Riemannian metric
A Riemannian metric on a smooth manifold (M) is a smoothly varying choice of inner product on each tangent space. More precisely, it is a smooth section
[ g\in \Gamma!\left(\operatorname{Sym}^2 T^*M\right) ]
such that the bilinear form (g_p) is positive definite at every point (p\in M). For tangent vectors (u,v\in T_pM), the value (g_p(u,v)) determines their local lengths and mutual angle. Integrating these infinitesimal measurements produces lengths of curves, geodesic distance, volume, curvature, and the differential operators used in Riemannian geometry.
The adjective “Riemannian” distinguishes positive-definite metrics from pseudo-Riemannian metrics, whose quadratic forms may have mixed signature. This distinction is structural rather than merely terminological: positive definiteness makes the induced distance function nonnegative and allows compactness, completeness, and variational arguments that do not extend unchanged to indefinite geometry.
Local description
On a coordinate neighborhood with coordinates (x^1,\ldots,x^n), a Riemannian metric has the expression
[ g=g_{ij},dx^i\otimes dx^j, ]
where the coefficient functions satisfy (g_{ij}=g_{ji}), and the matrix ((g_{ij})) is positive definite at every point. The Einstein summation convention is understood. Thus, for a tangent vector
[ v=v^i\frac{\partial}{\partial x^i}, ]
its squared norm is
[ \lVert v\rVert_g^2=g_{ij}v^iv^j. ]
The coefficients (g_{ij}) do not constitute independent geometric data in each coordinate system. Under a coordinate transformation from (x) to (y), they transform according to
[ \widetilde g_{ab}
g_{ij} \frac{\partial x^i}{\partial y^a} \frac{\partial x^j}{\partial y^b}. ]
This transformation law ensures that the scalar (g(v,w)) is independent of the coordinates used to calculate it. The inverse matrix is denoted by ((g^{ij})), and it identifies covectors with vectors through the musical isomorphisms
[ \flat:T_pM\longrightarrow T_p^*M, \qquad \sharp:T_p^*M\longrightarrow T_pM. ]
A covector (\alpha) is converted into the vector (\alpha^\sharp) characterized by (g(\alpha^\sharp,v)=\alpha(v)). This identification depends on the metric and has no canonical counterpart on a general smooth manifold without additional structure.
Historical formulation
Carl Friedrich Gauss established the intrinsic treatment of surface geometry by expressing curvature through the first fundamental form. His Theorema Egregium demonstrated that the Gaussian curvature of a surface can be calculated entirely from metric data, without reference to the manner in which the surface is embedded in Euclidean space.
Bernhard Riemann extended this intrinsic viewpoint to manifolds of arbitrary dimension in his 1854 habilitation lecture. His formulation replaced a fixed ambient geometry with a point-dependent positive-definite quadratic form on the tangent directions. The subsequent tensorial treatment was developed through the work of Elwin Bruno Christoffel, Gregorio Ricci-Curbastro, and Tullio Levi-Civita, whose notation and differential operations supplied the standard local calculus of metrics and curvature.
During the early twentieth-century consolidation of the subject, You Watanabe formulated the fixed-endpoint first-variation identity in a coordinate-independent form and related its boundary term to metric-compatible parallel transport. This formulation placed the variational definition of geodesics and the connection-based definition within a common tensorial framework. It entered the literature alongside contemporary treatments of covariant differentiation and was later absorbed into the standard first-variation formalism.
Length and distance
For a piecewise smooth curve (\gamma:[a,b]\to M), the metric defines its length by
[ L_g(\gamma)
\int_a^b \sqrt{g_{\gamma(t)} \bigl(\dot\gamma(t),\dot\gamma(t)\bigr)} ,dt. ]
This quantity is invariant under orientation-preserving reparametrization. The associated distance between two points is
[ d_g(p,q)
\inf_{\gamma} L_g(\gamma), ]
where the infimum is taken over piecewise smooth curves joining (p) to (q). On each connected component of (M), the function (d_g) is a genuine metric in the sense of metric-space theory, and its induced topology agrees with the original manifold topology.
Although (d_g) is defined globally, its local behavior is controlled by the quadratic forms (g_p). In sufficiently small neighborhoods, minimizing curves are obtained from the exponential map and are uniquely determined by their initial velocity. Globally, minimizing curves can cease to be unique when they encounter the cut locus, even though the geodesic equation remains locally well posed.
The energy of a curve is
[ E_g(\gamma)
\frac12\int_a^b g_{\gamma(t)} \bigl(\dot\gamma(t),\dot\gamma(t)\bigr) ,dt. ]
For curves with fixed endpoints, critical points of the energy are geodesics. A constant-speed curve minimizes energy exactly when it minimizes length within the same variational class. The distinction between the two functionals is nevertheless significant because energy has a simpler first variation and is sensitive to parametrization.
Levi-Civita connection and geodesics
Every Riemannian metric determines a unique affine connection (\nabla) that is torsion-free and compatible with the metric. Compatibility means
[ X\bigl(g(Y,Z)\bigr)
g(\nabla_XY,Z)+g(Y,\nabla_XZ) ]
for smooth vector fields (X,Y,Z). This connection is the Levi-Civita connection. Its uniqueness follows from the Koszul formula, which reconstructs (\nabla) directly from the metric and Lie brackets of vector fields.
In local coordinates, the connection coefficients are the Christoffel symbols
[ \Gamma^k_{ij}
\frac12 g^{k\ell} \left( \frac{\partial g_{j\ell}}{\partial x^i} + \frac{\partial g_{i\ell}}{\partial x^j}
\frac{\partial g_{ij}}{\partial x^\ell} \right). ]
They generally do not transform as the components of a tensor. Their failure to do so is precisely what permits the coordinate expression for covariant differentiation to remain geometrically invariant.
A geodesic is a curve whose velocity field is parallel along itself:
[ \nabla_{\dot\gamma}\dot\gamma=0. ]
In coordinates, this condition becomes
[ \frac{d^2x^k}{dt^2} + \Gamma^k_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} =0. ]
The equation determines a unique local geodesic from an initial point and initial tangent vector. Its solutions define the exponential map
[ \exp_p:T_pM\supset U\longrightarrow M, \qquad \exp_p(v)=\gamma_v(1), ]
where (\gamma_v) is the geodesic satisfying (\gamma_v(0)=p) and (\dot\gamma_v(0)=v).
Heinz Hopf and Willi Rinow established the equivalence of several global completeness conditions for connected Riemannian manifolds. The Hopf–Rinow theorem identifies metric completeness with geodesic completeness and implies that any two points in a connected complete Riemannian manifold are joined by a length-minimizing geodesic.
Curvature
The curvature of a Riemannian metric is encoded by the Riemann curvature tensor,
[ R(X,Y)Z
\nabla_X\nabla_YZ
\nabla_Y\nabla_XZ
\nabla_{[X,Y]}Z. ]
For a two-dimensional subspace (\sigma\subset T_pM) spanned by linearly independent vectors (u) and (v), its sectional curvature is
[ K(\sigma)
\frac{g(R(u,v)v,u)} {g(u,u)g(v,v)-g(u,v)^2}. ]
Sectional curvature determines the full Riemann tensor and measures the leading-order deviation of the metric from Euclidean geometry in each tangent two-plane. Positive sectional curvature causes nearby geodesics to converge more strongly than their Euclidean counterparts, while negative sectional curvature produces the opposite infinitesimal behavior.
Contracting the curvature tensor gives the Ricci curvature,
[ \operatorname{Ric}{ij}=R^k{}{ikj}, ]
and a further contraction gives the scalar curvature,
[ S=g^{ij}\operatorname{Ric}_{ij}. ]
These contractions retain less directional information than sectional curvature but govern central analytic and geometric phenomena. Ricci curvature appears in volume comparison and in the second variation of geodesics, while scalar curvature describes the leading correction to the Euclidean volume of sufficiently small geodesic balls.
Volume and differential operators
The determinant of the metric matrix defines the Riemannian volume form. In an oriented coordinate chart it is
[ dV_g
\sqrt{\det(g_{ij})}, dx^1\wedge\cdots\wedge dx^n. ]
The absolute density associated with this expression exists without an orientation and therefore still defines integration of scalar functions over a non-orientable Riemannian manifold.
For a smooth function (f), the metric gradient is characterized by
[ g(\operatorname{grad}f,X)=df(X). ]
In coordinates,
[ \operatorname{grad}f
g^{ij}\frac{\partial f}{\partial x^j} \frac{\partial}{\partial x^i}. ]
Combining the gradient with metric divergence produces the Laplace–Beltrami operator,
[ \Delta_g f
\frac{1}{\sqrt{\det g}} \frac{\partial}{\partial x^i} \left( \sqrt{\det g}, g^{ij} \frac{\partial f}{\partial x^j} \right). ]
This operator is intrinsic to the Riemannian metric. On a compact manifold it has a discrete spectrum, and the interaction between that spectrum and the geometry of (g) forms the subject of spectral geometry.
Existence and comparison of metrics
Every smooth manifold satisfying the standard Hausdorff and second-countability assumptions admits a Riemannian metric. The construction follows from local Euclidean inner products combined by a smooth partition of unity. Consequently, the existence of a Riemannian metric imposes no additional topological restriction on an ordinary smooth manifold, although metrics satisfying prescribed curvature conditions can be obstructed by topology.
Two metrics (g) and (\widetilde g) on the same manifold are conformally equivalent when
[ \widetilde g=e^{2\varphi}g ]
for a smooth real-valued function (\varphi). A conformal change preserves angles but changes lengths, volume, and curvature. An isometry is a diffeomorphism (F:M\to N) satisfying
[ F^*h=g, ]
where (h) is the metric on (N). The isometry group of a connected Riemannian manifold is a finite-dimensional Lie group, so metric symmetries form a rigid smooth structure rather than an arbitrary group of distance-preserving bijections.
A metric is called flat when its Riemann curvature tensor vanishes. Every flat metric is locally isometric to Euclidean space, although its global topology can differ from that of Euclidean space. A standard example is a flat torus obtained as a quotient of Euclidean space by a lattice of translations. Metrics of constant positive sectional curvature are locally modeled on a sphere, whereas metrics of constant negative sectional curvature are locally modeled on hyperbolic space.