Submanifolds and the Second Fundamental Form

One-line: The second fundamental form measures how a submanifold curves inside its ambient space by recording the normal component of the ambient covariant derivative of tangent vector fields. Key result: For an isometric immersion

f : M \to \tilde{M}

, the Gauss formula

\tilde{\nabla}_{X} Y = \nabla_{X} Y + II (X, Y)

splits the ambient connection into intrinsic (tangential) and extrinsic (normal) parts;

II

is a symmetric, bilinear, vector-valued form whose contractions yield the Gauss, Codazzi, and Ricci equations relating intrinsic and ambient curvature. Where it applies: Differential geometry of curves/surfaces, Riemannian submanifolds, minimal surface theory, isometric embedding, general relativity (hypersurfaces, ADM formalism), geometric PDE. Who/when: Gauss (1827, surfaces), Codazzi & Mainardi (1850s–60s), Ricci & Levi-Civita (1900s tensor calculus), modern formulation via connections (mid-20th century).

A submanifold $M$ of a Riemannian manifold $(\tilde{M}, \tilde{g})$ inherits a metric by restriction, but it also sits inside $\tilde{M}$ in a way that the intrinsic metric alone cannot capture. The second fundamental form $II$ is the precise object encoding this extrinsic bending: it is the obstruction to tangent vector fields staying tangent under the ambient covariant derivative. It matters because it is the bridge between intrinsic geometry (what an ant on the surface can measure) and extrinsic geometry (how the surface is embedded), and via the Gauss equation it shows that part of the curvature is intrinsic. The surprising fact — Gauss's Theorema Egregium — is that for a surface in $R^{3}$ , the product of the principal curvatures (the determinant of $II$ ), an apparently extrinsic quantity, equals the Gaussian curvature, which is intrinsic and survives bending without stretching.

The Core Insight

Given an isometric immersion $f : M \to \tilde{M}$ , take tangent vector fields $X, Y$ on $M$ . The ambient derivative $\tilde{\nabla}_{X} Y$ need not be tangent to $M$ . Split it:

\tilde{\nabla}_{X} Y = intrinsic: \nabla_{X} Y (\tilde{\nabla}_{X} Y)^{⊤} + extrinsic: II (X, Y) (\tilde{\nabla}_{X} Y)^{⊥} .

The tangential part is the Levi-Civita connection of the induced metric (this is a theorem, not a definition), and the normal part is a symmetric tensor. Everything about how

M

curves in

\tilde{M}

flows from

II

Formal Statement

Isometric immersion and induced metric

Let $(\tilde{M}, \tilde{g})$ be a Riemannian manifold of dimension $n + k$ and let $M$ be a smooth manifold of dimension $n$ . A smooth map $f : M \to \tilde{M}$ is an immersion if $d f_{p} : T_{p} M \to T_{f (p)} \tilde{M}$ is injective for all $p$ . It is an isometric immersion if the induced metric (first fundamental form)

g (X, Y) := \tilde{g} (df (X), df (Y))

equals a prescribed metric on

M

. We identify

T_{p} M

with its image

d f_{p} (T_{p} M) \subset T_{f (p)} \tilde{M}

and write

\tilde{g}

for both metrics. The normal bundle is the orthogonal complement

N_{p} M := (T_{p} M)^{⊥} \subset T_{f (p)} \tilde{M}, T_{f (p)} \tilde{M} = T_{p} M \oplus N_{p} M .

⟳Injectivity of

df

guarantees the pullback in [1] is positive-definite; orthogonal decomposition [2] follows from non-degeneracy of

\tilde{g}

on the finite-dimensional fiber.

The second fundamental form

Let $\tilde{\nabla}$ be the Levi-Civita connection of $(\tilde{M}, \tilde{g})$ . For vector fields $X, Y \in Γ (T M)$ (extended arbitrarily to $\tilde{M}$ near $f (M)$ ), define the second fundamental form as the normal projection

II (X, Y) := (\tilde{\nabla}_{X} Y)^{⊥} \in Γ (N M),

and define the induced connection by the tangential projection

\nabla_{X} Y := (\tilde{\nabla}_{X} Y)^{⊤}

. The Gauss formula is then

\tilde{\nabla}_{X} Y = \nabla_{X} Y + II (X, Y) .

For a hypersurface (

k = 1

) with unit normal

ν

, the scalar second fundamental form is

h (X, Y) := \tilde{g} (II (X, Y), ν)

, so

II (X, Y) = h (X, Y) ν

⟳[3] is well-defined because

II (X, Y)

depends only on the values of

X, Y

M

, not on extensions — verified via tensoriality below.

Properties of II

The map $II : Γ (T M) \times Γ (T M) \to Γ (N M)$ is:

$C^{\infty} (M)$ -bilinear (a tensor): $II (φX, Y) = φ II (X, Y)$ and similarly in $Y$ .
Symmetric: $II (X, Y) = II (Y, X)$ .
The tangential connection $\nabla$ is the Levi-Civita connection of the induced metric $g$ .

Step 1 (Tensoriality in $X$ ). Since

\tilde{\nabla}

is a connection,

\tilde{\nabla}_{φX} Y = φ \tilde{\nabla}_{X} Y

; projecting normally gives

C^{\infty}

-linearity in

X

. Step 2 (Tensoriality in $Y$ ). Compute

\tilde{\nabla}_{X} (φ Y) = (X φ) Y + φ \tilde{\nabla}_{X} Y .

⟳Leibniz rule for the connection in the second slot.

The term

(X φ) Y

is tangent to

M

, so it dies under normal projection, leaving

II (X, φ Y) = φ II (X, Y)

. Hence

II

is tensorial; it depends only on pointwise values. Step 3 (Symmetry). Because

\tilde{\nabla}

is torsion-free,

\tilde{\nabla}_{X} Y - \tilde{\nabla}_{Y} X = [X, Y] .

⟳Torsion-freeness

T (X, Y) = \tilde{\nabla}_{X} Y - \tilde{\nabla}_{Y} X - [X, Y] = 0

For

X, Y

tangent to

M

, the bracket

[X, Y]

is again tangent to

M

. Projecting [6] normally annihilates

[X, Y]

, giving

II (X, Y) - II (Y, X) = 0

. Step 4 ( $\nabla$ is metric and torsion-free). Torsion: projecting [6] tangentially,

\nabla_{X} Y - \nabla_{Y} X = [X, Y]

, so

\nabla

is torsion-free. Metric compatibility: for tangent

X, Y, Z

X g (Y, Z) = \tilde{g} (\tilde{\nabla}_{X} Y, Z) + \tilde{g} (Y, \tilde{\nabla}_{X} Z) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z),

⟳

\tilde{\nabla}

is metric; the normal parts

II (X, Y), II (X, Z)

are orthogonal to the tangent fields

Z, Y

, so contribute nothing.

By the uniqueness in the fundamental theorem of Riemannian geometry,

\nabla

is the Levi-Civita connection of

g

■

Shape operator and Weingarten formula

For a normal field $ξ \in Γ (N M)$ , split the ambient derivative of $ξ$ along tangent $X$ :

\tilde{\nabla}_{X} ξ = - A_{ξ} X + \nabla_{X}^{⊥} ξ,

where

A_{ξ} X := - (\tilde{\nabla}_{X} ξ)^{⊤}

defines the shape operator (Weingarten map) and

\nabla^{⊥}

is the normal connection. The two forms are related by

\tilde{g} (II (X, Y), ξ) = \tilde{g} (A_{ξ} X, Y) .

⟳Differentiate

\tilde{g} (Y, ξ) = 0

along

X

0 = \tilde{g} (\tilde{\nabla}_{X} Y, ξ) + \tilde{g} (Y, \tilde{\nabla}_{X} ξ)

, then use [4] and [8].

Thus

A_{ξ}

is a self-adjoint endomorphism of

T_{p} M

; its eigenvalues are the principal curvatures

κ_{1}, \dots, κ_{n}

, its trace gives the mean curvature vector

H = \frac{1}{n} tr_{g} II

, and for hypersurfaces

det A_{ν}

is the Gauss–Kronecker curvature.

The fundamental equations of submanifolds

Let $\tilde{R}, R, R^{⊥}$ be the curvature tensors of $\tilde{\nabla}, \nabla, \nabla^{⊥}$ . For tangent $X, Y, Z, W$ and normal $ξ, η$ :

Gauss equation:

\tilde{g} (\tilde{R} (X, Y) Z, W) = g (R (X, Y) Z, W) + \tilde{g} (II (X, W), II (Y, Z)) - \tilde{g} (II (X, Z), II (Y, W)) .

Codazzi (Codazzi–Mainardi) equation:

(\tilde{R} (X, Y) Z)^{⊥} = (\overset{ˉ}{\nabla}_{X} II) (Y, Z) - (\overset{ˉ}{\nabla}_{Y} II) (X, Z),

where

\overset{ˉ}{\nabla}

is the connection on

T^{*} M \otimes T^{*} M \otimes N M

built from

\nabla

and

\nabla^{⊥}

. Ricci equation:

\tilde{g} (\tilde{R} (X, Y) ξ, η) = \tilde{g} (R^{⊥} (X, Y) ξ, η) - \tilde{g} ([A_{ξ}, A_{η}] X, Y) .

Gauss equation. Apply

\tilde{\nabla}

twice using the Gauss formula [4]:

\tilde{\nabla}_{X} \tilde{\nabla}_{Y} Z = \tilde{\nabla}_{X} (\nabla_{Y} Z + II (Y, Z)) = \nabla_{X} \nabla_{Y} Z + II (X, \nabla_{Y} Z) - A_{II (Y, Z)} X + \nabla_{X}^{⊥} II (Y, Z) .

⟳Use [4] on

\tilde{\nabla}_{X} \nabla_{Y} Z

and [8] on

\tilde{\nabla}_{X} II (Y, Z)

Subtract the same with

X, Y

swapped, subtract

\tilde{\nabla}_{[X, Y]} Z

, and take the

\tilde{g} (\cdot, W)

inner product with

W

tangent. The tangential terms reassemble

R (X, Y) Z

and the shape-operator terms contribute

- \tilde{g} (A_{II (Y, Z)} X, W) + \tilde{g} (A_{II (X, Z)} Y, W)

; converting via [9] yields [10]. Codazzi equation. Take the normal component of

\tilde{R} (X, Y) Z

from the same expansion [13]. The tangential

\nabla

-terms vanish under normal projection; the surviving terms are exactly the covariant derivatives of

II

, giving [11]. Ricci equation. Differentiate a normal field twice using the Weingarten formula [8] and collect normal components; the failure of the normal-connection curvature to match

\tilde{R}

is precisely the commutator

[A_{ξ}, A_{η}]

■

Carl Friedrich Gauss | 1777–1855

Gauss laid the foundations in his 1827 masterwork Disquisitiones generales circa superficies curvas. While supervising the geodetic survey of the Kingdom of Hannover — measuring real terrain with theodolites — he was led to ask which surface quantities are intrinsic, i.e. measurable from within the surface without reference to the ambient space. His answer, the Theorema Egregium ("remarkable theorem"), states that the Gaussian curvature $K = κ_{1} κ_{2} = det A$ is intrinsic, even though each principal curvature is extrinsic. This single insight is the conceptual seed of Riemannian geometry and, eventually, of general relativity. Gauss introduced both fundamental forms (without the modern names) and the curvature concept that bears his name.

Building Intuition

Think of $II$ as answering the question: "If I walk straight along the surface (a geodesic of $M$ ), how fast am I being forced to curve out of the ambient space?" When you drive a car along a curving road on a hillside, your intrinsic path may be a straight line painted on the asphalt, but your body still feels acceleration pushing you — that normal acceleration is $II (\overset{γ}{˙}, \overset{γ}{˙})$ .

The shape operator $A_{ξ}$ reframes this as a linear map: it tells you how the unit normal tilts as you move tangentially. A flat plane has $A = 0$ (normal never tilts). A sphere of radius $r$ has $A = \frac{1}{r} Id$ (normal tilts uniformly in all directions). A cylinder has $A$ with one zero eigenvalue (no tilt along the axis) and one nonzero (tilt around the circle) — which is why a cylinder, having $det A = 0 = K$ , is intrinsically flat and can be unrolled to a plane.

The Gauss equation [10] is the deepest payoff: it says the intrinsic curvature you'd measure by parallel-transporting around a tiny loop equals the ambient curvature plus a correction quadratic in $II$ . For surfaces in flat $R^{3}$ , $\tilde{R} = 0$ , so $K$ is entirely determined by $II$ — yet $K$ itself is intrinsic. That tension is the Theorema Egregium.

A surface in $\mathbb R^3$ with tangent plane, normal, and the splitting of the ambient derivative into tangential ($\nabla_XY$) and normal ($\mathrm{II}(X,Y)$) parts.

Worked Examples

Sphere of radius $r$ in $\mathbb R^3$

Given:

S_{r}^{2} = {x \in R^{3} : ∣ x ∣ = r}

with outward unit normal

ν (x) = x / r

. The ambient connection

\tilde{\nabla}

is the flat directional derivative

D

. Step 1 — shape operator. For tangent

X

A_{ν} X = - (\tilde{\nabla}_{X} ν)^{⊤}

. Since

ν = x / r

D_{X} ν = X / r

(which is already tangent), so

A_{ν} X = - \frac{1}{r} X .

⟳

ν (x) = x / r

is linear in position; its directional derivative is

X / r

(Sign depends on normal orientation; with inward normal you'd get

+ \frac{1}{r}

.) Step 2 — second fundamental form.

h (X, Y) = \tilde{g} (A_{ν} X, Y) = - \frac{1}{r} g (X, Y)

, so

II (X, Y) = - \frac{1}{r} g (X, Y) ν

. The surface is totally umbilic: both principal curvatures equal

- 1/ r

. Step 3 — Gaussian curvature via Gauss equation. With

\tilde{R} = 0

in flat

R^{3}

, [10] gives

K = κ_{1} κ_{2} = det A_{ν} = (\frac{1}{r})^{2} .

⟳

det (\frac{1}{r} Id_{2 \times 2}) = 1/ r^{2}

Boxed answer:

K = 1/ r^{2}, H = - 1/ r, II = - \frac{1}{r} g ν

(units:

K

length^{- 2}

H

length^{- 1}

Graph of a function $z=u(x,y)$ as a hypersurface

Given:

M = {(x, y, u (x, y))} \subset R^{3}

. Frame

f_{x} = (1, 0, u_{x}), f_{y} = (0, 1, u_{y})

. Step 1 — induced metric.

g_{ij} = δ_{ij} + u_{i} u_{j}

, i.e.

g = (1 + u_{x}^{2} u_{x} u_{y} u_{x} u_{y} 1 + u_{y}^{2}), det g = 1 + u_{x}^{2} + u_{y}^{2} .

⟳Dot the frame vectors;

f_{x} \cdot f_{x} = 1 + u_{x}^{2}

, etc.

Step 2 — unit normal.

ν = \frac{( - u _{x} , - u _{y} , 1 )}{1 + u _{x}^{2} + u _{y}^{2}}

. Step 3 — second fundamental form.

h_{ij} = \tilde{g} (D_{f_{i}} f_{j}, ν) = ν \cdot f_{ij}

where

f_{ij} = (0, 0, u_{ij})

h_{ij} = \frac{u _{ij}}{1 + u _{x}^{2} + u _{y}^{2}}, h = \frac{1}{W} (u_{xx} u_{x y} u_{x y} u_{y y}), W = 1 + ∣\nabla u ∣^{2} .

⟳Only the

z

-component of

f_{ij}

survives the dot product with

ν

Step 4 — curvatures. Gaussian and mean curvature follow from

K = det h / det g

and

H = \frac{1}{2} tr_{g} h

K = \frac{u _{xx} u _{y y} - u _{x y}^{2}}{( 1 + ∣\nabla u ∣ ^{2} ) ^{2}}, H = \frac{( 1 + u _{y}^{2} ) u _{xx} - 2 u _{x} u _{y} u _{x y} + ( 1 + u _{x}^{2} ) u _{y y}}{2 ( 1 + ∣\nabla u ∣ ^{2} ) ^{3/2}} .

⟳

K = det (g^{- 1} h) = det h / det g

;

H = \frac{1}{2} g^{ij} h_{ij}

with

g^{ij}

the inverse metric.

Boxed answer (minimal surface PDE):

H = 0 ⟺ (1 + u_{y}^{2}) u_{xx} - 2 u_{x} u_{y} u_{x y} + (1 + u_{x}^{2}) u_{y y} = 0

— the nonparametric minimal surface equation.

Common Misconceptions

Sign and orientation conventions vary wildly. Many texts define

II (X, Y) = (\tilde{\nabla}_{X} Y)^{⊥}

(as here), but others use the opposite sign, or define the scalar form with a built-in minus. The mean curvature

H

can be a vector (

\frac{1}{n} tr II

) or a scalar (

tr A_{ν}

\frac{1}{n} tr A_{ν}

). Always check (i) the sign in the Weingarten formula [8], (ii) whether

H

is normalized by

1/ n

, and (iii) the chosen normal direction. The product of principal curvatures (Gaussian curvature) is sign-convention-independent in even codimension; mean curvature is not. $II$ depends on the embedding, not just the metric. Two isometric immersions of the same

(M, g)

can have completely different

II

. The Gaussian curvature is intrinsic, but

II

itself is genuinely extrinsic data. A cylinder and a plane are intrinsically identical but extrinsically different. Consider the cylinder

C = {(cos θ, sin θ, z)} \subset R^{3}

and the plane

P = R^{2}

. Roll the plane into the cylinder: this is a local isometry, so both have the same first fundamental form and the same Gaussian curvature

K = 0

. Yet their second fundamental forms differ entirely:

II_{P} \equiv 0, A_{C} = (0001) (eigenvalues κ_{1} = 0, κ_{2} = 1) .

⟳The plane is totally geodesic; the cylinder curves with radius 1 around its circular direction and is flat along its axis.

This proves

II

cannot be recovered from

g

alone — the Theorema Egregium extracts only the determinant

det A

as intrinsic, not

II

itself. It also shows totally geodesic (

II = 0

) is strictly stronger than flat (

K = 0

Connections

The second fundamental form sits at the crossroads of several theories. Via the Gauss Equation and Codazzi-Mainardi Equations it links to Riemann Curvature Tensor of both $M$ and $\tilde{M}$ . The Bonnet theorem (fundamental theorem of surface theory) states that $g$ and $II$ satisfying the Gauss–Codazzi compatibility equations determine a surface in $R^{3}$ up to rigid motion — making $(g, II)$ the complete invariant set, analogous to Frenet-Serret Formulas for curves. The vanishing of $II$ defines Totally Geodesic Submanifolds; the vanishing of its trace defines Minimal Surfaces. In codimension $\geq 2$ , the Ricci Equation and normal connection $\nabla^{⊥}$ become essential, connecting to gauge theory on the normal bundle. The mean curvature flow $\partial_{t} f = H$ and the study of Constant Mean Curvature Surfaces are nonlinear-PDE descendants.

Real-World Applications

General relativity — initial data and ADM formalism. A spacelike hypersurface $Σ$ in spacetime carries an induced metric $g$ (the "spatial geometry") and a second fundamental form $K_{ij}$ playing the role of extrinsic curvature / the conjugate momentum. The Gauss–Codazzi equations become the Hamiltonian and momentum constraints $R + (tr K)^{2} - ∣ K ∣^{2} = 16 π ρ$ and $\nabla^{j} (K_{ij} - g_{ij} tr K) = 8 π J_{i}$ in the Einstein equations.

Soap films and minimal surfaces. Physical soap films minimize area, hence have $H = 0$ ; the mean-curvature condition from $II$ governs their shape and Plateau's problem.

Capillarity and CMC interfaces. Soap bubbles and liquid drops have constant mean curvature determined by the pressure difference via the Young–Laplace law $Δ p = 2 σ H$ .

Computer graphics and geometry processing. Discrete second fundamental forms drive curvature-based shading, mesh smoothing, feature detection, and the discrete Gauss–Bonnet machinery.

Architecture and structural engineering. Curved shell structures (Gehry-style, gridshells, tensile membranes) are designed using principal curvatures and mean curvature; doubly-curved ( $K > 0$ ) panels resist buckling.

Image processing and vision. Level-set methods evolve image-derived surfaces by mean curvature flow ( $H$ ) for denoising and segmentation.

Cell and membrane biophysics. The Helfrich bending energy of a lipid bilayer is $\int (2 κ H^{2} + \overset{κ}{ˉ} K) d A$ — quadratic in the curvatures extracted from $II$ — governing red-blood-cell and vesicle shapes.

Historical Development

1760s — Euler. Introduces principal curvatures and Euler's formula $κ_{n} (θ) = κ_{1} cos^{2} θ + κ_{2} sin^{2} θ$ for normal curvature of surface sections.
1776 — Meusnier. Proves Meusnier's theorem relating the curvature of oblique sections to the normal curvature; precursor to the scalar second fundamental form.
1827 — Gauss. Disquisitiones generales circa superficies curvas: defines both fundamental forms (in coordinates $E, F, G$ and $L, M, N$ ), Gaussian curvature, and proves the Theorema Egregium.
1853–1869 — Mainardi & Codazzi. Independently derive the compatibility equations [11] now called the Codazzi–Mainardi equations; Bonnet (1867) establishes the fundamental theorem of surface theory.
1880s–1900s — Darboux. Leçons sur la théorie générale des surfaces systematizes moving-frame methods and the structure equations.
1900 — Ricci & Levi-Civita. Méthodes de calcul différentiel absolu establishes the tensor calculus enabling intrinsic, coordinate-free treatment; foundation for higher dimensions and codimensions.
1917 — Levi-Civita. Introduces parallel transport and the connection, clarifying that the induced connection is the tangential projection of the ambient one.
1916 — Einstein. General relativity uses Riemannian curvature; the 1920s–60s ADM/Gauss–Codazzi reformulation makes $II$ central to the initial-value problem.
Mid-20th c. — Chern, Kobayashi–Nomizu. Modern vector-bundle/connection formulation: Gauss, Codazzi, Ricci equations for arbitrary codimension stated as in [10]–[12].

Subtleties & Limits

Isometric embedding (Weyl & Nash problems). Given an abstract Riemannian

(M^{n}, g)

, when does there exist an isometric embedding into some

R^{N}

realizing

g

— and with what regularity? The Nash–Kuiper theorem (1954–56) gives

C^{1}

isometric embeddings in surprisingly low dimension but with wild

II

(counterintuitive "crumpled" embeddings); the smooth case (Nash 1956, Günther's improvements) needs higher

N

. The Weyl problem — isometrically embedding

(S^{2}, g)

with

K > 0

into

R^{3}

— was solved by Nirenberg and Pogorelov, but the analogous rigidity/existence questions for

K

changing sign or for higher genus remain delicate. Optimal target dimension. The minimal

N

for smooth global isometric embedding of compact

n

-manifolds is not sharply known; the Nash bound has been lowered but a tight general answer is open. Singularities of curvature flows. For mean curvature flow

\partial_{t} f = H

(which is driven by

tr II

), classifying the singularity models, neck-pinches, and the structure of

II

blow-ups in higher codimension remains an active research frontier (Huisken, Colding–Minicozzi, and others).

Submanifolds and the Second Fundamental Form

The Core Insight

Formal Statement

Isometric immersion and induced metric

The second fundamental form

Properties of II

Shape operator and Weingarten formula

The fundamental equations of submanifolds

Carl Friedrich Gauss | 1777–1855

Building Intuition

A surface in $\mathbb R^3$ with tangent plane, normal, and the splitting of the ambient derivative into tangential ($\nabla_XY$) and normal ($\mathrm{II}(X,Y)$) parts.

Worked Examples

Sphere of radius $r$ in $\mathbb R^3$

Graph of a function $z=u(x,y)$ as a hypersurface

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

Before this

Where this leads