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Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is a symmetric, non-degenerate bilinear form defined on the tangent spaces of a smooth manifold, enabling the measurement of distances, angles, and volumes in a manner independent of the choice of coordinates.^[1] It generalizes the Euclidean dot product to curved spaces, expressed locally as ds^2 = g_{ij} \, dx^i \, dx^j, where g_{ij} are the components of the metric tensor, which are smooth functions on the manifold.^[2] For Riemannian metrics, the tensor is positive definite, providing an inner product on each tangent space that induces a geometry where lengths of curves are given by L(\gamma) = \int \sqrt{g_{ij} \dot{\gamma}^i \dot{\gamma}^j} \, dt, while pseudo-Riemannian metrics, such as those with indefinite signature (e.g., (1,3) for spacetime), allow for timelike, spacelike, and null separations.^[1] The concept was introduced by Bernhard Riemann in his 1854 habilitation lecture "On the Hypotheses Which Lie at the Foundations of Geometry," where he extended the notion of curvature to higher-dimensional manifolds using what would later be formalized as the metric tensor, laying the groundwork for intrinsic geometry.^[3] This work, published posthumously in 1868, provided 20 independent quantities to describe the curvature of four-dimensional space, influencing modern differential geometry.^[3] In Riemannian geometry, the metric tensor defines geodesics as shortest paths and enables the computation of the Riemann curvature tensor, which quantifies how the manifold deviates from flatness.^[1] Beyond pure mathematics, the metric tensor plays a central role in theoretical physics, particularly in Albert Einstein's general theory of relativity (1915), where the pseudo-Riemannian metric g_{\mu\nu} describes the geometry of spacetime, with its curvature sourced by mass and energy via the Einstein field equations G_{\mu\nu} = 8\pi T_{\mu\nu}.^[3] It also appears in special relativity through the Minkowski metric \eta_{\mu\nu} = \operatorname{diag}(-1,1,1,1), which measures proper time and interval in flat spacetime, and in various applications like general relativity simulations, cosmology, and gauge theories.^[1] Every smooth manifold admits a Riemannian metric, ensuring the framework's broad applicability.^[2]

Introduction

Arc length and line element

In the mid-19th century, Bernhard Riemann sought to generalize Euclidean geometry to spaces of arbitrary dimension and curvature, motivated by the need to describe geometries without relying on an embedding in a higher-dimensional Euclidean space.^[4] This approach, outlined in his 1854 habilitation lecture, introduced the concept of a metric that locally approximates distances, enabling the measurement of lengths and angles intrinsically on curved manifolds.^[4] The line element ds^2 = g_{ij} \, dx^i \, dx^j provides the first-order infinitesimal approximation of the squared distance between nearby points in local coordinates on a manifold, where g_{ij} are the components of the metric tensor and dx^i are coordinate differentials.^[5] In this expression, summation over repeated indices i, j is implied, and the metric tensor encodes the geometry by contracting the differentials to yield a scalar measure of separation.^[6] For a smooth curve \gamma: [a, b] \to M parameterized by t, the arc length L(\gamma) is defined as the integral

L(\gamma) = \int_a^b \sqrt{g_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt}} \, dt,

which sums the infinitesimal lengths along the path.^[5] This formula arises in the Riemannian case, where the metric is positive-definite, ensuring g_{ij} v^i v^j > 0 for any nonzero tangent vector v, so the integrand is real and positive, yielding a well-defined length.^[7] The positive-definiteness guarantees that the square root is meaningful, distinguishing Riemannian metrics from indefinite ones used in other contexts.^[5]

Invariance under coordinate transformations

The metric tensor plays a crucial role in ensuring that geometric measurements, such as the arc length along a curve on a manifold, remain independent of the choice of coordinate system. This invariance arises because the metric is defined intrinsically on the manifold, without reference to specific coordinates, and its components transform in a manner that preserves the underlying geometry under diffeomorphisms—smooth, invertible maps between coordinate charts.^[8] Under a coordinate transformation from coordinates x^i to new coordinates x'^k, the components of the metric tensor g_{ij} transform according to the law for a covariant (0,2)-tensor:

g'_{kl} = \frac{\partial x^i}{\partial x'^k} \frac{\partial x^j}{\partial x'^l} g_{ij}.

This transformation rule reflects the contravariant nature of the differentials dx^i (which transform as dx'^k = \frac{\partial x'^k}{\partial x^m} dx^m) and the covariant nature of the metric, ensuring that the bilinear form g(\cdot, \cdot) pulls back consistently under the diffeomorphism. The pullback operation \phi^* g, for a diffeomorphism \phi, defines the metric on the source manifold by (\phi^* g)(u,v) = g(d\phi(u), d\phi(v)) for tangent vectors u, v, thereby maintaining geometric consistency across charts.^[9]^[8] To see how this guarantees the invariance of arc length, consider a curve \gamma parametrized by t, with arc length given by \int \sqrt{g_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt}} \, dt. In the new coordinates, the line element becomes ds'^2 = g'_{kl} dx'^k dx'^l. Substituting the transformation laws yields

ds'^2 = \frac{\partial x^i}{\partial x'^k} \frac{\partial x^j}{\partial x'^l} g_{ij} \, dx'^k dx'^l = g_{ij} dx^i dx^j = ds^2,

since dx'^k = \frac{\partial x'^k}{\partial x^m} dx^m implies the Jacobians cancel appropriately. Thus, the integral for the total arc length \int ds is unchanged, confirming that the metric encodes a coordinate-independent notion of length. This property extends to the broader role of the metric in defining diffeomorphism-invariant structures in differential geometry.^[9]^[8]

Length, angle, and area elements

The Riemannian metric tensor g on a manifold induces a notion of length for tangent vectors at each point. For a tangent vector v \in T_p M at a point p \in M, the length is defined as \|v\| = \sqrt{g(v, v)}, or in local coordinates, \|v\| = \sqrt{g_{ij} v^i v^j}, where the summation convention over repeated indices is used.^[10] This length measure arises from the positive-definiteness of the metric, ensuring that g(v, v) > 0 for nonzero v.^[11] The metric also defines angles between tangent vectors, providing a local geometric structure analogous to Euclidean spaces. Specifically, for two nonzero tangent vectors u, v \in T_p M, the angle \theta between them satisfies \cos \theta = g(u, v) / (\|u\| \|v\|), where g(u, v) = g_{ij} u^i v^j.^[12] This formula leverages the inner product properties of the metric, with \theta \in [0, \pi] due to the symmetry and positive-definiteness of g.^[10] On two-dimensional submanifolds or surfaces, the metric induces an area element that measures infinitesimal areas. In local coordinates (x^1, x^2), this area element is given by the 2-form \sqrt{\det g} \, dx^1 \wedge dx^2, where g denotes the matrix of metric components g_{ij}.^[12] The positive square root is well-defined in the Riemannian case, as \det g > 0 everywhere.^[11] This construction extends naturally to higher-dimensional volume elements on n-manifolds. The induced volume form is \sqrt{\det g} \, dx^1 \wedge \cdots \wedge dx^n, which integrates to yield volumes invariant under the metric's positive-definite signature.^[10] In the Riemannian setting, the absolute value is unnecessary, as the determinant remains positive, ensuring a consistent orientation-independent measure.^[12]

Definition

Metric as a bilinear form

In differential geometry, the metric tensor at a point p on a manifold M, denoted g_p, is defined as a symmetric, non-degenerate bilinear map g_p: T_p M \times T_p M \to \mathbb{R}, where T_p M is the tangent space at p.^[13]^[14] This structure provides a way to measure lengths and angles locally, extending the intuitive arc length concept to abstract tangent spaces.^[15] The bilinearity of g_p means it is linear in each argument over the real numbers \mathbb{R}; that is, for all u, v, w \in T_p M and scalars a, b \in \mathbb{R},

g_p(au + bv, w) = a g_p(u, w) + b g_p(v, w), \quad g_p(u, av + bw) = a g_p(u, v) + b g_p(u, w).

Symmetry follows from g_p(u, v) = g_p(v, u) for all u, v \in T_p M.^[13]^[14] For a Riemannian metric, non-degeneracy is strengthened to positive-definiteness: g_p(v, v) > 0 for all nonzero v \in T_p M, ensuring g_p induces a norm \|v\|_p = \sqrt{g_p(v, v)} on the tangent space.^[15]^[13] Unlike general tensors, which may transform in arbitrary ways, the metric g_p functions specifically as an inner product on T_p M, equipping the vector space with a geometry that allows computation of dot products, lengths, and angles between tangent vectors at p.^[14] This inner product property distinguishes it as the fundamental algebraic tool for local Riemannian structure. For smooth vector fields X and Y on M, the metric extends pointwise via g(X, Y)_p = g_p(X_p, Y_p) for each p \in M, enabling global expressions while preserving the local bilinear nature.^[13]^[15]

Riemannian metric tensor field

A Riemannian metric tensor field on a smooth manifold M is defined as a smooth section g of the tensor bundle T^0_2(M), which assigns to each point p \in M a symmetric, positive-definite bilinear form g_p: T_p M \times T_p M \to \mathbb{R} on the tangent space T_p M.^[16] This structure extends the local bilinear form at individual points to a global field compatible with the manifold's topology and differentiable structure.^[16] The key requirement for g is its smoothness: for any smooth vector fields X, Y on M, the function p \mapsto g_p(X_p, Y_p) must be a smooth real-valued function on M, ensuring that the metric varies continuously and differentiably across the manifold.^[16] This smoothness condition aligns g with the C^\infty-structure of M, allowing the metric to interact seamlessly with differential operators and other geometric constructions.^[16] In terms of an atlas of charts on M, g qualifies as a Riemannian metric tensor field if it is C^\infty and, at every point p, the associated g_p satisfies the algebraic properties of a positive-definite symmetric bilinear form.^[16] This framework generalizes to pseudo-Riemannian metric tensor fields, where g_p is a non-degenerate symmetric bilinear form of indefinite signature (e.g., with index \nu indicating the number of negative eigenvalues), though the Riemannian case restricts to positive-definite signatures for Euclidean-like geometry.^[17]

Components and Algebraic Properties

Coordinate components and tensor notation

In a local coordinate chart (x^1, \dots, x^n) on a manifold M, the metric tensor g at a point p \in M is represented by components g_{ij}(p), which form the entries of a symmetric n \times n matrix (g_{ij}(p)) that is positive definite for Riemannian metrics.^[18]^[11] These components are defined by g_{ij}(p) = g(\partial_i, \partial_j)|_p, where \partial_i = \partial/\partial x^i denotes the coordinate basis vectors, and they vary smoothly with p as functions g_{ij}: U \to \mathbb{R} on an open set U \subset M.^[18] The metric tensor is expressed in tensor notation as

g = g_{ij} \, dx^i \otimes dx^j,

where the Einstein summation convention is employed, implying summation over repeated indices i, j = 1, \dots, n.^[11]^[18] This abstract index notation captures the bilinear form without explicit summation symbols, facilitating contractions such as the inner product g(X, Y) = g_{ij} X^i Y^j for vector fields X = X^i \partial_i and Y = Y^j \partial_j.^[11] Under a change of coordinates from (x^k) to (\tilde{x}^i), the components transform covariantly as

\tilde{g}_{ij} = \frac{\partial x^k}{\partial \tilde{x}^i} \, g_{kl} \, \frac{\partial x^l}{\partial \tilde{x}^j},

ensuring the metric remains independent of the coordinate choice.^[18]^[11] This law reflects the tensorial nature of g, with the transformation matrix related to the Jacobian of the coordinate map. Since the matrix (g_{ij}(p)) is symmetric and positive definite at each point p, it can be diagonalized by an orthogonal change of basis in the tangent space T_p M.^[11] In particular, there exist local orthonormal frames \{e_i\} at p such that g(e_i, e_j) = \delta_{ij}, rendering the components diagonal with entries 1 along the diagonal. For pseudo-Riemannian metrics, such frames yield g_{ij} = \operatorname{diag}(\pm 1, 1, \dots, 1), depending on the signature.^[19]^[11]

Signature and metric types

The signature of a metric tensor g on an n-dimensional manifold M is specified by the pair (p, q), where p denotes the number of positive eigenvalues and q the number of negative eigenvalues of the symmetric bilinear form defined by g, satisfying p + q = n.^[11] This classification arises from the diagonalization of the metric in an orthonormal basis, where the signs of the eigenvalues determine the type of inner product induced on the tangent spaces.^[11] Riemannian metrics correspond to the signature (n, 0), making the bilinear form positive definite, such that g(X, X) > 0 for all nonzero tangent vectors X.^[16] This positive definiteness ensures that lengths and angles are well-defined in a manner analogous to Euclidean space, enabling the study of geometric properties like curvature and geodesics on the manifold.^[16] Lorentzian metrics have signature (1, n-1) or (n-1, 1), resulting in an indefinite bilinear form with one eigenvalue of opposite sign to the rest.^[20] These metrics produce a hyperbolic geometry, distinguishing timelike, spacelike, and null directions in the tangent spaces, which is essential for frameworks involving indefinite geometries.^[20] Degenerate metrics, where the bilinear form has a zero eigenvalue and the tensor is not invertible, are nonstandard and generally excluded from pseudo-Riemannian definitions, as they fail to provide a nondegenerate inner product on the full tangent space.^[11]

Inverse metric and index raising/lowering

The inverse metric tensor, denoted by g^{ij}, is the matrix inverse of the covariant metric tensor g_{ij}. It satisfies the orthogonality relation g^{ik} g_{kj} = \delta^i_j, where \delta^i_j is the Kronecker delta, ensuring that the inverse precisely undoes the action of the metric on tensor components.^[21]^[22] This contravariant tensor of type (2,0) transforms under coordinate changes as g^{ij}(x') = \frac{\partial x'^i}{\partial x^k} \frac{\partial x'^j}{\partial x^l} g^{kl}(x), maintaining its tensorial character.^[23] The inverse metric enables the raising of indices on tensor components, converting covariant quantities to contravariant ones. For a covector with components v_i, the raised version is given by v^i = g^{ij} v_j, which identifies the covector with a vector in the tangent space via the metric's duality.^[21]^[22] Similarly, for higher-rank tensors, indices are raised componentwise using g^{ij}. Conversely, the covariant metric lowers indices: for a vector with components v^j, the lowered version is v_i = g_{ij} v^j, mapping to the cotangent space.^[21]^[23] These operations are invertible and preserve the tensor's algebraic structure, facilitating computations in curvilinear coordinates. A key property of the inverse metric follows from linear algebra: the determinant of g^{ij} is the reciprocal of the determinant of g_{ij}, i.e., \det(g^{ij}) = 1 / \det(g_{ij}).^[24] This relation underscores the non-degeneracy of the metric tensor, as \det(g_{ij}) \neq 0 guarantees the existence of the inverse, allowing consistent index manipulations without singularity.^[25]

Induced and Derived Structures

Induced metrics on submanifolds

When a submanifold N is immersed in a Riemannian manifold (M, g_M) via a smooth immersion f: N \to M, the ambient metric g_M induces a metric on N through the pullback operation, defined as g_N = f^* g_M.^[26] This pullback metric restricts the inner product from the tangent space T_p M to the subspace df_p(T_q N) for each q \in N, ensuring that lengths and angles measured on N are consistent with those in the ambient space along the image of the immersion.^[27] The resulting g_N equips N with its own Riemannian structure, preserving the intrinsic geometry inherited from M.^[26] In local coordinates, suppose N has coordinates (y^a) and M has coordinates (x^i), with the immersion expressed locally as x^i = x^i(y). The components of the induced metric g_N are then given by

g_{ab} = g_{ij} \frac{\partial x^i}{\partial y^a} \frac{\partial x^j}{\partial y^b},

where g_{ij} are the components of g_M, and the partial derivatives represent the Jacobian of the immersion.^[26] This formula computes the metric tensor on N by projecting the ambient metric onto the tangent directions of the submanifold, allowing explicit calculations in coordinate charts.^[27] A key application arises with hypersurfaces, which are submanifolds of codimension one embedded in M. For such a hypersurface S \subset M, the induced metric g_S = \iota^* g_M (where \iota: S \hookrightarrow M is the inclusion) directly inherits the ambient metric restricted to the tangent bundle of S, enabling the study of geometric properties like curvature along the surface.^[26] This restriction is fundamental in extrinsic geometry, where the induced metric determines the intrinsic geometry of the hypersurface, while the second fundamental form describes how it bends within the ambient space.^[26]

Canonical volume form

In a smooth manifold equipped with a metric tensor g, the canonical volume form provides a natural way to define volumes and integrate over the manifold, derived intrinsically from the geometry induced by g. On an oriented n-dimensional manifold, this form is a top-degree differential form that measures the "infinitesimal volume" in each tangent space, ensuring compatibility with the metric's structure.^[28] In local coordinates (x^1, \dots, x^n), the canonical volume form \omega_g is expressed as

\omega_g = \sqrt{|\det(g_{ij})|} \, dx^1 \wedge \cdots \wedge dx^n,

where g_{ij} are the components of the metric tensor and \det(g_{ij}) is its determinant. This expression arises from the volume of the parallelepiped spanned by the coordinate basis vectors \partial/\partial x^i, which is \sqrt{|\det(g_{ij})|}.^[28]^[29] The form \omega_g is independent of the choice of coordinates, as it transforms covariantly under coordinate changes. If (y^j) are new coordinates related by a Jacobian matrix J = (\partial y^j / \partial x^i) with \det(J) \neq 0, the determinant transforms as \det(\tilde{g}_{kl}) = \det(g_{ij}) / (\det(J))^2, ensuring \sqrt{|\det(\tilde{g}_{kl})|} \, |\det(J)| = \sqrt{|\det(g_{ij})|}, thus preserving the volume element across charts.^[28] For integration, the volume of a region U \subset M or the integral of a compactly supported function f \in C_c^\infty(M) is given by

\int_U \omega_g = \int_U \sqrt{|\det(g_{ij})|} \, dx^1 \cdots dx^n, \quad \int_M f \, \omega_g = \sum_\alpha \int_{\phi_\alpha(U_\alpha)} (f \circ \phi_\alpha) \rho_\alpha \sqrt{|\det(g_{ij}^\alpha)|} \, dx^1_\alpha \cdots dx^n_\alpha,

where \{\phi_\alpha\} is an atlas and \{\rho_\alpha\} a partition of unity subordinate to it; this defines a positive measure on the manifold.^[28]^[29] In the Riemannian case, where g is positive definite, \det(g_{ij}) > 0, so the form simplifies to \omega_g = \sqrt{\det(g_{ij})} \, dx^1 \wedge \cdots \wedge dx^n, yielding an oriented volume form without absolute value. For pseudo-Riemannian metrics, such as Lorentzian signatures in relativity, the absolute value |\det(g_{ij})| is retained to ensure a positive volume measure, accommodating indefinite signatures while maintaining the form's role as a density for integration.^[28]^[30]

Tangent-cotangent bundle isomorphism

The Riemannian metric tensor g on a smooth manifold M induces a pair of musical isomorphisms between the tangent bundle TM and the cotangent bundle T^*M. These isomorphisms, commonly referred to as the flat map \flat: TM \to T^*M and the sharp map \sharp: T^*M \to TM, arise pointwise from the bilinear form provided by g at each tangent space T_pM. Specifically, for a tangent vector v \in T_pM, the flat map is defined by \flat(v) = g(v, \cdot), which assigns to v the linear functional \omega \in T_p^*M given by \omega(w) = g_p(v, w) for all w \in T_pM. The sharp map, as the inverse, uses the inverse metric tensor g^{-1} to map a covector \omega \in T_p^*M back to a tangent vector via \sharp(\omega) = g^{-1}(\cdot, \omega), or in coordinates, \sharp(\omega)_i = g^{ij} \omega_j. These maps are smooth bundle morphisms because g is a smooth section of the tensor bundle, preserving the smooth structure of TM and T^*M.^[31] The non-degeneracy of the metric tensor—meaning that g_p(v, w) = 0 for all w \in T_pM implies v = 0—ensures that both \flat and \sharp are bijective at each fiber, yielding a global bundle isomorphism TM \cong T^*M. This isomorphism identifies tangent vectors with covectors globally over M, facilitating the transfer of geometric structures between the bundles.^[31] A key consequence of this identification is the simplification of tensor algebra on the manifold: multivectors and multivectors can be equated via repeated applications of \flat and \sharp, allowing expressions involving mixed tensor types to be unified under a single framework and easing computations in coordinates where the metric components raise and lower indices. This duality is fundamental in Riemannian geometry, enabling the treatment of differential forms and vector fields on equal footing without explicit dual pairings.

Geometric Interpretations

Geodesics via energy functional

The metric tensor on a Riemannian manifold provides a way to measure lengths and angles, enabling the study of geodesics as the "straightest" paths via variational methods. A key tool for this is the energy functional, which quantifies the "energy" of a curve and whose critical points correspond to geodesics. For a smooth curve \gamma: [a, b] \to M on a Riemannian manifold (M, g), the energy functional is defined as

E(\gamma) = \frac{1}{2} \int_a^b g(\gamma'(t), \gamma'(t)) \, dt,

where g(\gamma'(t), \gamma'(t)) is the squared speed of the curve at time t. Geodesics are precisely the critical points of this functional under variations that fix the endpoints \gamma(a) and \gamma(b). To find these critical points, one applies the calculus of variations: the first variation of E vanishes if and only if \gamma satisfies the Euler-Lagrange equations derived from the Lagrangian L(t, \dot{\gamma}) = \frac{1}{2} g(\dot{\gamma}, \dot{\gamma}). These equations simplify to the geodesic equation

\nabla_{\gamma'(t)} \gamma'(t) = 0,

where \nabla denotes the Levi-Civita connection compatible with g. This condition means that the covariant acceleration of \gamma is zero, capturing the notion of uniform motion in curved space. The energy functional relates directly to the arc length functional L(\gamma) = \int_a^b \sqrt{g(\gamma'(t), \gamma'(t))} \, dt, which measures the total length of the curve. For curves reparameterized to have constant speed (specifically, unit speed where g(\gamma', \gamma') = 1), the energy becomes E(\gamma) = \frac{1}{2} (b - a), and minimizing E is equivalent to minimizing L up to reparameterization, since local minimizers of E can be rescaled to unit-speed geodesics that locally minimize length. The Levi-Civita connection \nabla, unique as the torsion-free metric-compatible connection on (M, g), is expressed in local coordinates via the Christoffel symbols of the second kind:

\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \right),

where g^{kl} are components of the inverse metric and \partial_i denotes partial differentiation with respect to the i-th coordinate. These symbols encode how the metric varies, defining the coordinate form of the covariant derivative \nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k and thus the geodesic equation \frac{d^2 x^k}{dt^2} + \Gamma^k_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} = 0. This formulation, originating from the work of Elwin Bruno Christoffel and Tullio Levi-Civita, allows explicit computation of geodesics from the metric components.

Applications in differential geometry

The Levi-Civita connection is the unique torsion-free affine connection on a Riemannian manifold that is compatible with the metric tensor, meaning it preserves the metric under parallel transport.^[32] This connection, introduced by Tullio Levi-Civita in his seminal work on parallelism in general manifolds, enables the covariant differentiation of tensor fields in a manner consistent with the geometry defined by the metric.^[33] Its torsion-free property ensures that the connection aligns with the Lie bracket of vector fields, while metric compatibility guarantees that the inner product of parallel-transported vectors remains unchanged, facilitating the study of intrinsic geometry without reference to an embedding space.^[34] The Riemann curvature tensor arises naturally from the Levi-Civita connection as a measure of how much the connection deviates from being flat, quantifying the intrinsic curvature of the manifold at each point.^[35] Derived from the second covariant derivatives of vector fields, it captures the non-commutativity of mixed partial derivatives in curved spaces, with components expressed in terms of the metric and its Christoffel symbols.^[36] Contractions of the Riemann tensor yield the Ricci curvature tensor, which contracts the first and third indices to produce a symmetric (0,2)-tensor describing average sectional curvatures, and further contraction along the remaining indices gives the scalar curvature, a single scalar function representing the overall trace of the Ricci tensor.^[37] These derived tensors are fundamental for analyzing global properties like the Einstein field equations in geometry, though here they underscore the metric's role in encoding all curvature information.^[38] Metric completeness on a Riemannian manifold is defined via Cauchy sequences with respect to the distance induced by the metric tensor, where every such sequence converges to a point within the manifold.^[39] The Hopf-Rinow theorem establishes that a connected Riemannian manifold is complete if and only if it is geodesically complete, meaning all geodesic rays can be extended indefinitely, and equivalently, closed and bounded subsets are compact. This completeness criterion links local metric properties to global compactness, implying that complete manifolds with finite diameter are compact, a result pivotal for theorems on the existence of minimizing geodesics and the structure of bounded regions.^[40] Conformal metrics are obtained by scaling a given Riemannian metric g by a positive smooth function, yielding g' = e^{2\phi} g where \phi is a scalar field on the manifold, which preserves angles between curves while altering lengths.^[41] This transformation maintains the conformal class of the metric, ensuring that the Levi-Civita connection of g' relates to that of g through additional terms involving the gradient of \phi, thus preserving local shape information essential for applications in complex analysis and Teichmüller theory.^[42] Such metrics are crucial in studying equivalence classes of geometries up to angle-preserving diffeomorphisms, facilitating the uniformization theorem for Riemann surfaces.^[43]

Examples

Euclidean metric in flat space

The Euclidean metric on flat space \mathbb{R}^n provides the simplest example of a Riemannian metric, serving as the foundational model for understanding more general metric tensors. In Cartesian coordinates x^1, \dots, x^n, the metric tensor has constant components g_{ij} = \delta_{ij}, where \delta_{ij} is the Kronecker delta (equal to 1 if i = j and 0 otherwise).^[44] The associated line element is thus given by

ds^2 = \delta_{ij} \, dx^i \, dx^j = \sum_{i=1}^n (dx^i)^2,

which measures the infinitesimal squared distance between points in the space.^[11] This metric exhibits several key properties that highlight its flatness. The components are constant throughout the space, independent of position, which simplifies computations in differential geometry.^[11] Consequently, the Christoffel symbols of the Levi-Civita connection vanish identically: \Gamma^k_{ij} = 0 for all indices, implying no intrinsic connection terms beyond partial derivatives.^[11] The curvature tensor also vanishes, R = 0, confirming that the space has zero sectional curvature everywhere and is geodesically flat, with straight lines serving as the shortest paths.^[11] The group of isometries preserving this metric consists of all transformations that maintain distances and angles, forming the Euclidean group E(n), which is the semidirect product of translations in \mathbb{R}^n and rotations in the orthogonal group O(n).^[45] This group acts transitively on the space, reflecting its high degree of symmetry. In broader applications, the Euclidean metric on \mathbb{R}^n models the local geometry of any Riemannian manifold near a point, where the metric can be approximated by its value at that point via normal coordinates, providing a flat tangent space isomorphism.^[11]

Round metric on the sphere

The round metric on the 2-sphere S^2 of radius R provides a fundamental example of a curved Riemannian metric, illustrating how the metric tensor encodes intrinsic geometry independent of the embedding space. In spherical coordinates (\theta, \phi), where \theta \in [0, \pi] is the polar angle and \phi \in [0, 2\pi) is the azimuthal angle, the line element takes the form

ds^2 = R^2 (d\theta^2 + \sin^2 \theta \, d\phi^2).

This expression arises as the pullback of the Euclidean metric on \mathbb{R}^3 under the standard parametrization of the sphere.^[46] The metric is diagonal in these coordinates, with components g_{\theta\theta} = R^2, g_{\phi\phi} = R^2 \sin^2 \theta, and off-diagonal terms g_{\theta\phi} = g_{\phi\theta} = 0.^[47] These components reflect the varying "stretch" in the \phi-direction due to the sphere's latitude dependence, while the \theta-direction remains uniformly scaled by the radius. The geodesics of the round metric are precisely the great circles, which are the intersections of the sphere with planes through its center.^[48] When parameterized by arc-length, these geodesics exhibit constant speed, corresponding to uniform motion along the shortest paths on the surface.^[49] This property underscores the metric's role in defining distances and paths intrinsically, without reference to the ambient space. A key geometric feature of the round metric is its constant positive curvature, specifically the Gaussian curvature K = 1/R^2.^[50] This uniform curvature distinguishes the sphere from flat spaces and quantifies its global topology, as confirmed by the Gauss-Bonnet theorem relating total curvature to the Euler characteristic. The round metric thus serves as the canonical example of a space of constant sectional curvature in Riemannian geometry.

Lorentzian metric in special relativity

In special relativity, the Lorentzian metric is exemplified by the Minkowski metric, which describes the flat spacetime geometry in inertial coordinates. The metric tensor is given by \eta_{\mu\nu} = \operatorname{diag}(-1, 1, 1, 1), where Greek indices run from 0 to 3, with x^0 = ct the time coordinate and x^i (for i=1,2,3) the spatial coordinates.^[51] The line element is then ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = -c^2 dt^2 + dx^2 + dy^2 + dz^2, invariant under Lorentz transformations.^[51] This metric has the Lorentzian signature (-, +, +, +), distinguishing one timelike dimension from three spacelike ones, which is essential for modeling relativistic phenomena.^[52] The signature induces a causal structure via light cones at each spacetime event: the future and past light cones consist of null geodesics where ds^2 = 0, bounding the regions accessible to timelike (ds^2 < 0) and spacelike (ds^2 > 0) paths, ensuring no faster-than-light signaling.^[53] For timelike paths, the proper time \tau along a worldline is the invariant interval measured by a comoving clock, defined by d\tau^2 = -ds^2 / c^2 = dt^2 - (dx^2 + dy^2 + dz^2)/c^2, integrated as \tau = \int d\tau.^[54] Inertial observers, at rest in these coordinates, follow straight-line geodesics in Minkowski spacetime, which maximize proper time between events and correspond to uniform motion at constant velocity.^[51]

References

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Metric Tensor -- from Wolfram MathWorld
Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed ...
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[PDF] MATH 144 NOTES: RIEMANNIAN GEOMETRY Contents 1. Manifolds
Mar 13, 2014 · Definition. A Riemannian metric g is a symmetric, positive definite 2-covariant tensor field. That is,. (1) gp(X, Y ) ...
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The Riemann curvature tensor is simply a collection of numbers at every point in space that describes its curvature. Riemann went on to make valuable ...
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[PDF] On the Hypotheses which lie at the Bases of Geometry. Bernhard ...
On the Hypotheses which lie at the Bases of. Geometry. Bernhard Riemann. Translated by William Kingdon Clifford. CDFHJLMN Ool. OQQQ. Nos. 1X3N 1X4N pp. 14-1_N.Missing: original | Show results with:original
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3 Introducing Riemannian Geometry‣ General Relativity ... - DAMTP
A general Riemannian metric gives us a way to measure the length of a vector X at each point, It also allows us to measure the angle between any two vectors X ...
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[PDF] Chapter 4. The First Fundamental Form (Induced Metric)
Traditionally, one displays the metric (or, metric components gij) by writing out the arc length element. Notations: ds2 = g11(du1)2 + 2g12du1du2 + g22(du2)2.
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[PDF] Arc Length and Riemannian Metric Geometry
Indefinite metric (in dimension 2). Consider a new definition of “length”: Take the square of the distance from. (x1,y1) to (x2,y2) to be. ∆s2. ≡ (x1 − x2)2.
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5. More Geometry - Lecture Notes on General Relativity - S. Carroll
Note that tensors with both upper and lower indices can generally be neither pushed forward nor pulled back. This machinery becomes somewhat less imposing once ...
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[PDF] Week 6: Differential geometry I
By definition, a scalar field φ remains invariant under a coordinate transformation, i.e. ... We differentiate the definition of the metric tensor, gab = ea · eb, ...
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[PDF] MATH 215C: Differential Geometry Introduction 1 April 3, 2023
Jun 7, 2023 · We also get a pullback: if T is a (0, 2) tensor on ˜M and φ : M → ˜M is any map, then the pullback of the tensor is defined via φ∗T(u,v) = T(dφ ...
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[PDF] Lectures on Differential Geometry Math 240C
Jun 6, 2011 · which gives an alternate proof that left invariant vector fields are closed under. Lie brackets in this case. Thus the Lie algebra of GL(n, R) ...
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[PDF] Differential Geometry - Michael Taylor
This course covers surfaces, Riemannian metrics, geodesics, vector fields, differential forms, covariant derivatives, curvature, and the Gauss-Bonnet theorem.
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Riemannian metrics
A Riemannian metric is a symmetric, nondegenerate bilinear form on \(M\). A smooth manifold equipped with a Riemannian metric is called a Riemannian manifold.
[14]
[PDF] lee-smooth-manifolds.pdf - MIT Mathematics
... John M. Lee. Introduction to. Smooth Manifolds. With 157 Illustrations. , Springer. Page 4. John M. Lee. Department of Mathematics. University of Washington.
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[PDF] Class 9. Riemannian and hermitian manifolds (September 26)
Sep 26, 2024 · a Riemannian metric on M is a collection of positive definite symmetric bilinear forms gp : TR,pM ⌦ TR,pM ! R that vary smoothly with p 2 M ...
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[PDF] NOTES ON RIEMANNIAN GEOMETRY Contents 1. Smooth ...
Apr 1, 2015 · In local coordinates xi, a Riemannian metric can be written as a symmetric tensor g := gijdxidxj. The notion of a Riemannian metric is supposed ...
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[PDF] Differential geometry Lecture 12: Pseudo-Riemannian manifolds
Jun 5, 2020 · Pseudo-Riemannian manifolds. Definition. A pseudo-Riemannian metric with index 0 ≤ ν ≤ dim(M) on. a smooth mfd. M is a symmetric (0, 2)-tensor ...
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[PDF] lecture 2: the riemannian metric
A Riemannian metric g is a smooth symmetric (0,2)-tensor field that is positive definite. We remark that many geometric structures on smooth manifold M are ...
[19]
[PDF] 3. Introducing Riemannian Geometry
The metric /±g provides a measure on the manifold that tells us what regions of the manifold are weighted more strongly than the others in the integral.
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[PDF] An overview of (completeness in) Lorentzian geometry
Oct 9, 2023 · A Lorentzian metric on a smooth manifold M is a smooth assignment of Lorentzian scalar products gx on each tangent space Tx M, for x ∈ M. We ...
[21]
[PDF] Notes on Differential Geometry
May 3, 2004 · By virtue of Eqn. (1.4) the metric tensor can be used to raise and lower indices in tensor equations. Technically, “indices up or down” means ...
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[PDF] Introduction to Tensor Calculus for General Relativity - MIT
The scalar product of two vectors requires the metric tensor while that of two one-forms requires the inverse metric tensor. In general, gµν 6= gµν. The ...
[23]
[PDF] 13 General Relativity - The University of New Mexico
The inverse gi` of the metric tensor g, like the inverse. (1.222) of any matrix, is the transpose of the cofactor matrix divided by its determinant det(g) gi ...
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Math 21b: Determinants
The determinant of the inverse of an invertible matrix is the inverse of the determinant: det(A-1) = 1 / det(A) [6.2.6, page 265]. Similar matrices have the ...
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[PDF] Basic Concepts in Differential Geometry - OU Math
In particular, this paper will focus on Riemannian geometry, the study of real, smooth manifolds equipped with a metric tensor. Like the inner product of linear ...
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[PDF] Riemannian Manifolds: An Introduction to Curvature
lengths, and distances on a Riemannian manifold. A primary goal is to show that all length-minimizing curves are geodesics, and that all geodes- ics are ...
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[PDF] Lecture Notes on Differential Geometry
Lecture Notes on Differential Geometry ... In particular, what are the corresponding tensors for the. Riemannian metric g? 4.1.4 Induced metric on tensors.
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[PDF] lecture 4: the riemannian measure
One can check that ωg is a well-defined global volume form on M, which is called the Riemannian volume form for the oriented Riemannian manifold (M,g). Remark.
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[PDF] Introduction to Riemannian Geometry and Geometric Statistics
Feb 2, 2023 · We cover the basics of differentiable manifolds (Section 2), Riemannian manifolds (Section 3) and Lie groups (Section 4). Then we delve into ...
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The Riemannian metric | Mathematics for Physics
A (pseudo) Riemannian metric (AKA metric) is a (pseudo) metric tensor field on a manifold M M , making M M a (pseudo) Riemannian manifold. The metric and ...<|control11|><|separator|>
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[PDF] arXiv:1806.05440v1 [math.DG] 14 Jun 2018
Jun 14, 2018 · This can be achieved by using the musical isomorphism between the tangent bundle and the cotangent bundle. In this article we use the ...
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Fundamental Theorem of Riemannian Geometry -- from Wolfram MathWorld
### Summary of Fundamental Theorem of Riemannian Geometry
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[PDF] On the history of Levi-Civita's parallel transport - arXiv
Aug 6, 2016 · [35] T. Levi-Civita, ”Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana”, ...
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Levi-Civita Connection -- from Wolfram MathWorld
As a connection on the tangent bundle, it provides a well-defined method for differentiating vector fields, forms, or any other kind of tensor.
[35]
Riemann Tensor -- from Wolfram MathWorld
The Riemann tensor is in some sense the only tensor that can be constructed from the metric tensor and its first and second derivatives.Missing: Bernhard | Show results with:Bernhard
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[PDF] The Riemann Curvature Tensor - Louisiana Tech Digital Commons
May 6, 2019 · This paper will provide an overview of tensors and tensor operations. In particular, properties of the Riemann tensor will be examined.Missing: seminal | Show results with:seminal
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[PDF] Chapter 14 Curvature in Riemannian Manifolds - CIS UPenn
However, Riemann's seminal paper published in 1868 two years after his death ... Spaces for which the Ricci tensor is proportional to the metric are called ...
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[PDF] arXiv:1106.2037v1 [gr-qc] 10 Jun 2011
Jun 10, 2011 · Riemann curvature from the affine connection. In this section, we will derive the Riemann curvature tensor in terms of the affine connection.
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[PDF] LECTURE 15: COMPLETENESS 1. The Hopf-Rinow Theorem
Apr 20, 2024 · So as metric spaces, Riemannian manifolds are special (and nice) metric spaces. We list a couple immediate consequences of Hopf-Rinow theorem.Missing: compactness | Show results with:compactness
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[PDF] THE HOPF-RINOW THEOREM Contents 1. Introduction 1 2. Tensors ...
Aug 3, 2016 · Abstract. This paper is an introduction to Riemannian geometry, with an aim towards proving the Hopf-Rinow theorem on complete Riemannian ...Missing: original | Show results with:original
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[0805.2235] Conformal Metrics - arXiv
May 15, 2008 · This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and ...Missing: definition | Show results with:definition
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conformal geometry in nLab
Jul 2, 2024 · A conformal structure on a manifold is the structure of a Riemannian metric modulo rescalings of the metric tensor by some real valued function on the manifold.
[43]
[PDF] Conformal Metrics - Discrete Differential Geometry (600.657)
A conformal map is an angle-preserving transformation. Liouville's Theorem: In dimensions greater than 2, the Möbius transformations (translations, rotations, ...
[44]
Euclidean Metric -- from Wolfram MathWorld
### Summary of Euclidean Metric
[45]
isometry group in nLab
### Definition of Isometry Group for Euclidean Space
[46]
[PDF] Physics 161: Black Holes: Lecture 19: 19 Feb 2010
ds2 = R2dθ2 + R2 sin2 θdφ2, where R is a constant (the radius of the sphere) ... We can also set dθ = dφ = 0 to consider only radial moving light. The ...
[47]
[PDF] Chapter 11 Riemannian Metrics, Riemannian Manifolds - CIS UPenn
Given any metric g on M, if ϕ is a local diffeomorphism, we define the pull-back metric, ϕ⇤g, on N induced by g as follows: For all p 2 N, for all u, v 2 TpN,.
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Chapter 3: Section 7: Part 4
Thus, a geodesic on a sphere is a great circle, which is a circle formed by the intersection of the sphere with a plane through the sphere's center.
[49]
[PDF] Euler Equation and Geodesics - UNCW
Feb 2, 2018 · hθ = R. So, ds2 = R2dθ. 2 + R2 sin2 θdφ. 2. Thinking of the path in ... = c. Solving for φ. 0, dφ dθ. = c sin θ p sin2 θ − c2 . One can ...<|control11|><|separator|>
[50]
Chapter 3: Section 8: Part 3: Gaussian Curvature
... R and thus a curvature of kn(q) = 1/R. Thus, the principal curvatures must be k1 = 1/R and k2 = 1/R, so that the curvature of a sphere of radius R is. K = 1. R2 ...
[51]
1 Geodesics in Spacetime‣ General Relativity by David Tong
In Minkowski space, it is simple to check that the proper time between two timelike-separated points is maximised by a straight line, a fact known as the twin ...
[52]
[PDF] Part II General Relativity - DAMTP
A metric with signature σ = n is called a “Riemannian metric” and a metric with signature σ = n − 2 is called “Lorentzian”. For example, the four ...
[53]
light-cone - Einstein-Online
In special as well as in general relativity, the speed of light sets the upper limit for the transmission of influences and signals.
[54]
Proper Time, Coordinate Systems, Lorentz Transformations
The essence of the Special Theory of Relativity (STR) is that it connects three distinct quantities to each other: space, time, and proper time.Proper Time · The STR Relationship... · Choice of Inertial Reference...