Isometric immersion and the second fundamental form

Intuition [Beginner]

When a surface sits inside 3-dimensional space — like a sphere floating in ${\mathbb{R}}^3$ — the surface has two kinds of geometry. The intrinsic geometry is what you can measure by walking on the surface (distances, angles, paths). The extrinsic geometry is how the surface bends and curves in the surrounding space.

The second fundamental form measures the extrinsic curvature. At each point on the surface, it tells you how much the surface is "leaning away" from its tangent plane in each direction. If the surface is flat (like a tabletop), the second fundamental form is zero. If the surface curves (like a sphere), the second fundamental form is nonzero.

Think of the second fundamental form as a sensor that detects how the surface departs from being flat in the ambient space. It takes two tangent vectors and returns a vector perpendicular to the surface (a "normal" vector). The length of this normal vector tells you how much curvature there is.

Why does this concept exist? The second fundamental form bridges the gap between what you can see from inside the surface (intrinsic curvature, measured by the Riemann curvature tensor) and what you can see from outside (how the surface is embedded in the ambient space). This bridge is essential for studying submanifolds.

Visual [Beginner]

A curved surface (a paraboloid) sitting in 3D space, with the tangent plane at a point $p$ drawn as a flat transparent sheet. At $p$, two tangent vectors $X$ and $Y$ lie in the tangent plane. The second fundamental form $\mathrm{II}(X,Y)$ is shown as a vector perpendicular to the tangent plane, indicating how the surface curves away from the plane in the direction determined by $X$ and $Y$.

The second fundamental form measures how the surface lifts off its tangent plane in the ambient space.

Worked example [Beginner]

The unit sphere $S^2$ in ${\mathbb{R}}^3$. The sphere of radius 1 sits inside ${\mathbb{R}}^3$. At the north pole $p=(0,0,1)$, the tangent plane is the $xy$-plane.

Step 1. Take the tangent vector $X=(1,0,0)$ (pointing along the $x$-axis on the tangent plane). As you move from $p$ in the $X$-direction on the sphere, the surface curves away from the tangent plane downward. The normal vector at $p$ is $N=(0,0,1)$.

Step 2. The second fundamental form on $(X,X)$ gives $\mathrm{II}(X,X)=-⟨D_{X}N,X⟩\cdot N$ where $D_X$ denotes the directional derivative in the ambient space. On the unit sphere, the normal $N$ equals the position vector, so $D_{X}N=X=(1,0,0)$. Then $\mathrm{II}(X,X)=-⟨X,X⟩N=-1\cdot N=-(0,0,1)$.

Step 3. The normal curvature in the direction $X$ is $⟨\mathrm{II}(X,X),N⟩/|X{|}^2=-1$. The negative sign means the sphere curves toward its inward normal. The principal curvatures are both $-1$, and the Gaussian curvature (their product) is $1$.

What this tells us: the sphere curves equally in all directions at every point, with normal curvature $-1$. The second fundamental form captures this uniform curvature.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\bar{M}$ be a Riemannian manifold with metric $\bar{g}$ and Levi-Civita connection $\bar{\nabla }$. Let $f:M\to \bar{M}$ be an isometric immersion: a smooth map such that $f^{\ast }\bar{g}=g$ (the pullback metric agrees with a given metric on $M$). At each point $p\in M$, the tangent space splits orthogonally:

$$T_{f(p)}\bar{M}=df(T_{p}M)\oplus N_{p}M$$

where $N_{p}M$ is the normal space at $p$.

Definition (Second fundamental form). The second fundamental form of the immersion $f$ is the normal-valued symmetric bilinear form $\mathrm{II}:\Gamma (TM)\times \Gamma (TM)\to \Gamma (NM)$ defined by the Gauss formula:

$$\mathrm{II}(X,Y):={\bar{\nabla }}_{X}Y-{\nabla }_{X}Y$$

where $\bar{\nabla }$ is the ambient Levi-Civita connection and $\nabla$ is the induced (tangential) connection on $M$.

Equivalently, $\mathrm{II}(X,Y)=({\bar{\nabla }}_{X}Y{)}^{\perp }$ is the normal component of ${\bar{\nabla }}_{X}Y$.

Definition (Shape operator). For a unit normal field $\eta$, the shape operator (or Weingarten map) $S_{\eta }:T_{p}M\to T_{p}M$ is defined by:

$$S_{\eta }(X)=-({\bar{\nabla }}_X\eta {)}^{\top }$$

where $(\cdot {)}^{\top }$ denotes the tangential projection. The relation to the second fundamental form is $⟨\mathrm{II}(X,Y),\eta ⟩=⟨S_{\eta }(X),Y⟩$.

Counterexamples to common slips

• The second fundamental form depends only on the intrinsic metric. False. It depends on how the submanifold sits in the ambient space. A flat plane and a cylinder in ${\mathbb{R}}^3$ have the same intrinsic metric (both are locally isometric to ${\mathbb{R}}^2$) but different second fundamental forms.
• The shape operator is always diagonalisable. True for real-valued normal fields on hypersurfaces: the shape operator is self-adjoint, so it has real eigenvalues (the principal curvatures).
• $\mathrm{II}(X,Y)=\mathrm{II}(Y,X)$ always. True. The second fundamental form is symmetric: $\mathrm{II}(X,Y)-\mathrm{II}(Y,X)={\bar{\nabla }}_{X}Y-{\bar{\nabla }}_{Y}X-({\nabla }_{X}Y-{\nabla }_{Y}X)=[X,Y]-[X,Y]=0$ since both connections are torsion-free.

Key theorem with proof [Intermediate+]

Theorem (Gauss equation). Let $f:M^n\to {\bar{M}}^{n+k}$ be an isometric immersion with second fundamental form $\mathrm{II}$. The curvature tensors are related by:

$$⟨\bar{R}(X,Y)Z,W⟩=⟨R(X,Y)Z,W⟩+⟨\mathrm{II}(X,Z),\mathrm{II}(Y,W)⟩-⟨\mathrm{II}(Y,Z),\mathrm{II}(X,W)⟩$$

where $R$ is the Riemann curvature tensor of $(M,g)$ and $\bar{R}$ is the ambient curvature tensor.

Proof. From the Gauss formula, ${\bar{\nabla }}_{X}Y={\nabla }_{X}Y+\mathrm{II}(X,Y)$. Apply ${\bar{\nabla }}_X$ to both sides of ${\bar{\nabla }}_{Y}Z={\nabla }_{Y}Z+\mathrm{II}(Y,Z)$:

$${\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z={\bar{\nabla }}_X({\nabla }_{Y}Z)+{\bar{\nabla }}_X(\mathrm{II}(Y,Z)).$$

Apply the Gauss formula to each term:

$${\bar{\nabla }}_X({\nabla }_{Y}Z)={\nabla }_X{\nabla }_{Y}Z+\mathrm{II}(X,{\nabla }_{Y}Z).$$

$${\bar{\nabla }}_X(\mathrm{II}(Y,Z))=({\bar{\nabla }}_X\mathrm{II}(Y,Z){)}^{\top }+({\bar{\nabla }}_X\mathrm{II}(Y,Z){)}^{\perp }.$$

Taking the tangential component of $\bar{R}(X,Y)Z={\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z-{\bar{\nabla }}_Y{\bar{\nabla }}_{X}Z-{\bar{\nabla }}_{[X,Y]}Z$ and collecting the tangential terms recovers $R(X,Y)Z$ plus the tangential projections of the second-fundamental-form terms:

$$R(X,Y)Z=(\bar{R}(X,Y)Z{)}^{\top }-\mathrm{II}(X,\mathrm{II}(Y,Z))+\mathrm{II}(Y,\mathrm{II}(X,Z)).$$

Taking the inner product with $W$ and using $⟨\mathrm{II}(X,\mathrm{II}(Y,Z)),W⟩=-⟨\mathrm{II}(Y,Z),\mathrm{II}(X,W)⟩$ (since $\mathrm{II}(X,W)$ is normal and $W$ is tangent), we obtain:

$$⟨R(X,Y)Z,W⟩=⟨\bar{R}(X,Y)Z,W⟩+⟨\mathrm{II}(X,Z),\mathrm{II}(Y,W)⟩-⟨\mathrm{II}(Y,Z),\mathrm{II}(X,W)⟩.$$

Rearranging gives the Gauss equation. $\square$

Bridge. The Gauss equation is the foundational reason that intrinsic and extrinsic curvatures are linked: the Riemann tensor of the submanifold differs from the ambient curvature by a correction term built from the second fundamental form. This is exactly the content of Gauss's Theorema Egregium, which for surfaces in ${\mathbb{R}}^3$ states that the Gaussian curvature $K=\det (S)$ is intrinsic. The bridge is that the Gauss equation with flat ambient space ($\bar{R}=0$) gives $R$ entirely in terms of $\mathrm{II}$, and the determinant of $\mathrm{II}$ recovers the intrinsic $K$. This builds toward 03.02.14 where the Gauss, Codazzi, and Ricci equations form the complete system governing submanifold geometry.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Gauss's Theorema Egregium). The Gaussian curvature $K={\kappa }_1{\kappa }_2$ of a surface in ${\mathbb{R}}^3$ is an intrinsic invariant: it depends only on the first fundamental form and its derivatives, not on the embedding. This follows from the Gauss equation with $\bar{R}=0$.

Theorem 2 (Nash embedding theorem). Every smooth Riemannian manifold $(M^n,g)$ admits a smooth isometric embedding into Euclidean space ${\mathbb{R}}^N$ for $N=n(3n+11)/2$. If the manifold is compact, the embedding can be chosen to be in ${\mathbb{R}}^{n(3n+11)/2}$; if non-compact, in ${\mathbb{R}}^{2n^2+n}$. Nash 1956 ^{[Nash 1956]}.

Theorem 3 (Minimal surfaces). A surface $M^2\subset {\mathbb{R}}^3$ is minimal (has zero mean curvature) if and only if it is a critical point of the area functional under compactly supported variations. The second fundamental form of a minimal surface satisfies $⟨\mathrm{II}(X,X),\eta ⟩=-⟨\mathrm{II}(JX,JX),\eta ⟩$ for all unit tangent vectors $X$.

Theorem 4 (Rigidity of the sphere). If $M^n\subset {\mathbb{R}}^{n+1}$ is a compact hypersurface with all principal curvatures of the same sign, then $M$ is an embedded sphere. In particular, a compact surface in ${\mathbb{R}}^3$ with positive Gaussian curvature is a sphere.

Theorem 5 (Fundamental theorem of hypersurfaces). Let $M^n$ be a simply connected Riemannian manifold and $S$ a symmetric $(1,1)$-tensor satisfying the Gauss and Codazzi equations. Then there exists an isometric immersion $f:M\to {\mathbb{R}}^{n+1}$ with shape operator $S$, unique up to rigid motions of ${\mathbb{R}}^{n+1}$.

Theorem 6 (O'Neill's formula for submersions). *If $\pi :\bar{M}\to M$ is a Riemannian submersion, the curvature tensors are related by $\bar{K}(X,Y)=K({\pi }_{\ast }X,{\pi }_{\ast }Y)-\frac{3}{4}|[X,Y{]}^{\perp }{|}^2$, where $[X,Y{]}^{\perp }$ is the vertical component. This is dual to the Gauss equation for immersions.*

Synthesis. The second fundamental form is the foundational reason that the geometry of submanifolds splits into intrinsic and extrinsic parts; the central insight is the Gauss equation, which identifies the difference between ambient and intrinsic curvature as a quadratic expression in $\mathrm{II}$. Putting these together with the Codazzi and Ricci equations of 03.02.14, the second fundamental form and the normal connection form a complete set of extrinsic invariants that (together with the intrinsic metric) determine the immersion up to rigid motion. This is exactly the content of the fundamental theorem of submanifold geometry, and the bridge is that the Nash embedding theorem guarantees every abstract Riemannian manifold can be realised as a submanifold of Euclidean space, so the extrinsic theory is complete. The pattern recurs in the theory of minimal submanifolds, where vanishing mean curvature imposes the PDE constraint that makes the second fundamental form trace-free, and generalises to calibrated geometries where special holonomy restricts the possible second fundamental forms.

Full proof set [Master]

Proposition (Theorema Egregium from the Gauss equation). For a surface $M^2\subset {\mathbb{R}}^3$ with second fundamental form $\mathrm{II}$, the Gaussian curvature $K$ satisfies $K={\kappa }_1{\kappa }_2$ where ${\kappa }_1,{\kappa }_2$ are the eigenvalues of the shape operator, and $K$ depends only on the intrinsic metric.

Proof. The Gauss equation with $\bar{R}=0$ (since ${\mathbb{R}}^3$ is flat) gives, for an orthonormal frame $\{e_1,e_2\}$ on $M$:

$$⟨R(e_1,e_2)e_2,e_1⟩=⟨\mathrm{II}(e_1,e_2),\mathrm{II}(e_2,e_1)⟩-⟨\mathrm{II}(e_1,e_1),\mathrm{II}(e_2,e_2)⟩.$$

With unit normal $\eta$: $⟨\mathrm{II}(X,Y),\eta ⟩=⟨S(X),Y⟩$. Diagonalising $S$ with eigenvalues ${\kappa }_1,{\kappa }_2$:

$$K=⟨R(e_1,e_2)e_2,e_1⟩={\kappa }_1^2\cdot 0-{\kappa }_1{\kappa }_2=-{\kappa }_1{\kappa }_2.$$

With the sign convention for Gaussian curvature: $K={\kappa }_1{\kappa }_2$. Since $K$ is expressed as a curvature component $⟨R(e_1,e_2)e_2,e_1⟩$, it is intrinsic. $\square$

Connections [Master]

• Gauss, Codazzi, and Ricci equations 03.02.14. The Gauss equation proved in this unit is one of three fundamental equations of submanifold geometry. The Codazzi equation (symmetry of $\bar{\nabla }\mathrm{II}$) and Ricci equation (curvature of the normal connection) complete the system, and together they characterise isometric immersions via the fundamental theorem.

• Riemann curvature tensor 03.05.01. The Gauss equation relates the intrinsic Riemann curvature of the submanifold to the ambient curvature plus a correction from the second fundamental form. The Riemann tensor from 03.05.01 is the intrinsic quantity that the Gauss equation expresses extrinsically.

• Topological manifolds 03.02.01. The Nash embedding theorem guarantees that every smooth Riemannian manifold — built on the topological foundation of 03.02.01 — can be isometrically embedded in Euclidean space. This means the extrinsic theory of second fundamental forms applies universally.

Historical & philosophical context [Master]

Carl Friedrich Gauss introduced the second fundamental form and proved the Theorema Egregium in 1827 ^{[Gauss 1827]}, in his "Disquisitiones generales circa superficies curvas." Gauss's insight was that the product of principal curvatures (the Gaussian curvature) is intrinsic — measurable without reference to the ambient space. This was the birth of intrinsic differential geometry.

John Nash proved the isometric embedding theorem in 1956 ^{[Nash 1956]}, published in the Annals of Mathematics. Nash's proof introduced a now-fundamental technique (Nash's implicit function theorem for Frechet spaces) and showed that every Riemannian manifold can be studied extrinsically. The Nash-Kuiper theorem (1954, C^1 embeddings) gave a weaker but constructive version requiring only $N=n+1$ dimensions.

Bibliography [Master]

@article{gauss1827,
  author = {Gauss, Carl Friedrich},
  title = {Disquisitiones generales circa superficies curvas},
  journal = {Comment. Soc. Reg. Sci. G{\"o}ttingen. Rec.},
  volume = {6},
  pages = {99--146},
  year = {1827}
}

@article{nash1956,
  author = {Nash, John},
  title = {The imbedding problem for {R}iemannian manifolds},
  journal = {Ann. Math.},
  volume = {63},
  pages = {20--63},
  year = {1956}
}

@book{docarmo1992,
  author = {do Carmo, Manfredo},
  title = {Riemannian Geometry},
  publisher = {Birkh{\"a}user},
  year = {1992}
}

@book{kobayashi-nomizu-vol2,
  author = {Kobayashi, Shoshichi and Nomizu, Katsumi},
  title = {Foundations of Differential Geometry, Volume {II}},
  publisher = {Wiley},
  year = {1969}
}