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On a closed Riemannian manifold with nonnegative Ricci curvature, integration of the Bochner-Weitzenbock formula for one-forms for a harmonic one-form givesBoth terms are nonnegative, so $\nabla \alpha =0$. Riemannian volume density makes the argument valid even without orientation.

On a connected compact oriented Riemannian manifold without boundary, a top-degree form is $f{\mathrm{vol}}_g$. Its exterior derivative is automatically zero, and the Hodge codifferential vanishes exactly when $df=0$. Thus it is harmonic exactly when $f$ is constant. Equivalently the Hodge star operator identifies this space with harmonic functions. Without connectedness the dimension is the number of connected components.

For the positive Laplace-Beltrami operator on a closed compact Riemannian manifold, the heat trace is $Z_M(t)={\sum }_j{e}^{-t{\lambda }_j}={\int }_M{K}_M(t,x,x)dV(x)$. It records every eigenvalue with multiplicity. In a finite free isometric quotient it equals $|U{|}^{-1}{\sum }_{u\in U}{\int }_N{K}_N(t,x,ux)dV(x)$. The orbital integrals form a class function of the ambient finite isometry group, so Gassmann equivalence gives equal quotient heat traces and hence Sunada theorem.

On an oriented boundaryless Riemannian manifold, apply Stokes theorem to the compactly supported form $f\ast \alpha$. The graded Leibniz rule gives $\int df\wedge \ast \alpha =-\int fd\ast \alpha$. The Hodge star operator identifies these top forms with the scalar integrands displayed above. This proves that the codifferential on one-forms is $\delta =-\ast d\ast$, the formal adjoint of the exterior derivative. Compact support of $f$ suffices.

On a closed oriented Riemannian manifold, let $\alpha$ have degree $r-1$ and $\beta$ degree $r$. The graded Leibniz rule and Stokes theorem give $\int d\alpha \wedge \ast \beta =(-1{)}^r\int \alpha \wedge d\ast \beta$. The codifferential convention $\delta \beta =(-1{)}^{n(r+1)+1}\ast d\ast \beta$, combined with the square of the Hodge star, gives $\ast \delta \beta =(-1{)}^{r}d\ast \beta$. This proves that the codifferential is the formal adjoint of the exterior derivative. A boundary requires the corresponding boundary term or specified boundary conditions.

The Hodge Laplacian acts on differential forms by $\Delta =d\delta +\delta d$. On a compact manifold,

It is one of the Laplace operators in differential geometry.

On an oriented Riemannian manifold, the codifferential convention $\delta =(-1{)}^{n(p+1)+1}\ast d\ast$ and ${\ast }^2=(-1{)}^{p(n-p)}$ give $d\ast \alpha =(-1{)}^p\ast \delta \alpha$ and $\delta \ast \alpha =(-1{)}^{p+1}\ast d\alpha$. Applying these twice proves the displayed identity. Thus the Hodge star operator preserves harmonic differential forms even without compactness.

Hodge theory connects de Rham cohomology with harmonic differential forms through the Hodge Laplacian on a compact Riemannian manifold. The Hodge decomposition theorem supplies a unique harmonic representative of each cohomology class. On a compact Kähler manifold, the decomposition additionally respects the complex bidegrees of differential forms.

A Riemannian manifold is homogeneous when its isometry group acts transitively on points. For example, a Lie group with a left-invariant metric is homogeneous under left translations. Homogeneity gives the same local metric-ball geometry at every point, which yields the completeness of homogeneous Riemannian manifolds. It does not imply two-point homogeneity: that stronger property also controls directions and pairs at equal distance.

If the Riemannian exponential map at one point $p$ of a connected finite-dimensional Riemannian manifold is defined on all of $T_{p}M$, every point can be joined to $p$ by a minimizing geodesic. The distance splitting through a small geodesic sphere selects a radial direction attaining $d(p,q)=r+d(\gamma (r),q)$. If this equality stopped before the target, repeat the splitting at its last point. A broken minimizing path cannot have a corner, so the new segment continues the same geodesic, contradicting maximality. This is the minimizing-geodesic lemma used in the radial continuation proof of Hopf-Rinow.

// Target: riemannian-geometry.bigb

For a connected finite-dimensional Riemannian manifold, completeness of the Riemannian distance, geodesic completeness, and compactness of every closed bounded subset are equivalent. These conditions imply that any two points are joined by a minimizing geodesic. Pairwise existence of minimizing geodesics alone does not imply completeness: an open Euclidean ball is incomplete, yet the line segment between any two of its points stays in the ball and minimizes length.

Hyperbolic geometry studies spaces of constant negative curvature.

The complete simply connected Riemannian manifold of dimension $n$ with constant sectional curvature $-1$ is hyperbolic space. In the upper half-space model its metric is $d{s}^2=(d{x}_1^2+\cdots +d{x}_n^2)/x_n^2$ for $x_n>0$. Its ideal boundary in dimension three is the Riemann sphere.

A complete hyperbolic three-manifold is a complete Riemannian manifold of dimension three with constant sectional curvature $-1$. Its universal cover is hyperbolic space ${\mathbb{H}}^3$ and its deck group acts freely and discretely by Riemannian isometries. In the orientable case it is a torsion-free Kleinian group.

An immersion $F:M\to N$ into a Riemannian manifold $(N,h)$ inherits the Riemannian metric $g=F^{\ast }h$: measure two tangent vectors after mapping them into the surrounding tangent space. This is a pullback of a Riemannian metric, withThe injective derivative of an immersion makes this inner product positive-definite. For a parametrized surface in Euclidean space, its coefficients are simply the scalar products of the coordinate tangent vectors: the first fundamental form.

For the circular cylinder $F(\theta ,z)=(R\cos \theta ,R\sin \theta ,z)$, the tangent vectors $F_{\theta }$ and $F_z$ are orthogonal, with squared norms $R^2$ and $1$. ConsequentlyThe unrolled coordinate $x=R\theta$ gives $d{s}^2=d{x}^2+d{z}^2$. This calculation explains intrinsic flatness of a circular cylinder and makes Riemannian distance on a circular cylinder a planar problem with repeated endpoint lifts.

Preserving a Riemannian metric locally preserves the arc length of every mapped curve, but it need not preserve distances measured by unrestricted paths in the surrounding space. The great-circle distance between two sphere points, for example, exceeds their straight chord length unless they coincide. ^[ref].

On an oriented even-dimensional Riemannian manifold of strictly positive sectional curvature, a nonconstant closed geodesic has a length-decreasing smooth variation. Its parallel transport fixes the tangent and acts on the odd-dimensional normal space by a special orthogonal group element. An odd-dimensional special orthogonal transformation has a fixed vector, so there is a nonzero periodic parallel normal field $V$. The second variation of geodesic energy is then strictly negative. The Cauchy-Schwarz inequality converts lower energy into strictly lower length because the original geodesic has constant speed. For an embedded geodesic the small variation is a smooth isotopy; otherwise it is a deformation through immersed loops.

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 ${\int }_{\partial M}f{\iota }_X{\omega }_g$.

An isometric extension of a connected metric manifold is a proper embedding onto an open subset of a larger connected manifold of the same dimension, preserving the metric tensor. Smooth complete connected Riemannian manifolds admit no such proper extension; Lorentzian geodesic completeness has different subtleties.

An isometric immersion is an immersion $f:M\to N$ between Riemannian manifolds whose derivative preserves the inner products of tangent vectors; equivalently, $f^{\ast }{g}_N=g_M$. Its derivative is injective and the target may have larger dimension. A local isometry additionally requires a local diffeomorphism, so its tangent maps are linear isomorphisms. For example, a sphere with its induced Riemannian metric is isometrically immersed in Euclidean space, although its positive intrinsic sectional curvature differs from that of the ambient space.

For a connected three-dimensional Riemannian manifold, the Myers-Steenrod theorem bounds the dimension of its isometry group by six. Dimension five cannot occur: isotropy embeds in $O(3)$, whose Lie subalgebras have dimensions zero, one or three. A five-dimensional action would require three-dimensional isotropy and a two-dimensional orbit; full rotational isotropy has no invariant tangent two-plane. The maximal dimension six forces transitivity and isotropy, hence constant sectional curvature. Each of the six listed dimensions is attained by a suitable metric.