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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 pseudo-Riemannian manifold is a smooth manifold with a smooth metric tensor that is a nondegenerate bilinear form on each tangent space. Positivity is not required. A Riemannian manifold is the positive-definite case, while a Lorentzian manifold has exactly one temporal direction in its metric signature.

On a compact oriented Riemannian manifold without boundary, every real linear functional on ${\mathcal{H}}^p$ has a unique $L^2$ representing harmonic form $h_f$: $f(\phi )=⟨\phi ,h_f{⟩}_{L^2}$. If $h_a$ is an orthonormal harmonic basis, then $h_f={\sum }_{a}f(h_a)h_a$. The Hodge star gives a harmonic $(n-p)$-form ${\psi }_0=\ast h_f$ satisfying $f(\phi )={\int }_M\phi \wedge {\psi }_0$. The ambiguity of harmonic wedge-pairing representatives describes all smooth representatives with this same functional.

A covering map between Riemannian manifolds that is a local isometry. The metric on the covering manifold is the pullback of the base metric; its deck transformations are isometries. This local metric agreement makes the Laplace-Beltrami operator commute with pullback of smooth functions.

The distance on a connected Riemannian manifold is the infimum of the arc lengths of piecewise smooth curves joining the endpoints:It answers “how far must I travel while staying in this geometry?” The Riemannian metric measures an instantaneous tangent vector, whereas Riemannian distance is the global result of comparing complete routes.

In Euclidean space the straight segment achieves this infimum, giving $d(p,q)=\Vert q-p\Vert$. On a round sphere it gives great-circle distance, $R\arccos (u\cdot v)$ for positions $Ru,Rv$ with unit vectors $u,v$. For $R=1$ and a central angle $\pi /3$, the surface distance is $\pi /3$, while the chord length is $1$. On a circular cylinder, routes with different winding lead to the minimum in Riemannian distance on a circular cylinder.

The word infimum matters: a shortest route need not exist. In the Euclidean plane with the origin removed, the points $(-1,0)$ and $(1,0)$ still have Riemannian distance $2$, approached by paths detouring around an arbitrarily small circle. Every allowed route has length strictly greater than $2$, since the only Euclidean straight segment passes through the missing point. The Hopf-Rinow theorem guarantees a minimizing geodesic between any two points when a connected Riemannian manifold is complete.

On an oriented Riemannian manifold with boundary, the divergence of a Riemannian vector field obeys the displayed integral identity for smooth compactly supported vector fields $X$, or for arbitrary smooth $X$ if the manifold is compact. Here $N$ is the outward unit normal vector and the boundary has the outward-normal-first boundary orientation. Its induced Riemannian volume form is $j^{\ast }{\iota }_N{\omega }_g$. Splitting $X$ into its normal and tangential components gives $j^{\ast }{\iota }_X{\omega }_g=g(X,N){\omega }_{\tilde{g}}$, since a top form vanishes on $n$ boundary-tangent arguments. Apply the Stokes theorem to $d{\iota }_X{\omega }_g=(\mathrm{div}X){\omega }_g$.

Without a support or decay condition, flux at infinity can invalidate the result even if the integrals converge. On $[0,\infty )$ with Riemannian metric $d{x}^2$, $X=(\arctan x){\partial }_x$ has boundary flux zero but integral of divergence $\pi /2$.

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.

For a connected Riemannian manifold, the full holonomy group consists of parallel transport maps around all piecewise smooth loops based at $p$. Metric compatibility makes them orthogonal. Basepoint changes conjugate this group by parallel transport. This is full holonomy, not just the subgroup using contractible loops.

A Riemannian manifold is a pair $(M,g)$ consisting of a smooth manifold $M$ and a Riemannian metric $g$. For each point $p$, the positive-definite inner product $g_p$ on its tangent space tells us the size and angle of possible instantaneous movements. The smooth manifold describes which local coordinates are possible; the Riemannian metric adds a choice of measurement.

For example, the plane can carry $d{s}^2=d{x}^2+d{y}^2$ or $d{s}^2=4d{x}^2+d{y}^2$. In the second Riemannian metric, a horizontal displacement costs twice as much length as the same coordinate displacement in the first. The underlying smooth manifold is unchanged. The coordinate transformation $X=2x,Y=y$ identifies the second geometry with an ordinary Euclidean plane, illustrating why coordinate appearance alone does not determine curvature.

A round sphere with its induced metric is another Riemannian manifold. Its tangent spaces each have an ordinary Euclidean inner product, although its global geometry is different from a plane's. This is the useful tension: measurements are Euclidean to first order at a point, while how those measurements fit together produces curvature and global restrictions on geodesics.

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].

A surjective submersion between Riemannian manifolds is Riemannian when its derivative is a linear isometry from the orthogonal complement of its kernel onto the target tangent space. Its vertical space is the kernel and its horizontal space is the orthogonal complement.

For a metric vector bundle connection over a Riemannian manifold, the rough Laplacian is the formally nonnegative connection operatorwhere $e_i$ is a local orthonormal frame. The correction term makes the expression independent of the orthonormal frame. On a closed manifold, integration by parts gives $⟨{\nabla }^{\ast }\nabla s,s{⟩}_{L^2}=\Vert \nabla s{\Vert }_{L^2}^2$.

On a connected Riemannian manifold of dimension at least three, pointwise independence of sectional curvature from the tangent two-plane forces the curvature to be constant. The contracted Bianchi identity differentiates the relation $\mathrm{Ric}=(n-1)Kg$ and gives $(n-2)dK=0$.

For an embedded submanifold $M\subseteq N$ of a Riemannian manifold, the vector-valued second fundamental form isHere $\overline{\nabla }$ is the ambient Levi-Civita connection and the superscript denotes projection onto the normal bundle. For a hypersurface in Euclidean space with unit normal $N$, the scalar form is $II(X,Y)=⟨A(X,Y),N⟩=⟨-dN(X),Y⟩$. In a surface parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.

On a compact oriented Riemannian manifold of dimension four without boundary, every exact three-form has a self-dual two-form primitive. Write $\beta =d\omega$ and use the Hodge decomposition theorem to split $\omega =h+d\xi +\delta \psi$. Then $\beta =d\delta \psi$. For $a=\delta \psi$, the codifferential identity $\delta =-\ast d\ast$ and the fact ${\ast }^2=1$ on two-forms give $\ast a=-d\ast \psi$, so $d\ast a=0$. Thus $\sigma =a+\ast a$ satisfies $\ast \sigma =\sigma$ and $d\sigma =\beta$. It is twice the self-dual projection of $a$, rather than the projection itself.

On a closed oriented Riemannian manifold of dimension four, the Hodge star operator preserves harmonic two-forms and splits them into orthogonal spaces ${\mathcal{H}}^{+}$ and ${\mathcal{H}}^{-}$ with eigenvalues $+1$ and $-1$. The real de Rham intersection form in dimension four equals the positive $L^2$ inner product on ${\mathcal{H}}^{+}$ and its negative on ${\mathcal{H}}^{-}$. It is nondegenerate, since $Q(h,\ast h)=\Vert h{\Vert }_2^2$. Its signature pair is $(\dim {\mathcal{H}}^{+},\dim {\mathcal{H}}^{-})$.

The Poisson equation for differential forms on a closed oriented Riemannian manifold is solvable exactly when its source is $L^2$-orthogonal to every harmonic differential form of the same degree. Necessity follows from the formal adjoint identity $⟨\Delta \alpha ,h⟩=⟨\alpha ,\Delta h⟩=0$. For sufficiency, the Green operator of the Hodge Laplacian gives $\alpha =G\beta$, since $\Delta G=I-P_{\mathcal{H}}$.

On a closed two-dimensional Riemannian manifold, the first three integrated scalar heat invariants are $A$, $\frac{1}{3}\int KdA$, and $\frac{1}{15}\int K^{2}dA$, where $K$ is Gaussian curvature. If its scalar spectrum agrees with that of a hyperbolic surface of area $A$, these integrals are $A,-A/3,A/15$. Expanding the displayed square then forces $K=-1$ everywhere. Thus constant negative curvature is spectrally determined in this setting, although the global hyperbolic metric need not be determined up to isometry.

Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.

Geometric splitting theorems give hypotheses under which a manifold decomposes as a metric product. The Cheeger-Gromoll splitting theorem gives a Euclidean line factor for a complete connected Riemannian manifold with nonnegative Ricci curvature containing a globally minimizing geodesic line. Other splitting theorems use different curvature or causal hypotheses.