Unit 7: Special Geometries, Flat Spaces & Hypersurfaces

1. Definition and Characterization of Flat Manifolds

A Riemannian space $R_n$ is called flat (or Euclidean / pseudo-Euclidean) if there exists a coordinate system $(y^1, \dots, y^n)$ in which the metric tensor components are everywhere constant:

By a constant linear transformation, such a metric can always be brought to diagonal form with entries $\pm 1$:

2. The Theorem of Flatness

Theorem 7.1 (Riemann Flatness Criterion): A Riemannian manifold $(M, g)$ is locally flat if and only if its Riemann curvature tensor vanishes identically everywhere:

Proof (Necessity):

If coordinates $y^k$ exist such that $g_{ij} = \text{const}$, then all first and second partial derivatives of the metric vanish identically:

Since $R^l_{ijk}$ is a genuine $(1, 3)$ tensor, if its components vanish in the $y$-frame, they must vanish in every coordinate system $x^i$:

Thus $R_{hijk} = 0$ is strictly necessary.

Proof (Sufficiency):

Suppose $R^l_{ijk} = 0$ throughout a simply connected neighborhood. We seek $n$ independent scalar functions $y^a(x)$ such that their gradients form a parallel orthonormal frame:

This is an overdetermined system of second-order linear PDEs for $y^a$. By Frobenius's integrability theorem, a solution exists if and only if the cross-covariant derivatives commute:

Applying the Ricci identity:

Since $R^m_{ijk} = 0$, the integrability condition is satisfied identically! Thus, there exist $n$ independent coordinate functions $y^a$ satisfying $\nabla_j (\partial_i y^a) = 0$. In these coordinates, $\partial_k g_{ab} = \nabla_k g_{ab} = 0$, so $g_{ab}$ is constant everywhere. $\blacksquare$

1. Conformal Transformations

Two metrics $g$ and $\tilde{g}$ on a manifold $M$ are called conformally equivalent if they differ only by a positive scale factor:

Conformal transformations preserve all angles between intersecting curves, but alter lengths and geodesic trajectories. A Riemannian manifold is conformally flat if it is conformally equivalent to flat Euclidean space:

2. The Weyl Curvature Tensor

Hermann Weyl (1918) asked: What geometric tensor measures the intrinsic curvature that is invariant under conformal rescalings?

Definition 7.1 (Weyl Conformal Curvature Tensor): For an $n$-dimensional Riemannian manifold with $n \ge 3$, the Weyl tensor $C^l_{ijk}$ is the totally trace-free part of the Riemann curvature tensor:

Fundamental Properties of the Weyl Tensor:

1. Trace-Free: Contracting any pair of indices with $g$ yields zero identically:

2. Conformal Invariance: Under $\tilde{g}_{ij} = e^{2\sigma} g_{ij}$, the mixed Weyl tensor is strictly invariant:

while the covariant tensor scales conformally: $\tilde{C}_{hijk} = e^{2\sigma} C_{hijk}$.

3. Conformal Flatness Theorem (Weyl, 1918):

• For $n \ge 4$: A manifold is conformally flat if and only if $C^l_{ijk} = 0$.
• For $n = 3$: The Weyl tensor vanishes identically for all metrics ($C_{hijk} \equiv 0$). Conformal flatness in 3D is instead governed by the vanishing of the Cotton-York tensor:

• For $n = 2$: Every 2D Riemannian manifold is locally conformally flat (isothermal coordinates always exist).

1. Geometry of Embedded Hypersurfaces

Let $M^n$ be an $n$-dimensional hypersurface embedded in an $(n+1)$-dimensional Riemannian manifold $(\tilde{M}^{n+1}, G_{AB})$. Let $x^A$ ($A = 1, \dots, n+1$) be coordinates on $\tilde{M}$, and let $u^\alpha$ ($\alpha = 1, \dots, n$) be intrinsic coordinates on $M^n$. The hypersurface is defined parametrically by:

2. The Induced Metric (First Fundamental Form)

The tangent vectors to coordinate curves on $M^n$ are:

The infinitesimal displacement on $M^n$ is $dx^A = B^A_\alpha du^\alpha$. The arc length element restricted to the hypersurface defines the induced metric (first fundamental form) $a_{\alpha\beta}$:

where:

$a_{\alpha\beta}$ is an intrinsic symmetric rank-2 covariant tensor on the hypersurface $M^n$.

3. The Unit Normal Vector and Gauss's Formula

At each point of $M^n$, there exists a unique (up to sign) unit normal vector field $N^A$ satisfying:

($\epsilon = +1$ for spacelike normals in Riemannian geometry).

Now differentiate the tangent vectors $B^A_\alpha = \frac{\partial x^A}{\partial u^\alpha}$ along the surface: The ambient covariant derivative $\tilde{\nabla}_\beta B^A_\alpha$ decomposes into components tangent to the surface and normal to the surface:

Theorem 7.2 (Gauss's Formula):

where:

• $\bar{\Gamma}^\mu_{\alpha\beta}$ are the intrinsic Christoffel symbols computed from the induced metric $a_{\alpha\beta}$.
• $b_{\alpha\beta}$ is the second fundamental form of the hypersurface.

1. The Second Fundamental Form $b_{\alpha\beta}$

Contracting Gauss's formula with the normal vector $N_A = G_{AB} N^B$:

Since $G_{AB} N^A B^B_\alpha = 0$, differentiating with respect to $u^\beta$:

Because $\tilde{\nabla}_\beta B^B_\alpha = \tilde{\nabla}_\alpha B^B_\beta$ in torsion-free ambient space, the second fundamental form is symmetric:

2. Weingarten Equations (The Shape Operator)

How does the normal vector $N^A$ change as we move along the hypersurface? Since $G_{AB} N^A N^B = 1$, differentiating yields $N_A \tilde{\nabla}_\alpha N^A = 0$, meaning $\tilde{\nabla}_\alpha N^A$ is purely tangent to $M^n$:

Theorem 7.3 (Weingarten Equations):

where $b^\beta_\alpha = a^{\beta\mu} b_{\mu\alpha}$ is the shape operator (or Weingarten map).

3. Principal, Mean, and Gaussian Curvatures

The shape operator $b^\beta_\alpha$ is a symmetric linear endomorphism of the tangent space $T_p M^n$.

• Its eigenvalues $\kappa_1, \kappa_2, \dots, \kappa_n$ are called the principal curvatures.
• The corresponding eigenvectors are the principal directions.
• The Mean Curvature $H$ is the normalized trace:

• For a 2D surface in $\mathbb{R}^3$, the extrinsic Gaussian curvature is the determinant:

A surface is a minimal surface if $H = 0$ (e.g. soap films, catenoids, helicoids).

1. Ambient vs Intrinsic Curvature

By evaluating the commutator of ambient covariant derivatives on the hypersurface tangent vectors $B^A_\alpha$, we establish the exact connection between the intrinsic Riemann curvature $R_{\alpha\beta\gamma\delta}$ of $M^n$ and the ambient curvature $\tilde{R}_{ABCD}$ of $\tilde{M}^{n+1}$.

2. The Gauss Equation

Theorem 7.4 (The Gauss Equation):

If the ambient space is flat Euclidean space ($\tilde{R}_{ABCD} = 0$):

Gauss's Theorema Egregium (Remarkable Theorem, 1827):

For a 2D surface embedded in $\mathbb{R}^3$, setting $(\alpha, \beta, \gamma, \delta) = (1, 2, 1, 2)$:

Dividing by the metric determinant $a = \det(a_{\alpha\beta})$:

The Gaussian curvature $K$ depends ONLY on the intrinsic metric $a_{\alpha\beta}$ and its derivatives! Bending a surface without stretching or tearing (isometric deformation) preserves $K$ identically.

3. The Codazzi-Mainardi Equations

The normal component of the commutator curvature identity yields the differential compatibility conditions on the second fundamental form:

Theorem 7.5 (The Codazzi-Mainardi Equations):

In flat Euclidean ambient space ($\tilde{R} = 0$):

The covariant derivative of the second fundamental form is completely symmetric in all three indices $\alpha, \beta, \gamma$!

Together, the Gauss and Codazzi-Mainardi equations are the necessary and sufficient integrability conditions (Fundamental Theorem of Surface Theory) for a pair of symmetric tensors $(a_{\alpha\beta}, b_{\alpha\beta})$ to uniquely realize a surface in $\mathbb{R}^{n+1}$ up to rigid motions.