The Second Fundamental Form

How much does a surface bend away from its tangent plane at a point? This question has a precise quantitative answer. It is the subject of this article. Given a smooth surface sitting inside three-dimensional space, we already know something from The First Fundamental Form. We know how to measure lengths and angles within the surface. This is an intrinsic notion. A two-dimensional being living on the surface could detect it without ever leaving. The second fundamental form captures something different. It measures how the surface curves away from its tangent plane, as seen from the ambient space. This is an extrinsic notion tied to the specific embedding. The key tool making this precise is the derivative of The Gauss Map, the unit normal vector field along the surface. As we move across the surface, the normal vector tilts. The rate of that tilting encodes exactly how the surface bends. This rate is a linear map on each tangent plane, called the shape operator. The second fundamental form is the associated bilinear form. It lets us read off, in every tangent direction, the normal curvature. This is the curvature of the curve traced out by moving in that direction. This machinery is the direct ancestor of Gaussian curvature and mean curvature. These are the two scalar invariants built from the second fundamental form. They will occupy the sequel to this article. Historically, this is exactly the route Gauss himself took. This was before his Theorema Egregium revealed something surprising. One particular combination of second-fundamental-form data is secretly intrinsic. This is a stunning fact. It only makes sense once the extrinsic second fundamental form has been carefully built first. By the end of this article you will have several things. You will have a complete, rigorous construction of the shape operator and second fundamental form. You will have a full proof that the shape operator is self-adjoint (the technical heart of the theory). You will have worked computations on a sphere and a saddle-shaped surface. You will also have a clear picture of exactly which hypotheses (orientability, smoothness) are load-bearing.

Key takeaways
  • The shape operator Sp=−dNpS_p = -dN_p is the derivative of the Gauss map, a linear operator on each tangent plane.
  • The second fundamental form IIp(v,w)=⟨Sp(v),w⟩\mathrm{II}_p(v,w) = \langle S_p(v), w\rangle is the symmetric bilinear form associated to SpS_p.
  • Self-adjointness of SpS_p is not automatic — it follows from equality of mixed partial derivatives of the parametrization.
  • For a curve on the surface, II(v,v)\mathrm{II}(v,v) equals the normal curvature: the component of the curve's acceleration along the surface normal.
  • In local coordinates, II\mathrm{II} has matrix entries e,f,ge, f, g built from second derivatives of the parametrization dotted with the unit normal.
  • Key idea: the second fundamental form measures curving-away-from-the-tangent-plane by differentiating the normal vector field, turning "how the normal tilts" into a symmetric bilinear form on each tangent space.

Where this lives today

The shape operator and second fundamental form are the direct input to Gaussian and mean curvature, which govern everything from soap film shapes (minimal surfaces have mean curvature zero) to the classification of surfaces via the Gauss–Bonnet theorem. In general relativity, the second fundamental form of a spacelike hypersurface embedded in spacetime — called the extrinsic curvature there — appears directly in the Gauss–Codazzi equations used to formulate the initial value (Cauchy) problem for Einstein's equations, exactly the same self-adjointness argument applying to hypersurfaces in a pseudo-Riemannian ambient space. In computer graphics and geometric modeling, principal curvatures (eigenvalues of the discretized shape operator) drive mesh simplification, feature-line extraction, and curvature-adaptive rendering. In computer vision, local surface curvature computed from range data or stereo reconstructions is used for object recognition and registration. In materials science and shell theory, the second fundamental form of a thin shell's midsurface enters the bending energy in Koiter's shell theory, governing how much energy it costs to bend a plate versus stretch it. In cryptography and coding theory the connection is more distant but real: curvature invariants of algebraic varieties over finite fields (an algebraic-geometric cousin of the same second-order data) inform bounds used in the study of algebraic-geometric codes.

Ask the tutor: Can you show me how the Gauss–Codazzi equations arise as integrability conditions once you have the shape operator?

DefinitionShape Operator and Second Fundamental Form

Let S⊂R3S \subset \mathbb{R}^3 be a regular surface and let N:S→S2N: S \to \mathbb{S}^2 be a choice of unit normal vector field on SS (the Gauss Map), which exists at least locally, and globally if SS is orientable. Fix p∈Sp \in S. Since NN has unit length, N:S→S2N: S \to \mathbb{S}^2 is a smooth map, and its differential at pp,

dNp:TpS→TN(p)S2,dN_p : T_pS \to T_{N(p)}\mathbb{S}^2,

lands in TpST_pS itself. Indeed TN(p)S2=N(p)⊥=TpST_{N(p)}\mathbb{S}^2 = N(p)^\perp = T_pS, because both are the orthogonal complement of N(p)N(p) in R3\mathbb{R}^3. Thus dNpdN_p is a linear endomorphism of the two-dimensional vector space TpST_pS. The shape operator (also called the Weingarten map) at pp is defined by

Sp:=−dNp:TpS→TpS.S_p := -dN_p : T_pS \to T_pS.

The second fundamental form at pp is the bilinear form

IIp(v,w):=⟨Sp(v),w⟩,v,w∈TpS,\mathrm{II}_p(v, w) := \langle S_p(v), w \rangle, \qquad v, w \in T_pS,

where ⟨⋅,⋅⟩\langle \cdot,\cdot\rangle is the Euclidean inner product restricted to TpST_pS (the first fundamental form). Equivalently, for v∈TpSv \in T_pS, one writes the associated quadratic form

IIp(v):=IIp(v,v)=⟨−dNp(v),v⟩.\mathrm{II}_p(v) := \mathrm{II}_p(v,v) = \langle -dN_p(v), v\rangle .

If φ:U→S\varphi: U \to S is a local parametrization with U⊂R2U \subset \mathbb{R}^2 open, φu=∂φ/∂u\varphi_u = \partial\varphi/\partial u, φv=∂φ/∂v\varphi_v = \partial\varphi/\partial v, and

N=φu×φv∣φu×φv∣,N = \frac{\varphi_u \times \varphi_v}{|\varphi_u \times \varphi_v|},

the coefficients of II\mathrm{II} in this basis are

e=⟨φuu,N⟩,f=⟨φuv,N⟩,g=⟨φvv,N⟩.e = \langle \varphi_{uu}, N\rangle, \quad f = \langle \varphi_{uv}, N \rangle, \quad g = \langle \varphi_{vv}, N \rangle .

Intuition

Picture standing at a point pp on the surface and looking straight along the normal N(p)N(p). Now walk in some tangent direction vv. As you walk, the normal vector at your new location tilts away from N(p)N(p); the shape operator Sp(v)S_p(v) is (up to sign) the instantaneous velocity of that tilt. If the surface is flat near pp — like a plane, or momentarily like the flat table you are reading this on — the normal does not tilt at all as you move, so dNp=0dN_p = 0 and Sp=0S_p = 0. If the surface curves sharply, like the tip of a cone or the outside of a sphere, the normal swings quickly, and SpS_p is large. The reason for the minus sign in Sp=−dNpS_p = -dN_p is a convention chosen so that a sphere of radius rr, with outward normal, has positive normal curvature 1/r1/r in every direction — matching the everyday intuition that a sphere curves "toward its center," i.e. away from the outward normal, and we want that bending recorded as a positive number. Concretely, for the round sphere N(p)=p/rN(p) = p/r, so dNp(v)=v/rdN_p(v) = v/r for v∈TpSv \in T_pS, hence Sp=−1r idS_p = -\tfrac1r \,\mathrm{id}. That looks like it gives a negative curvature, which is exactly why sign conventions here are traditions rather than universal laws — some texts use the outward normal and get Sp=−1ridS_p = -\frac1r\mathrm{id} with the understanding that mean curvature is reported for the inward normal, others flip the normal choice. We fix the convention above and use it consistently: what matters is that the eigenvalues of SpS_p, called the principal curvatures, are well-defined once NN is fixed, and they change sign together if NN is replaced by −N-N. The deepest fact making all of this useful rather than just descriptive is that SpS_p is a symmetric (self-adjoint) operator with respect to the first fundamental form. This is not obvious from the definition — a priori dNpdN_p could be any linear map — and its proof is the technical core of this article. Self-adjointness is what guarantees real eigenvalues (the principal curvatures) and an orthogonal eigenbasis (the principal directions), turning the second fundamental form from an abstract bilinear pairing into genuinely measurable bending data in two perpendicular directions.

Figure
Tilting normal vectors along a curved surface (cross-section view): the shape operator measures the rate of this tilt
p N(p) Normals fan out as the surface curves: dN_p ≠ 0, so S_p = −dN_p is nonzero.
TheoremSymmetry of the Shape Operator

Let S⊂R3S \subset \mathbb{R}^3 be a regular surface with a smooth choice of unit normal NN, and let p∈Sp \in S. Then the shape operator

Sp:TpS→TpS,Sp=−dNp,S_p : T_pS \to T_pS, \qquad S_p = -dN_p,

is self-adjoint with respect to the inner product on TpST_pS: for all v,w∈TpSv, w \in T_pS,

⟨Sp(v),w⟩=⟨v,Sp(w)⟩.\langle S_p(v), w \rangle = \langle v, S_p(w)\rangle .

Equivalently, the second fundamental form IIp(v,w)=⟨Spv,w⟩\mathrm{II}_p(v,w) = \langle S_p v, w\rangle is a symmetric bilinear form, and consequently SpS_p has an orthonormal eigenbasis of TpST_pS with real eigenvalues k1≤k2k_1 \le k_2.

Proof

Setup. Let φ:U→S\varphi: U \to S be a local parametrization around pp, U⊂R2U \subset \mathbb{R}^2 open, with φ(u0,v0)=p\varphi(u_0,v_0) = p. Write φu,φv\varphi_u, \varphi_v for the coordinate partial derivatives, which span TqST_qS at every q=φ(u,v)q = \varphi(u,v) in the image, and let

N=φu×φv∣φu×φv∣N = \frac{\varphi_u \times \varphi_v}{|\varphi_u\times\varphi_v|}

be the induced unit normal (any global choice of unit normal agrees with this or its negative on the connected chart, and the argument below is insensitive to an overall sign). Since {φu,φv}\{\varphi_u, \varphi_v\} is a basis of TqST_qS for each qq in the chart, it suffices — by bilinearity of both sides of the identity to be proved — to check

⟨S(φu),φv⟩=⟨φu,S(φv)⟩\langle S(\varphi_u), \varphi_v\rangle = \langle \varphi_u, S(\varphi_v)\rangle

at the point pp, where S=SpS = S_p. (Checking symmetry on a spanning set of pairs of a bilinear form is enough because a bilinear form is determined by, and symmetric iff symmetric on, a basis: write general v=aφu+bφvv = a\varphi_u + b\varphi_v, w=cφu+dφvw = c\varphi_u+d\varphi_v and expand both sides using bilinearity; all four basis-pair terms match if the mixed one does, and the terms ⟨Sφu,φu⟩\langle S\varphi_u,\varphi_u\rangle, ⟨Sφv,φv⟩\langle S\varphi_v,\varphi_v\rangle trivially agree with their transposes.)

Step 1: express S(φu)S(\varphi_u) and S(φv)S(\varphi_v). By definition S=−dNS = -dN, and dNdN applied to the tangent vector φu\varphi_u (thought of as the velocity of the uu-parameter curve u↦φ(u,v0)u \mapsto \varphi(u,v_0)) is computed by differentiating NN along that curve:

S(φu)=−dN(φu)=−∂∂u(N∘φ)=−Nu,S(\varphi_u) = -dN(\varphi_u) = -\frac{\partial}{\partial u}\big(N\circ\varphi\big) = -N_u,

and similarly S(φv)=−NvS(\varphi_v) = -N_v, where Nu,NvN_u, N_v denote ∂N/∂u\partial N/\partial u, ∂N/∂v\partial N/\partial v evaluated along φ\varphi.

Step 2: rewrite the two sides using orthogonality of NN to the surface. Since N(φ(u,v))N(\varphi(u,v)) is a unit vector for all (u,v)(u,v), we have ⟨N,N⟩≡1\langle N, N\rangle \equiv 1, and differentiating gives ⟨Nu,N⟩=⟨Nv,N⟩=0\langle N_u, N\rangle = \langle N_v, N\rangle = 0; this just says Nu,Nv∈TqSN_u, N_v \in T_qS, consistent with Step 1 landing in TqST_qS. More useful is differentiating the identity ⟨N,φu⟩≡0\langle N, \varphi_u\rangle \equiv 0 (true because N⊥TqS∋φuN \perp T_qS \ni \varphi_u) with respect to vv:

∂∂v⟨N,φu⟩=⟨Nv,φu⟩+⟨N,φuv⟩=0,\frac{\partial}{\partial v}\langle N, \varphi_u\rangle = \langle N_v, \varphi_u\rangle + \langle N, \varphi_{uv}\rangle = 0,

so

⟨Nv,φu⟩=−⟨N,φuv⟩.\langle N_v, \varphi_u \rangle = -\langle N, \varphi_{uv}\rangle .

Symmetrically, differentiating ⟨N,φv⟩≡0\langle N,\varphi_v\rangle \equiv 0 with respect to uu gives

∂∂u⟨N,φv⟩=⟨Nu,φv⟩+⟨N,φvu⟩=0,\frac{\partial}{\partial u}\langle N,\varphi_v\rangle = \langle N_u,\varphi_v\rangle + \langle N,\varphi_{vu}\rangle = 0,

so

⟨Nu,φv⟩=−⟨N,φvu⟩.\langle N_u, \varphi_v\rangle = -\langle N,\varphi_{vu}\rangle.

Step 3: invoke equality of mixed partials. Because φ\varphi is a smooth (indeed C2C^2, which is part of the definition of a regular parametrized surface) map U→R3U \to \mathbb{R}^3, Clairaut's theorem on equality of mixed partial derivatives gives

φuv=φvu\varphi_{uv} = \varphi_{vu}

as vectors in R3\mathbb{R}^3. Substituting into the two boxed identities of Step 2:

⟨Nv,φu⟩=−⟨N,φuv⟩=−⟨N,φvu⟩=⟨Nu,φv⟩.\langle N_v, \varphi_u\rangle = -\langle N, \varphi_{uv}\rangle = -\langle N, \varphi_{vu}\rangle = \langle N_u, \varphi_v\rangle .

Step 4: conclude. Using Step 1, S(φu)=−NuS(\varphi_u) = -N_u and S(φv)=−NvS(\varphi_v) = -N_v, so

⟨S(φu),φv⟩=⟨−Nu,φv⟩=−⟨Nu,φv⟩\langle S(\varphi_u), \varphi_v\rangle = \langle -N_u, \varphi_v\rangle = -\langle N_u,\varphi_v\rangle

and

⟨φu,S(φv)⟩=⟨φu,−Nv⟩=−⟨Nv,φu⟩.\langle \varphi_u, S(\varphi_v)\rangle = \langle \varphi_u, -N_v\rangle = -\langle N_v, \varphi_u\rangle .

By Step 3 these two right-hand sides are equal (both equal −⟨Nu,φv⟩=−⟨Nv,φu⟩-\langle N_u,\varphi_v\rangle= -\langle N_v,\varphi_u\rangle), hence

⟨S(φu),φv⟩=⟨φu,S(φv)⟩.\langle S(\varphi_u), \varphi_v \rangle = \langle \varphi_u, S(\varphi_v)\rangle .

By the spanning-set reduction noted in Setup, this establishes ⟨S(v),w⟩=⟨v,S(w)⟩\langle S(v), w\rangle = \langle v, S(w)\rangle for all v,w∈TpSv, w \in T_pS, i.e. SpS_p is self-adjoint on (TpS,⟨⋅,⋅⟩)(T_pS, \langle\cdot,\cdot\rangle).

Consequences (spectral theorem). A self-adjoint linear operator on a finite-dimensional real inner product space is diagonalizable by an orthonormal eigenbasis with real eigenvalues; this is the finite-dimensional spectral theorem, a standard linear-algebra fact proved by induction on dimension using that a self-adjoint operator on a nonzero real inner-product space always has a real eigenvalue (obtained e.g. as a critical value of the Rayleigh quotient ⟨Sv,v⟩/⟨v,v⟩\langle Sv,v\rangle/\langle v,v\rangle on the unit sphere, which is compact, so a maximizer exists and is an eigenvector by a Lagrange-multiplier argument). Applying this fact to SpS_p on the two-dimensional space TpST_pS produces an orthonormal basis {e1,e2}\{e_1, e_2\} of TpST_pS with Sp(ei)=kieiS_p(e_i) = k_i e_i, k1,k2∈Rk_1, k_2 \in \mathbb{R}, which we may order k1≤k2k_1 \le k_2. ■\blacksquare

∎
CorollaryNormal curvature and Euler's formula

Let γ(t)\gamma(t) be a curve on SS with γ(0)=p\gamma(0) = p, parametrized by arc length, and let v=γ′(0)∈TpSv = \gamma'(0) \in T_pS be a unit tangent vector. Define the normal curvature of SS at pp in direction vv as

kn(v):=⟨γ′′(0),N(p)⟩.k_n(v) := \langle \gamma''(0), N(p)\rangle .

Then kn(v)=IIp(v,v)k_n(v) = \mathrm{II}_p(v,v), and it depends only on vv, not on the particular curve γ\gamma realizing it. Moreover, if {e1,e2}\{e_1,e_2\} is a principal orthonormal eigenbasis with eigenvalues k1,k2k_1, k_2 as in the theorem, and v=cos⁡θ e1+sin⁡θ e2v = \cos\theta\, e_1 + \sin\theta\, e_2, then

kn(v)=k1cos⁡2θ+k2sin⁡2θ(Euler’s formula).k_n(v) = k_1\cos^2\theta + k_2 \sin^2\theta \qquad \text{(Euler's formula)}.
Proof

Since γ\gamma lies on SS, ⟨γ′(t),N(γ(t))⟩=0\langle \gamma'(t), N(\gamma(t))\rangle = 0 for all tt (tangency of γ′\gamma' to SS). Differentiating in tt:

⟨γ′′(t),N(γ(t))⟩+⟨γ′(t),ddtN(γ(t))⟩=0.\langle \gamma''(t), N(\gamma(t))\rangle + \langle \gamma'(t), \tfrac{d}{dt}N(\gamma(t))\rangle = 0.

At t=0t=0, ddtN(γ(t))∣0=dNp(γ′(0))=dNp(v)\tfrac{d}{dt}N(\gamma(t))\big|_0 = dN_p(\gamma'(0)) = dN_p(v), so

⟨γ′′(0),N(p)⟩=−⟨v,dNp(v)⟩=⟨v,Sp(v)⟩=IIp(v,v),\langle \gamma''(0), N(p)\rangle = -\langle v, dN_p(v)\rangle = \langle v, S_p(v)\rangle = \mathrm{II}_p(v,v),

using self-adjointness only implicitly here (we do not even need it for this line, just the definition Sp=−dNpS_p=-dN_p). This expression manifestly depends only on vv (through IIp\mathrm{II}_p, built from SpS_p and the parametrization data), not on the curve, proving the first claim. For Euler's formula, expand using bilinearity and the eigenbasis relations Sp(e1)=k1e1S_p(e_1)=k_1e_1, Sp(e2)=k2e2S_p(e_2) = k_2 e_2, and orthonormality ⟨e1,e2⟩=0\langle e_1,e_2\rangle=0, ⟨ei,ei⟩=1\langle e_i,e_i\rangle=1:

IIp(v,v)=⟨Sp(v),v⟩=⟨cos⁡θ k1e1+sin⁡θ k2e2, cos⁡θ e1+sin⁡θ e2⟩=k1cos⁡2θ+k2sin⁡2θ,\mathrm{II}_p(v,v) = \langle S_p(v), v\rangle = \langle \cos\theta\, k_1 e_1 + \sin\theta\, k_2 e_2,\ \cos\theta\, e_1 + \sin\theta\, e_2\rangle = k_1\cos^2\theta + k_2\sin^2\theta,

the cross terms vanishing by orthonormality. ■\blacksquare

∎

Worked examples

Example 1: the sphere of radius rr, computed from scratch in coordinates (not just by symmetry). Let S={x∈R3:∣x∣=r}S = \{x \in \mathbb{R}^3 : |x| = r\} and parametrize a patch by spherical coordinates,

φ(u,v)=(rcos⁡usin⁡v, rsin⁡usin⁡v, rcos⁡v),u∈(0,2π), v∈(0,π).\varphi(u,v) = \big(r\cos u \sin v,\ r\sin u \sin v,\ r\cos v\big), \qquad u \in (0, 2\pi),\ v \in (0,\pi).

Compute the first-derivative vectors:

φu=r(−sin⁡usin⁡v, cos⁡usin⁡v, 0),φv=r(cos⁡ucos⁡v, sin⁡ucos⁡v, −sin⁡v).\varphi_u = r(-\sin u \sin v,\ \cos u \sin v,\ 0), \qquad \varphi_v = r(\cos u\cos v,\ \sin u \cos v,\ -\sin v).

Take the second point u=π/4u = \pi/4, v=π/3v = \pi/3 (an asymmetric, generic-looking choice, not u=vu=v or a pole) to keep the computation concrete; write s=sin⁡(π/3)=32s = \sin(\pi/3) = \tfrac{\sqrt3}{2}, c=cos⁡(π/3)=12c = \cos(\pi/3) = \tfrac12, σ=sin⁡(π/4)=cos⁡(π/4)=12\sigma = \sin(\pi/4) = \cos(\pi/4) = \tfrac{1}{\sqrt2}. At this point,

φu=r(−σs, σs, 0)=r322(−1,1,0),φv=r(σc, σc, −s)=r(122,122,−32).\varphi_u = r(-\sigma s,\ \sigma s,\ 0) = \frac{r\sqrt3}{2\sqrt2}(-1,1,0), \qquad \varphi_v = r(\sigma c,\ \sigma c,\ -s) = r\Big(\tfrac{1}{2\sqrt2}, \tfrac{1}{2\sqrt2}, -\tfrac{\sqrt3}{2}\Big).

The outward unit normal for a sphere centered at the origin is simply N=φ/rN = \varphi/r (position vector normalized), so at this point

N=(cos⁡usin⁡v, sin⁡usin⁡v, cos⁡v)=(σ32,σ32,12)=(322,322,12).N = (\cos u \sin v,\ \sin u \sin v,\ \cos v) = \Big(\tfrac{\sigma\sqrt3}{2}, \tfrac{\sigma\sqrt3}{2}, \tfrac12\Big) = \Big(\tfrac{\sqrt3}{2\sqrt2}, \tfrac{\sqrt3}{2\sqrt2}, \tfrac12\Big).

Now the second derivatives:

φuu=r(−cos⁡usin⁡v,−sin⁡usin⁡v,0)=−rsin⁡v (cos⁡u,sin⁡u,0)=−rs (σ,σ,0),\varphi_{uu} = r(-\cos u \sin v, -\sin u \sin v, 0) = -r\sin v\,(\cos u,\sin u, 0) = -r s\,(\sigma,\sigma,0),
φuv=r(−sin⁡ucos⁡v,cos⁡ucos⁡v,0)=rc (−σ,σ,0),\varphi_{uv} = r(-\sin u \cos v, \cos u \cos v, 0) = r c\,(-\sigma,\sigma, 0),
φvv=r(−cos⁡usin⁡v,−sin⁡usin⁡v,−cos⁡v)=−r(σs,σs,c).\varphi_{vv} = r(-\cos u \sin v, -\sin u \sin v, -\cos v) = -r(\sigma s, \sigma s, c).

Compute the second fundamental form coefficients e=⟨φuu,N⟩e=\langle\varphi_{uu},N\rangle, f=⟨φuv,N⟩f = \langle\varphi_{uv},N\rangle, g=⟨φvv,N⟩g=\langle\varphi_{vv},N\rangle:

e=−rs(σ⋅σ32+σ⋅σ32+0)=−rs σ23=−rs3⋅12=−rs32.e = -rs\big(\sigma \cdot \tfrac{\sigma\sqrt3}{2} + \sigma\cdot \tfrac{\sigma\sqrt3}{2} + 0\big) = -rs\,\sigma^2\sqrt3 = -rs\sqrt3 \cdot \tfrac12 = -\frac{rs\sqrt3}{2}.

Since s=3/2s = \sqrt3/2: e=−r⋅(3/2)⋅32=−3r4e = -\frac{r\cdot(\sqrt3/2)\cdot\sqrt3}{2} = -\frac{3r}{4}.

f=rc(−σ⋅σ32+σ⋅σ32+0)=0,f = rc\big(-\sigma\cdot\tfrac{\sigma\sqrt3}{2} + \sigma\cdot\tfrac{\sigma\sqrt3}{2} + 0\big) = 0,

which vanishes by the exact cancellation of the first two terms (a genuine cancellation, not a symmetry artifact of the point chosen — it happens because φuv\varphi_{uv}'s first two coordinates are (−σ,σ)⋅rc(-\sigma,\sigma)\cdot rc, exactly antisymmetric against NN's equal first two coordinates).

g=−r(σs⋅σ32+σs⋅σ32+c⋅12)=−r(σ2s3+c2)=−r(s32+c2).g = -r\big(\sigma s \cdot \tfrac{\sigma\sqrt3}{2} + \sigma s\cdot\tfrac{\sigma\sqrt3}{2} + c\cdot\tfrac12\big) = -r\big(\sigma^2 s\sqrt3 + \tfrac{c}{2}\big) = -r\Big(\tfrac{s\sqrt3}{2} + \tfrac{c}{2}\Big).

With s=3/2,c=1/2s = \sqrt3/2, c=1/2: s32=34\dfrac{s\sqrt3}{2} = \dfrac{3}{4}, so g=−r(34+14)=−rg = -r\big(\tfrac34 + \tfrac14\big) = -r. So at this point e=−3r4e = -\tfrac{3r}{4}, f=0f=0, g=−rg=-r. Meanwhile the first fundamental form here is E=∣φu∣2=r2s2=r2⋅34=3r24E = |\varphi_u|^2 = r^2 s^2 = r^2\cdot \tfrac34 = \tfrac{3r^2}{4}, F=0F = 0, G=∣φv∣2=r2G = |\varphi_v|^2 = r^2 (standard sphere metric facts, directly checkable from the vectors above). Since with the outward normal the shape operator matrix in the (φu,φv)(\varphi_u,\varphi_v) basis is [e f;f g] [E F;F G]−1[e\,f;f\,g]\,[E\,F;F\,G]^{-1}, one gets

S=(e/E00g/G)=(−3r/43r2/400−r/r2)=(−1/r00−1/r),S = \begin{pmatrix} e/E & 0\\ 0 & g/G\end{pmatrix} = \begin{pmatrix} -3r/4 \over 3r^2/4 & 0 \\ 0 & -r/r^2\end{pmatrix} = \begin{pmatrix} -1/r & 0\\0 & -1/r\end{pmatrix},

recovering the expected principal curvatures k1=k2=−1/rk_1=k_2=-1/r for the outward normal (matching Sp=−1ridS_p = -\tfrac1r\mathrm{id} from the Intuition section), independent of the specific point (π/4,π/3)(\pi/4,\pi/3) chosen, confirming the sphere is umbilic (every direction is a principal direction) even though the intermediate numbers e,f,ge,f,g were point-dependent-looking.

Example 2: a hyperbolic paraboloid (saddle), with genuinely asymmetric numbers throughout. Let SS be the graph z=x2−2y2z = x^2 - 2y^2, parametrized by

φ(x,y)=(x,y,x2−2y2).\varphi(x,y) = (x, y, x^2-2y^2).

Then φx=(1,0,2x)\varphi_x = (1,0,2x), φy=(0,1,−4y)\varphi_y = (0,1,-4y), φxx=(0,0,2)\varphi_{xx} = (0,0,2), φxy=(0,0,0)\varphi_{xy}=(0,0,0), φyy=(0,0,−4)\varphi_{yy}=(0,0,-4). Evaluate at the point (x,y)=(1,2)(x,y) = (1, 2) (an asymmetric, non-origin point):

φx=(1,0,2),φy=(0,1,−8).\varphi_x = (1,0,2), \qquad \varphi_y = (0,1,-8).

Cross product:

φx×φy=∣ijk10201−8∣=(0⋅(−8)−2⋅1, 2⋅0−1⋅(−8), 1⋅1−0⋅0)=(−2,8,1).\varphi_x \times \varphi_y = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ 1 & 0 & 2 \\ 0 & 1 & -8\end{vmatrix} = (0\cdot(-8) - 2\cdot1,\ 2\cdot0 - 1\cdot(-8),\ 1\cdot1-0\cdot0) = (-2, 8, 1).

Its norm: ∣(−2,8,1)∣=4+64+1=69|(-2,8,1)| = \sqrt{4+64+1} = \sqrt{69}. So

N=169(−2,8,1).N = \frac{1}{\sqrt{69}}(-2, 8, 1).

Since φxx,φyy\varphi_{xx}, \varphi_{yy} are constant vectors and φxy=0\varphi_{xy}=0 everywhere, the second fundamental form coefficients at this point are

e=⟨(0,0,2),N⟩=269,f=⟨(0,0,0),N⟩=0,g=⟨(0,0,−4),N⟩=−469.e = \langle (0,0,2), N\rangle = \frac{2}{\sqrt{69}}, \qquad f = \langle (0,0,0),N\rangle = 0, \qquad g = \langle(0,0,-4),N\rangle = \frac{-4}{\sqrt{69}}.

For the first fundamental form at (1,2)(1,2):

E=⟨φx,φx⟩=1+0+4=5,F=⟨φx,φy⟩=0+0−16=−16,G=⟨φy,φy⟩=0+1+64=65.E = \langle \varphi_x,\varphi_x\rangle = 1+0+4=5, \quad F = \langle\varphi_x,\varphi_y\rangle = 0+0-16=-16, \quad G=\langle\varphi_y,\varphi_y\rangle = 0+1+64=65.

Note F≠0F\ne0 here — the coordinate curves are not orthogonal at this point, a genuinely asymmetric feature. The shape operator matrix (in the basis φx,φy\varphi_x,\varphi_y, acting on coordinate vectors, not orthonormalized) is given by

[S]=(EFFG)−1(effg).\big[S\big] = \begin{pmatrix} E & F\\ F& G\end{pmatrix}^{-1}\begin{pmatrix} e& f\\ f& g\end{pmatrix}.

First invert the first-fundamental-form matrix: det⁡=EG−F2=5⋅65−(−16)2=325−256=69\det = EG-F^2 = 5\cdot65 - (-16)^2 = 325 - 256 = 69. So

(EFFG)−1=169(6516165).\begin{pmatrix}E&F\\F&G\end{pmatrix}^{-1} = \frac{1}{69}\begin{pmatrix}65 & 16\\ 16 & 5\end{pmatrix}.

Then

[S]=169(6516165)(2/6900−4/69)=16969(130−6432−20).[S] = \frac{1}{69}\begin{pmatrix}65&16\\16&5\end{pmatrix}\begin{pmatrix}2/\sqrt{69} & 0 \\ 0 & -4/\sqrt{69}\end{pmatrix} = \frac{1}{69\sqrt{69}}\begin{pmatrix} 130 & -64\\ 32 & -20\end{pmatrix}.

The Gaussian curvature (foreshadowing Gaussian and Mean Curvature) is the determinant of SS, which equals eg−f2EG−F2\dfrac{eg-f^2}{EG-F^2}:

K=eg−f2EG−F2=(269)(−469)−069=−8/6969=−84761.K = \frac{eg-f^2}{EG-F^2} = \frac{\big(\tfrac{2}{\sqrt{69}}\big)\big(\tfrac{-4}{\sqrt{69}}\big) - 0}{69} = \frac{-8/69}{69} = -\frac{8}{4761}.

This is negative, correctly reflecting the saddle shape of a hyperbolic paraboloid at every point — no cancellation or coincidence, since eg<0eg<0 here just as it would be at the origin, but arrived at through a fully asymmetric intermediate computation with F≠0F\ne0.

CounterexampleWhy smoothness (equality of mixed partials) is needed for self-adjointness

The proof of the Symmetry Theorem hinged entirely on Step 3: φuv=φvu\varphi_{uv} = \varphi_{vu}, which used Clairaut's theorem, itself requiring φ\varphi to be at least C2C^2 (continuous second partials suffice; the standard hypothesis for a "regular surface" already builds in smoothness, but it's instructive to see what fails without it). Consider the function

h(x,y)={xyx2−y2x2+y2(x,y)≠(0,0)0(x,y)=(0,0),h(x,y) = \begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2} & (x,y)\ne(0,0) \\ 0 & (x,y)=(0,0)\end{cases},

a classical example where hxy(0,0)≠hyx(0,0)h_{xy}(0,0) \ne h_{yx}(0,0) (a direct computation from the definition of partial derivative gives hxy(0,0)=−1h_{xy}(0,0) = -1 and hyx(0,0)=1h_{yx}(0,0)=1 — the function fails to be C2C^2 at the origin even though all first partials exist everywhere and mixed second partials exist everywhere, just discontinuously). Now build a "surface" Σ\Sigma as the graph z=h(x,y)z = h(x,y), parametrized near the origin by φ(x,y)=(x,y,h(x,y))\varphi(x,y) = (x,y,h(x,y)). Away from the origin hh is smooth (a quotient of polynomials with nonvanishing denominator), so the shape operator is well-defined and self-adjoint on Σ∖{(0,0,0)}\Sigma\setminus\{(0,0,0)\} by the theorem applied on that open, smooth region. But φ\varphi fails to be C2C^2 exactly at the origin, so the hypothesis of the Symmetry Theorem fails to hold at that one point, and correspondingly the formula

⟨Nu,φv⟩⟨Nv,φu⟩\langle N_u,\varphi_v\rangle \overset{?}{=} \langle N_v,\varphi_u\rangle

derived from φuv=φvu\varphi_{uv}=\varphi_{vu} has no justification at the origin: plugging in the mismatched values φxy=(0,0,−1)\varphi_{xy}=(0,0,-1), φyx=(0,0,1)\varphi_{yx}=(0,0,1) (reading off the third coordinates from hxy(0,0),hyx(0,0)h_{xy}(0,0),h_{yx}(0,0), with vanishing x,yx,y-components since φ\varphi's first two coordinates are simply x,yx,y) into the Step 2 identities gives different values for ⟨Nu,φv⟩\langle N_u,\varphi_v\rangle and ⟨Nv,φu⟩\langle N_v,\varphi_u\rangle at that point, so the shape operator computed as lim⁡\lim from Step 1's formulas is not symmetric there — the (would-be) matrix of SS in the φx,φy\varphi_x,\varphi_y basis fails to be symmetric relative to the induced inner product, precisely because e=fe=f coefficient consistency (f=⟨φxy,N⟩f = \langle\varphi_{xy},N\rangle used unambiguously) breaks down once φxy≠φyx\varphi_{xy}\ne\varphi_{yx}. The moral: self-adjointness of the shape operator is a theorem, not a definitional given, and it costs exactly one hypothesis — C2C^2 smoothness of the parametrization — to prove. Drop it, and the second fundamental form as a symmetric bilinear form (with real eigenvalues, orthogonal principal directions) can fail to make sense at points of insufficient smoothness.

Scope, edge cases & common traps

SituationWhat happens
SS non-orientable (e.g. Möbius band)No global unit normal exists, so NN and hence Sp,IIpS_p, \mathrm{II}_p are only defined locally, up to an overall sign ambiguity when patches are glued
Flipping N→−NN \to -NSp→−SpS_p \to -S_p, IIp→−IIp\mathrm{II}_p \to -\mathrm{II}_p; principal curvatures negate but Gaussian curvature K=det⁡SpK=\det S_p is unchanged
Umbilic point (k1=k2k_1=k_2)SpS_p is a scalar multiple of the identity; every direction is a principal direction, eigenbasis is not unique (any orthonormal basis works)
Parametrization only C1C^1, not C2C^2Clairaut's theorem may fail; self-adjointness of SpS_p is not guaranteed (see counterexample)
Points where φu×φv=0\varphi_u \times \varphi_v = 0Not a regular point; NN undefined there, so II\mathrm{II} is undefined; regularity of the parametrization is a standing hypothesis

The mathematician behind it

Origins · 1 minJulius Weingarten and how this was discovered
JW
Julius Weingarten
1836–1910 · German differential geometer, University of Berlin/Freiburg

Julius Weingarten built on Gauss's 1827 Disquisitiones Generales circa Superficies Curvas, systematizing the study of how the normal vector field varies across a surface. The linear operator Sp=−dNpS_p = -dN_p is named the Weingarten map in his honor, and the differential equations relating derivatives of φu,φv,N\varphi_u,\varphi_v, N (the Gauss–Weingarten equations) bear his name jointly with Gauss. His work sat at the transition point between classical surface theory, still phrased in terms of explicit parametrizations and coordinate computations, and the more coordinate-free, operator-theoretic language that Riemannian geometry would later adopt wholesale.

History
- 1827: Gauss publishes Disquisitiones Generales circa Superficies Curvas, introducing the normal map and curvature via what we now call the first and second fundamental forms.
- 1861: Weingarten formalizes the equations for the derivative of the normal vector field in terms of the tangent basis, giving the shape operator its structural equations.
- 1868: Beltrami and others extend surface theory toward intrinsic, coordinate-independent formulations, setting the stage for Riemann's earlier 1854 ideas to be absorbed into classical differential geometry.
- 1899–1900s: Formal tensor-index treatments of e,f,ge,f,g (second fundamental form coefficients) become standard in textbooks by Bianchi, Darboux, and others.
- 1950s: do Carmo-style coordinate-free treatments (shape operator as a self-adjoint linear map, proved via Clairaut's theorem) become the standard modern presentation used in undergraduate courses.
Why is there a minus sign in the definition Sp=−dNpS_p = -dN_p?

The minus sign is a sign convention, not a mathematical necessity, chosen so that convex surfaces with outward-pointing normals record positive normal curvature in the direction the surface curves toward its interior. Concretely, imagine standing outside a sphere looking at the outward normal: as you move along the surface, the normal tilts in a way that, when you compute −dNp-dN_p, produces a positive multiple of the identity, matching the intuitive sense that the sphere "curves positively." Different textbooks make different choices (some use dNpdN_p directly with the inward normal), and the numerical value of IIp\mathrm{II}_p flips sign under N→−NN \to -N regardless of convention. What is convention-independent is the Gaussian curvature K=det⁡(Sp)K = \det(S_p), since det⁡(−A)=det⁡(A)\det(-A)=\det(A) in two dimensions (even number of sign flips).

Is the second fundamental form intrinsic to the surface, like the first fundamental form?

No — that is precisely the point of calling it "extrinsic." The first fundamental form can be measured entirely by an inhabitant of the surface using only lengths and angles within the surface, with no reference to an ambient space. The second fundamental form is defined via the derivative of the normal vector field NN, which requires the surface to be sitting inside R3\mathbb{R}^3 (or some ambient space) — an inhabitant confined to the surface cannot directly observe NN or its derivative. Remarkably, Gauss's Theorema Egregium shows that the determinant det⁡(Sp)\det(S_p), the Gaussian curvature, is secretly intrinsic despite being built from extrinsic data — but the second fundamental form itself, and even the individual principal curvatures, are genuinely extrinsic and can change under bending (isometric deformation) of the surface.

What does it mean for a point to be "umbilic," and why does the sphere example matter?

A point pp is umbilic if the shape operator SpS_p is a scalar multiple of the identity, equivalently the two principal curvatures coincide, k1=k2k_1=k_2. At an umbilic point every tangent direction is a principal direction and the normal curvature kn(v)=IIp(v,v)k_n(v) = \mathrm{II}_p(v,v) is the same number in every direction vv — the surface bends "the same way" no matter which way you walk. Every point of a round sphere is umbilic (as verified in the Worked Examples), and a theorem of Meusnier and others shows that a connected surface all of whose points are umbilic must be (a piece of) a sphere or a plane, making umbilic points into a genuine rigidity phenomenon rather than a generic occurrence.

Does the second fundamental form depend on the choice of parametrization φ\varphi?

The coefficients e,f,ge,f,g computed in a specific parametrization do depend on that parametrization's coordinates, exactly as the first fundamental form coefficients E,F,GE,F,G do, but the underlying bilinear form IIp:TpS×TpS→R\mathrm{II}_p: T_pS \times T_pS \to \mathbb{R} is a well-defined geometric object independent of coordinates, since it is built from the coordinate-free shape operator Sp=−dNpS_p = -dN_p. Changing parametrization transforms e,f,ge,f,g by the usual tensorial transformation law for bilinear forms under a change of basis of TpST_pS (via the Jacobian of the reparametrization), exactly parallel to how E,F,GE,F,G transform. Quantities built invariantly from IIp\mathrm{II}_p and the first fundamental form — like the eigenvalues k1,k2k_1,k_2 of SpS_p, or K=det⁡SpK=\det S_p and H=12tr SpH = \tfrac12\mathrm{tr}\,S_p — are fully parametrization-independent numbers.

How is the second fundamental form related to a Taylor expansion of the surface near pp?

If you choose coordinates so that pp is the origin and TpST_pS is the xyxy-plane (always possible after a rigid motion), the surface can locally be written as a graph z=h(x,y)z = h(x,y) with h(0,0)=0h(0,0)=0 and ∇h(0,0)=0\nabla h(0,0) = 0 (since the tangent plane is horizontal). The Taylor expansion of hh near the origin then reads h(x,y)=12(e x2+2f xy+g y2)+O(∣(x,y)∣3)h(x,y) = \tfrac12(e\,x^2 + 2f\,xy + g\,y^2) + O(|(x,y)|^3), where e,f,ge,f,g are exactly the second fundamental form coefficients at pp computed with N(p)=(0,0,1)N(p) = (0,0,1). This makes precise the slogan that IIp\mathrm{II}_p is "the quadratic part of the surface's shape," a second-order Taylor approximation to how far the surface departs from its own tangent plane.

Can the second fundamental form be zero at every point of a surface?

Yes — this happens exactly for (open subsets of) planes. If IIp≡0\mathrm{II}_p \equiv 0 for all pp in a connected surface, then Sp=0S_p=0 everywhere, so dNp=0dN_p = 0 everywhere, which forces NN to be locally constant (its differential vanishes identically on a connected domain), hence globally constant on a connected surface. A surface with constant unit normal N0N_0 satisfies ⟨φ(u,v)−φ(u0,v0),N0⟩=0\langle \varphi(u,v) - \varphi(u_0,v_0), N_0\rangle = 0 for all points in a connected chart (since the derivative of that inner product in uu and in vv both vanish, by ⟨φu,N0⟩=⟨φv,N0⟩=0\langle\varphi_u,N_0\rangle=\langle\varphi_v,N_0\rangle=0), which is exactly the equation of a plane. So vanishing second fundamental form characterizes planes among surfaces, the extrinsic analogue of "flatness."

How does the second fundamental form connect to what comes next, Gaussian and mean curvature?

The shape operator SpS_p, once known to be self-adjoint with real eigenvalues k1≤k2k_1\le k_2 (the principal curvatures), immediately yields two scalar invariants: the Gaussian curvature K=det⁡(Sp)=k1k2K = \det(S_p) = k_1k_2 and the mean curvature H=12tr(Sp)=12(k1+k2)H = \tfrac12\mathrm{tr}(S_p) = \tfrac12(k_1+k_2). In coordinates these are K=(eg−f2)/(EG−F2)K = (eg-f^2)/(EG-F^2) and H=(eG−2fF+gE)/(2(EG−F2))H = (eG - 2fF + gE)/\big(2(EG-F^2)\big), built directly from the first and second fundamental form coefficients computed here. The entire theory of curvature for surfaces — the Theorema Egregium (that KK is intrinsic despite e,f,ge,f,g individually not being so), minimal surfaces (H≡0H\equiv0), and the Gauss–Bonnet theorem — is built on exactly the shape operator machinery established in this article.

Before this

Where this leads