Riemannian metric ($g$)

A Riemannian metric is a smoothly varying positive-definite inner product $g_p$ on each tangent space $T_{p}M$ of a smooth manifold. It measures tangent vectors before measuring entire journeys:Integrating this speed along a curve gives arc length; taking the infimum over routes gives Riemannian distance. A smooth manifold together with this measuring rule is a Riemannian manifold.

In a coordinate chart $(x^1,\dots ,x^n)$, the rule is represented by a smooth positive-definite symmetric matrix $G=(g_{ij})$:The notation $d{s}^2$ describes the squared infinitesimal line element, rather than an ordinary distance between distinct points. Changing coordinates changes the matrix entries; it preserves the measured norm of the same tangent vector.

For the plane with $d{s}^2=4d{x}^2+d{y}^2$, the vector $(1,0)$ has norm $2$ and $(0,1)$ has norm $1$. The coordinate change $X=2x,Y=y$ turns this into $d{s}^2=d{X}^2+d{Y}^2$: coefficients that look different can describe the same geometry. On a circular cylinder the induced metric $R^{2}d{\theta }^2+d{z}^2$ similarly becomes Euclidean when unrolled. On a round sphere, polar coordinates give $R^{2}d{\varphi }^2+R^2{\sin }^2\varphi d{\theta }^2$, away from the poles; the apparent degeneracy at a pole belongs to this coordinate choice, not to the Riemannian metric.

The point of the construction is that measurements make sense intrinsically, even without a chosen surrounding Euclidean space. It gives a common starting point for geodesics, curvature encoded by the Riemann curvature tensor, and the Laplace-Beltrami operator. ^[ref].

An intrinsic two-dimensional metric locally written $d{s}^2=d{r}^2+f(r{)}^{2}d{\varphi }^2$ with positive $f$. Rotation of the angular coordinate is an isometry. Such a metric need not have a global realization as a surface of revolution in three-dimensional Euclidean space.

For the affinely parametrized geodesic Lagrangian $\mathcal{L}=({\dot{r}}^2+f(r{)}^2{\dot{\varphi }}^2)/2$, the conserved cyclic momentum is $J=f(r{)}^2\dot{\varphi }$. Its Euler-Lagrange equations are $\ddot{r}-f{f}^{\prime }{\dot{\varphi }}^2=0$ and $\ddot{\varphi }+2f^{\prime }\dot{r}\dot{\varphi }/f=0$.

The line element of a Riemannian metric is its local expression for squared infinitesimal length. Along a parametrized curve $x(t)$, the arc length is $\int \sqrt{g_{ij}(x(t)){\dot{x}}^i{\dot{x}}^j}dt$. For a circular cylinder of radius $R$ parametrized by angle $\theta$ and axial position $z$, the induced metric gives $d{s}^2=R^{2}d{\theta }^2+d{z}^2$. Unrolling with $u=R\theta$ yields the Euclidean form $d{u}^2+d{z}^2$, explaining the straight unrolled descriptions of geodesics on a circular cylinder.

In spherical coordinates on a sphere of radius $R$, the position is $X=R(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$. Its coordinate tangent vectors satisfy $X_{\theta }\cdot X_{\varphi }=0$, $|X_{\theta }{|}^2=R^2$, and $|X_{\varphi }{|}^2=R^2{\sin }^2\theta$. Expanding $|dX{|}^2$ proves the displayed line element, the first fundamental form of the sphere. Along a parametrized path it gives the arc length by integrating $R\sqrt{{\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2}$. The vanishing azimuthal coefficient at a pole is a coordinate singularity, not a singularity of the sphere.

A smooth local basis of vector fields whose inner products under a Riemannian metric are the identity matrix. Tracing the Riemannian Hessian in such a frame gives the Laplace-Beltrami operator, with the chosen sign.

At any point of a Riemannian manifold, choose geodesic normal coordinates and orthonormalize their coordinate fields smoothly. At the point, the metric is the identity and its first derivatives vanish, so the orthonormalization coefficients have zero first derivatives. The Christoffel symbols also vanish there. This gives a local Riemannian orthonormal frame with the displayed property, useful for invariant identities evaluated at one point.

A smooth Riemannian metric with no nonidentity local isometry between open subsets. Equivalently distinct nonempty open subsets never have isometric induced metrics. This strong local rigidity is also called bumpy in some spectral-geometry treatments; it is distinct from the closed-geodesic nondegeneracy convention for bumpy metrics.

On a compact smooth manifold without boundary of dimension at least two, a residual set of smooth Riemannian metrics consists of nowhere locally homogeneous metrics. Thus such metrics are dense in the smooth topology by the Baire category theorem. In dimension one, arclength coordinates always provide nontrivial local isometries.

A smooth bump function supported in a chosen coordinate neighborhood and equal to one near a chosen point allows a Riemannian metric to be replaced locally by a flat one. The displayed convex combination stays positive definite and glues smoothly to the original metric outside the support. This is a local modification of the metric, not a claim that the original metric is flat.

For a smooth function $f$ on a Riemannian manifold, its Riemannian gradient is the unique vector field satisfying $g({\nabla }_{g}f,V)=df(V)$ for every vector field $V$. It depends on the Riemannian metric, and its negative generates the downward gradient flow.

The Euclidean metric is the constant Riemannian metric on ${\mathbb{R}}^n$ given in Cartesian coordinates by $g={\sum }_i(d{x}^i{)}^2$. Its Levi-Civita connection has zero Christoffel symbols in those coordinates, so it is flat and its geodesics are straight lines. The Euclidean norm is the pointwise length determined by this metric.

The product of Riemannian manifolds has metricThe tangent summands are orthogonal, and positivity follows because a nonzero product tangent vector has a nonzero component. Its Levi-Civita connection is the product affine connection of the two factor Levi-Civita connections: mixed torsion is zero and metric compatibility follows from independence of the two coordinate sets.

A Riemannian metric gives inverse vector bundle isomorphismsIn coordinates they have matrices $g_{ij}$ and $g^{ij}$. Every smooth manifold admits a Riemannian metric by a partition of unity, so its tangent and cotangent bundles are isomorphic as real smooth vector bundles. This isomorphism depends on the metric, not merely on the manifold.

For a smooth map $f:M\to N$, the pullback of a Riemannian metric is $(f^{\ast }g{)}_p(u,v)=g_{f(p)}(d{f}_{p}u,d{f}_{p}v)$. It is positive definite precisely when $d{f}_p$ is injective at every point, that is, when $f$ is an immersion. In coordinates, $(f^{\ast }g{)}_{ij}=g_{ab}(f(y)){\partial }_i{f}^a{\partial }_j{f}^b$.

A conformal rescaling multiplies a Riemannian metric by a positive scalar function. It preserves angles while changing lengths and volumes; in two dimensions, $\tilde{g}=e^{2\omega }g$ gives ${\Delta }_{\tilde{g}}=e^{-2\omega }{\Delta }_g$ on scalar functions.

Two Riemannian metrics lie in the same conformal class when one is a positive smooth scalar multiple of the other. They measure the same angles, while lengths may change from point to point. On an oriented Riemannian surface, rotation of each tangent plane through $\pi /2$ determines a complex structure which depends only on this conformal class.

For $\tilde{g}=e^{2\sigma }g$, the displayed correction is a symmetric tensor, so adding it to $\nabla$ gives a torsion-free connection. The two extra metric-compatibility pairings sum to $2d\sigma (X)g(Y,Z)$; this is exactly the derivative of the conformal factor. Uniqueness of the Levi-Civita connection proves the formula. It also yields the critical-point criterion for equal conformal connections.

On a positive-dimensional Riemannian manifold, equality means equality of the two covariant derivatives at $p$ for every pair of vector fields. A critical point of $\sigma$ annihilates the conformal correction. Conversely, writing that correction as $A$, the identity $g(A(v,v),v)=g(v,v)d\sigma (v)$ for all tangent vectors forces $d\sigma =0$ if $A=0$. A smooth function on a nonempty compact boundaryless manifold has a critical point, so conformally related connections coincide somewhere. Compact manifolds with boundary need not have one.

In dimension $N$, the conformal rescaling of a Riemannian metric $\tilde{g}=e^{2\omega }g$ changes the Ricci scalar by the displayed formula. In two dimensions the gradient-square term vanishes; for a flat base metric, $\tilde{R}=-2e^{-2\omega }\Delta \omega$.

On an oriented Riemannian $n$-manifold, the volume form is the unique positive $n$-form taking value one on every positively oriented orthonormal frame. In positive local coordinates,

The total Riemannian volume is the integral of the Riemannian volume form, or its density on a nonorientable manifold. In local coordinates the density is $\sqrt{\det g}d{x}^1\cdots d{x}^d$. It is preserved by Riemannian isometries. On a closed manifold it is the leading coefficient among the integrated heat invariants, and on a surface it is the surface area.

For positive coordinate charts put $J=\partial y/\partial x$. The metric tensor transforms by $g_x=J^T{g}_{y}J$, giving $\det g_x=(\det J{)}^2\det g_y$. The top wedge product of differential forms transforms by $d{y}^1\wedge \cdots \wedge d{y}^n=(\det J)d{x}^1\wedge \cdots \wedge d{x}^n$. Since $\det J>0$, the two factors match. Thus the local expressions define one global Riemannian volume form.

In a positive Riemannian orthonormal frame $X={\sum }_i{f}_i{E}_i$ with dual coframe ${\omega }_i$, the Riemannian volume form is ${\omega }_1\wedge \cdots \wedge {\omega }_n$. Inserting $E_i$ in its first slot omits ${\omega }_i$ and produces sign $(-1{)}^{i-1}$, giving the displayed interior product formula.

At a point in a normal orthonormal frame, torsion-freeness makes the frame brackets vanish there, hence the dual one-forms have zero exterior derivative there. Differentiating the interior product formula then leaves only ${\sum }_i{E}_i(f_i){\omega }_g$, the divergence of a Riemannian vector field. Since the point is arbitrary, the identity holds globally.

On a compact oriented Riemannian manifold without boundary, the Generalized Stokes theorem applied to $f{\iota }_X{\omega }_g$ gives the displayed identity. The product expansion is $d(f{\iota }_X{\omega }_g)=[X(f)+f\mathrm{div}X]{\omega }_g$. With boundary, the omitted boundary integral is ${\int }_{\partial M}f{\iota }_X{\omega }_g$.

The Levi-Civita connection of an oriented Riemannian manifold preserves its Riemannian volume form. In a positively oriented orthonormal frame its connection matrix is skew-symmetric. Differentiating the wedge of the dual coframe gives minus the trace of that matrix times the volume form, which is zero. Equivalently parallel transport preserves the metric and orientation, so it preserves the normalized volume form.

For a nonnegative smooth bump function supported in an open set, multiplying a Riemannian metric by $1+t\eta$ increases volume there for $t>0$ while leaving the metric elsewhere fixed. For distinct regular compact domains a perturbation supported in the interior of one outside the other breaks equal volume, and therefore breaks any isometry between the domains. The perturbations tend to the original metric in the smooth topology.

The $(n-1)$-dimensional volume of the Euclidean unit sphere equals $n$ times the $n$-dimensional volume of its unit ball. If $R={\sum }_i{x}^i{\partial }_i$ is the radial vector field and $\omega =d{x}^1\wedge \cdots \wedge d{x}^n$, then $d({\iota }_R\omega )=n\omega$, and the Generalized Stokes theorem proves the formula.

A differential form is parallel when its covariant derivative vanishes. Parallel transport then preserves it, and every parallel form is harmonic.

The form is preserved by parallel transport; this is the alternating-tensor case of vanishing covariant derivative.

For a function $u$ on a compact Riemannian manifold, its Dirichlet energy is $\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$. Adding a linear source term $-{\int }_{M}fud{\mathrm{vol}}_g$ gives Euler-Lagrange equation $-{\Delta }_{g}u=f$.

For a scalar function $u$ on a Riemannian manifold, the Dirichlet energy is $E(u)=\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$.

For a smooth map between Riemannian manifolds, its Dirichlet energy is half the integral of the squared norm of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a J-holomorphic curve into a symplectic manifold with a compatible almost complex structure, the Energy identity for a J-holomorphic curve identifies it with the pulled-back symplectic area. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.