3 Introducing Riemannian Geometry‣ General Relativity by David Tong](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S3.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Differential geometry*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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