Riemannian metric ($g$)

A Riemannian metric assigns a positive-definite inner product to every tangent space, varying smoothly from point to point.

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.

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.

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 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.

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 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.