Submanifolds and the Second Fundamental Form

⚡ Key Takeaways

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 $\tilde\nabla_X Y = \nabla_X Y + \mathrm{II}(X,Y)$ splits the ambient connection into intrinsic (tangential) and extrinsic (normal) parts; $\mathrm{II}$ 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).

💡 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 = \underbrace{(\tilde\nabla_X Y)^\top}_{\text{intrinsic: } \nabla_X Y} + \underbrace{(\tilde\nabla_X Y)^\perp}_{\text{extrinsic: } \mathrm{II}(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 $\mathrm{II}$ .

DefinitionIsometric 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 $df_p:T_pM\to T_{f(p)}\tilde M$ is injective for all $p$ . It is an isometric immersion if the induced metric (first fundamental form)

1

\quad g(X,Y) := \tilde g\big(df(X),\,df(Y)\big)

equals a prescribed metric on $M$ . We identify $T_pM$ with its image $df_p(T_pM)\subset T_{f(p)}\tilde M$ and write $\tilde g$ for both metrics. The normal bundle is the orthogonal complement

2

\quad N_pM := (T_pM)^\perp \subset T_{f(p)}\tilde M,\qquad T_{f(p)}\tilde M = T_pM \oplus N_pM.

Derivation hint: 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.

DefinitionThe second fundamental form

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

3

\quad \mathrm{II}(X,Y) := \big(\tilde\nabla_X Y\big)^{\perp} \in \Gamma(NM),

and define the induced connection by the tangential projection $\nabla_X Y := (\tilde\nabla_X Y)^\top$ . The Gauss formula is then

4

\quad \tilde\nabla_X Y = \nabla_X Y + \mathrm{II}(X,Y).

For a hypersurface ( $k=1$ ) with unit normal $\nu$ , the scalar second fundamental form is $h(X,Y) := \tilde g(\mathrm{II}(X,Y),\nu)$ , so $\mathrm{II}(X,Y) = h(X,Y)\,\nu$ . Derivation hint: [3] is well-defined because $\mathrm{II}(X,Y)$ depends only on the values of $X,Y$ on $M$ , not on extensions — verified via tensoriality below.

TheoremProperties of II

The map $\mathrm{II}:\Gamma(TM)\times\Gamma(TM)\to\Gamma(NM)$ is: 1. $C^\infty(M)$ -bilinear (a tensor): $\mathrm{II}(\varphi X,Y)=\varphi\,\mathrm{II}(X,Y)$ and similarly in $Y$ . 2. Symmetric: $\mathrm{II}(X,Y)=\mathrm{II}(Y,X)$ . 3. The tangential connection $\nabla$ is the Levi-Civita connection of the induced metric $g$ .

Proof

Step 1 (Tensoriality in $X$ ). Since $\tilde\nabla$ is a connection, $\tilde\nabla_{\varphi X}Y = \varphi\,\tilde\nabla_X Y$ ; projecting normally gives $C^\infty$ -linearity in $X$ .

Step 2 (Tensoriality in $Y$ ). Compute

5

\quad \tilde\nabla_X(\varphi Y) = (X\varphi)\,Y + \varphi\,\tilde\nabla_X Y.

Derivation hint: Leibniz rule for the connection in the second slot. The term $(X\varphi)Y$ is tangent to $M$ , so it dies under normal projection, leaving $\mathrm{II}(X,\varphi Y)=\varphi\,\mathrm{II}(X,Y)$ . Hence $\mathrm{II}$ is tensorial; it depends only on pointwise values.

Step 3 (Symmetry). Because $\tilde\nabla$ is torsion-free,

6

\quad \tilde\nabla_X Y - \tilde\nabla_Y X = [X,Y].

Derivation hint: Torsion-freeness $T(X,Y)=\tilde\nabla_XY-\tilde\nabla_YX-[X,Y]=0$ . For $X,Y$ tangent to $M$ , the bracket $[X,Y]$ is again tangent to $M$ . Projecting [6] normally annihilates $[X,Y]$ , giving $\mathrm{II}(X,Y)-\mathrm{II}(Y,X)=0$ .

Step 4 ( $\nabla$ is metric and torsion-free). Torsion: projecting [6] tangentially, $\nabla_XY-\nabla_YX = [X,Y]$ , so $\nabla$ is torsion-free. Metric compatibility: for tangent $X,Y,Z$ ,

7

\quad X\,g(Y,Z) = \tilde g(\tilde\nabla_XY,Z)+\tilde g(Y,\tilde\nabla_XZ) = g(\nabla_XY,Z)+g(Y,\nabla_XZ),

Derivation hint: $\tilde\nabla$ is metric; the normal parts $\mathrm{II}(X,Y),\mathrm{II}(X,Z)$ 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 $g$ . $\blacksquare$

∎

DefinitionShape operator and Weingarten formula

For a normal field $\xi\in\Gamma(NM)$ , split the ambient derivative of $\xi$ along tangent $X$ :

8

\quad \tilde\nabla_X\xi = -A_\xi X + \nabla^\perp_X\xi,

where $A_\xi X := -(\tilde\nabla_X\xi)^\top$ defines the shape operator (Weingarten map) and $\nabla^\perp$ is the normal connection. The two forms are related by

9

\quad \tilde g\big(\mathrm{II}(X,Y),\xi\big) = \tilde g\big(A_\xi X,\,Y\big).

Derivation hint: Differentiate $\tilde g(Y,\xi)=0$ along $X$ : $0 = \tilde g(\tilde\nabla_XY,\xi)+\tilde g(Y,\tilde\nabla_X\xi)$ , then use [4] and [8]. Thus $A_\xi$ is a self-adjoint endomorphism of $T_pM$ ; its eigenvalues are the principal curvatures $\kappa_1,\dots,\kappa_n$ , its trace gives the mean curvature vector $H=\tfrac1n\operatorname{tr}_g\mathrm{II}$ , and for hypersurfaces $\det A_\nu$ is the Gauss–Kronecker curvature.

TheoremThe fundamental equations of submanifolds

Let $\tilde R,R,R^\perp$ be the curvature tensors of $\tilde\nabla,\nabla,\nabla^\perp$ . For tangent $X,Y,Z,W$ and normal $\xi,\eta$ :

Gauss equation:

10

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

Codazzi (Codazzi–Mainardi) equation:

11

\quad \big(\tilde R(X,Y)Z\big)^\perp = (\bar\nabla_X\mathrm{II})(Y,Z) - (\bar\nabla_Y\mathrm{II})(X,Z),

where $\bar\nabla$ is the connection on $T^*M\otimes T^*M\otimes NM$ built from $\nabla$ and $\nabla^\perp$ .

Ricci equation:

12

\quad \tilde g(\tilde R(X,Y)\xi,\eta) = \tilde g(R^\perp(X,Y)\xi,\eta) - \tilde g\big([A_\xi,A_\eta]X,Y\big).

Proof

Gauss equation. Apply $\tilde\nabla$ twice using the Gauss formula [4]:

13

\quad \tilde\nabla_X\tilde\nabla_YZ = \tilde\nabla_X\big(\nabla_YZ + \mathrm{II}(Y,Z)\big) = \nabla_X\nabla_YZ + \mathrm{II}(X,\nabla_YZ) - A_{\mathrm{II}(Y,Z)}X + \nabla^\perp_X\mathrm{II}(Y,Z).

Derivation hint: Use [4] on $\tilde\nabla_X\nabla_YZ$ and [8] on $\tilde\nabla_X\mathrm{II}(Y,Z)$ . 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 $-\tilde g(A_{\mathrm{II}(Y,Z)}X,W)+\tilde g(A_{\mathrm{II}(X,Z)}Y,W)$ ; 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 $\mathrm{II}$ , 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 $[A_\xi,A_\eta]$ . $\blacksquare$

∎

CF

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 $K=\kappa_1\kappa_2=\det A$ 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.

🧠 Intuition

Think of $\mathrm{II}$ 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 $\mathrm{II}(\dot\gamma,\dot\gamma)$ . The shape operator $A_\xi$ 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 $A=\tfrac1r\,\mathrm{Id}$ (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 $\mathrm{II}$ . For surfaces in flat $\mathbb R^3$ , $\tilde R=0$ , so $K$ is entirely determined by $\mathrm{II}$ — yet $K$ itself is intrinsic. That tension is the Theorema Egregium.

Figure

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.

ExampleSphere of radius

r

in

\mathbb R^3

Given: $S^2_r = \{x\in\mathbb R^3 : |x|=r\}$ with outward unit normal $\nu(x)=x/r$ . The ambient connection $\tilde\nabla$ is the flat directional derivative $D$ .

Step 1 — shape operator. For tangent $X$ , $A_\nu X = -(\tilde\nabla_X\nu)^\top$ . Since $\nu=x/r$ , $D_X\nu = X/r$ (which is already tangent), so

14

\quad A_\nu X = -\tfrac1r X.

Derivation hint: $\nu(x)=x/r$ is linear in position; its directional derivative is $X/r$ . (Sign depends on normal orientation; with inward normal you'd get $+\tfrac1r$ .)

Step 2 — second fundamental form. $h(X,Y)=\tilde g(A_\nu X,Y) = -\tfrac1r\,g(X,Y)$ , so $\mathrm{II}(X,Y) = -\tfrac1r g(X,Y)\,\nu$ . The surface is totally umbilic: both principal curvatures equal $-1/r$ .

Step 3 — Gaussian curvature via Gauss equation. With $\tilde R=0$ in flat $\mathbb R^3$ , [10] gives

15

\quad K = \kappa_1\kappa_2 = \det A_\nu = \big(\tfrac1r\big)^2.

Derivation hint: $\det\big(\tfrac1r\mathrm{Id}_{2\times2}\big)=1/r^2$ .

Boxed answer: $\boxed{K = 1/r^2,\quad H = -1/r,\quad \mathrm{II}=-\tfrac1r\,g\,\nu}$ (units: $K$ in $\text{length}^{-2}$ , $H$ in $\text{length}^{-1}$ ).

ExampleGraph of a function

z=u(x,y)

as a hypersurface

Given: $M=\{(x,y,u(x,y))\}\subset\mathbb R^3$ . Frame $f_x=(1,0,u_x),\ f_y=(0,1,u_y)$ .

Step 1 — induced metric. $g_{ij}=\delta_{ij}+u_iu_j$ , i.e.

16

\quad g = \begin{pmatrix} 1+u_x^2 & u_xu_y \\ u_xu_y & 1+u_y^2\end{pmatrix},\qquad \det g = 1+u_x^2+u_y^2.

Derivation hint: Dot the frame vectors; $f_x\cdot f_x = 1+u_x^2$ , etc.

Step 2 — unit normal. $\nu = \dfrac{(-u_x,-u_y,1)}{\sqrt{1+u_x^2+u_y^2}}$ .

Step 3 — second fundamental form. $h_{ij}=\tilde g(D_{f_i}f_j,\nu) = \nu\cdot f_{ij}$ where $f_{ij}=(0,0,u_{ij})$ :

17

\quad h_{ij} = \frac{u_{ij}}{\sqrt{1+u_x^2+u_y^2}},\qquad h = \frac{1}{W}\begin{pmatrix} u_{xx} & u_{xy} \\ u_{xy} & u_{yy}\end{pmatrix},\ \ W=\sqrt{1+|\nabla u|^2}.

Derivation hint: Only the $z$ -component of $f_{ij}$ survives the dot product with $\nu$ .

Step 4 — curvatures. Gaussian and mean curvature follow from $K=\det h/\det g$ and $H=\tfrac12\operatorname{tr}_g h$ :

18

\quad K = \frac{u_{xx}u_{yy}-u_{xy}^2}{(1+|\nabla u|^2)^2},\qquad H = \frac{(1+u_y^2)u_{xx} - 2u_xu_yu_{xy} + (1+u_x^2)u_{yy}}{2(1+|\nabla u|^2)^{3/2}}.

Derivation hint: $K=\det(g^{-1}h)=\det h/\det g$ ; $H=\tfrac12 g^{ij}h_{ij}$ with $g^{ij}$ the inverse metric.

Boxed answer (minimal surface PDE): $H=0 \iff \boxed{(1+u_y^2)u_{xx} - 2u_xu_yu_{xy} + (1+u_x^2)u_{yy}=0}$ — the nonparametric minimal surface equation.

⚠️ Common mistake

Sign and orientation conventions vary wildly. Many texts define $\mathrm{II}(X,Y)=(\tilde\nabla_X Y)^\perp$ (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 ( $\tfrac1n\operatorname{tr}\mathrm{II}$ ) or a scalar ( $\operatorname{tr} A_\nu$ or $\tfrac1n\operatorname{tr} A_\nu$ ). 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.

$\mathrm{II}$ depends on the embedding, not just the metric. Two isometric immersions of the same $(M,g)$ can have completely different $\mathrm{II}$ . The Gaussian curvature is intrinsic, but $\mathrm{II}$ itself is genuinely extrinsic data.

Counterexample

A cylinder and a plane are intrinsically identical but extrinsically different. Consider the cylinder $C=\{(\cos\theta,\sin\theta,z)\}\subset\mathbb R^3$ and the plane $P=\mathbb R^2$ . 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:

19

\quad \mathrm{II}_P \equiv 0, \qquad A_C = \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}\ (\text{eigenvalues } \kappa_1=0,\ \kappa_2=1).

Derivation hint: The plane is totally geodesic; the cylinder curves with radius 1 around its circular direction and is flat along its axis. This proves $\mathrm{II}$ cannot be recovered from $g$ alone — the Theorema Egregium extracts only the determinant $\det A$ as intrinsic, not $\mathrm{II}$ itself. It also shows totally geodesic ( $\mathrm{II}=0$ ) is strictly stronger than flat ( $K=0$ ).

connection

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 $\mathrm{II}$ satisfying the Gauss–Codazzi compatibility equations determine a surface in $\mathbb R^3$ up to rigid motion — making $(g,\mathrm{II})$ the complete invariant set, analogous to Frenet-Serret Formulas for curves. The vanishing of $\mathrm{II}$ defines Totally Geodesic Submanifolds; the vanishing of its trace defines Minimal Surfaces. In codimension $\ge 2$ , the Ricci Equation and normal connection $\nabla^\perp$ 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.

🔬 Where this appears

⏳ History

- 1760s — Euler. Introduces principal curvatures and Euler's formula

\kappa_n(\theta)=\kappa_1\cos^2\theta+\kappa_2\sin^2\theta

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

\mathrm{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].

open-problem

Isometric embedding (Weyl & Nash problems). Given an abstract Riemannian $(M^n,g)$ , when does there exist an isometric embedding into some $\mathbb R^N$ realizing $g$ — and with what regularity? The Nash–Kuiper theorem (1954–56) gives $C^1$ isometric embeddings in surprisingly low dimension but with wild $\mathrm{II}$ (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 $\mathbb R^3$ — 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 $\operatorname{tr}\mathrm{II}$ ), classifying the singularity models, neck-pinches, and the structure of $\mathrm{II}$ blow-ups in higher codimension remains an active research frontier (Huisken, Colding–Minicozzi, and others).

Submanifolds and the Second Fundamental Form

Formal Statement

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits