Intrinsic Geometry of Hypersurfaces

The intrinsic geometry of a hypersurface consists of measurements that can be made entirely within it: lengths, angles, its Levi-Civita derivative, and its intrinsic curvature. For a non-null hypersurface these structures come from pulling back the ambient metric, but once obtained they do not remember uniquely how the hypersurface sits in the ambient spacetime.

This separation lets one compare different embeddings of the same abstract geometry and identify exactly which conclusions follow from the induced metric alone.

Required background. Affine Connections and Covariant Derivatives supplies covariant derivatives, and The Levi–Civita Connection supplies the metric-compatible, torsion-free uniqueness theorem; The Riemann Curvature Tensor supplies intrinsic and ambient curvature; Pseudo-Riemannian Metrics and Signature supplies induced signature. Helpful background. Calculus and Linear Algebra for Relativity supplies pullback matrices and projections; Exterior Algebra and Differential Forms supplies induced forms, while Pullbacks and Integration over Parametrized Domains supplies their integration.

Pulling back the metric

Let $\iota:\Sigma\to M$ embed a non-null hypersurface. Its first fundamental form is

$$h=\iota^*g.$$

In local hypersurface coordinates $y^a$ and ambient coordinates $x^\mu(y)$,

$$h_{ab} =g_{\mu\nu} \frac{\partial x^\mu}{\partial y^a} \frac{\partial x^\nu}{\partial y^b}.$$

Equivalently, for a unit normal with $n^2=\epsilon$,

$$h_{\mu\nu}=g_{\mu\nu}-\epsilon n_\mu n_\nu$$

when $h$ is viewed as a spacetime tensor annihilating the normal. The coordinate expression makes clear that $h_{ab}$ transforms only under changes of the coordinates on $\Sigma$; the projected spacetime expression makes ambient covariance clear.

For a spacelike hypersurface, $h$ is Riemannian. For a timelike hypersurface it is Lorentzian. A null hypersurface is excluded from this construction because its pullback metric is degenerate and has no inverse with which to define an ordinary Levi-Civita connection.

The induced covariant derivative

For tangent vector fields $X$ and $Y$, define

$$D_XY=h(\nabla_XY),$$

where the $h$ on the right denotes tangential projection. This definition is independent of how tangent extensions are chosen away from $\Sigma$. Because the ambient connection is torsion free and compatible with $g$, projection shows that

$$D_XY-D_YX=[X,Y], \qquad D h=0.$$

The uniqueness theorem therefore identifies $D$ as the Levi-Civita connection of the induced metric. For a scalar, $D_a f$ is just the tangential derivative. For a tangential covector,

$$D_a v_b =h_a{}^\mu h_b{}^\nu\nabla_\mu v_\nu.$$

One projector is required for every free index. Forgetting one can leave an ambient normal component in an object being treated as intrinsic.

The intrinsic curvature is defined by commuting $D_a$ and $D_b$ on a tangential vector. It can be calculated from $h_{ab}$ without knowing the normal or the ambient metric away from the surface. Its relation to ambient curvature and extrinsic curvature is the content of the Gauss–Codazzi–Ricci Relations.

Intrinsic versus extrinsic data

Adopt the chapter convention

$$K(X,Y)=-g(\nabla_X n,Y).$$

Metric compatibility gives the Gauss formula

$$\nabla_XY=D_XY+\epsilon K(X,Y)n.$$

The tangential term is intrinsic; the normal term depends on the embedding and on the chosen normal. Reversing $n$ changes $K$ but leaves $h$ and $D$ unchanged.

A plane and a cylinder in Euclidean three-space illustrate the distinction. Both have zero intrinsic Gaussian curvature, and a sufficiently small resident using only distances cannot distinguish them. Yet the plane has vanishing second fundamental form, while the cylinder bends in one principal direction. A round sphere has nonzero intrinsic curvature and bends in both directions. These examples appear in Gourgoulhon 2007, §2.3.5, pp. 24–28, PDF; the relation between induced and ambient derivatives is developed in Gourgoulhon 2007, §2.4.3, pp. 32–34, PDF.

Intrinsic data also need not determine a unique embedding. Additional equations and global hypotheses govern whether an abstract metric can be embedded and how many inequivalent embeddings it admits. This page constructs the geometry of an existing embedding; it does not assert a general embedding theorem.

Coordinate check on a sphere

For a sphere of radius $R$ in Euclidean three-space,

$$h=R^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right).$$

Its nonzero Christoffel symbols include

$$\Gamma^\theta{}_{\phi\phi} =-\sin\theta\cos\theta, \qquad \Gamma^\phi{}_{\theta\phi}=\cot\theta.$$

Computing from the induced derivative gives scalar curvature $2/R^2$. No reference to the ambient radial coordinate is required after the induced metric is known. The coordinate singularities at the poles do not signal singular intrinsic geometry.

By contrast, the same intrinsic calculation on a cylinder gives zero curvature even though its outward normal varies. The difference is captured by the second fundamental form, not by changing the definition of the intrinsic connection.

Common pitfalls

Pullback is not restriction of components. Tangent vectors must be inserted into every covariant slot. Ambient coordinate indices do not automatically become hypersurface indices.

Intrinsic flatness does not mean an unbent embedding. The cylinder has zero intrinsic curvature but nonzero extrinsic curvature.

Null hypersurfaces require different machinery. Their degenerate metric does not possess the inverse assumed in Levi-Civita uniqueness. Screen or rigging constructions must state their additional choices.

Exercises

Parametrize the cylinder of radius $R$ by $x=R\cos\phi$, $y=R\sin\phi$, $z=z$. Compute its induced metric and show locally that it is flat.

Solution

The tangent vectors are $\partial_\phi x^\mu=(-R\sin\phi,R\cos\phi,0)$ and $\partial_z x^\mu=(0,0,1)$. Their Euclidean inner products give

$$h=R^2d\phi^2+dz^2.$$

With the local coordinate $s=R\phi$, this becomes $h=ds^2+dz^2$. Its connection coefficients and intrinsic Riemann tensor vanish in these coordinates. Globally $\phi$ is periodic, so the cylinder is locally flat but not globally identical to the plane.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035.