Second fundamental form ($II$)

For an embedded submanifold $M\subseteq N$ of a Riemannian manifold, the vector-valued second fundamental form isHere $\overline{\nabla }$ is the ambient Levi-Civita connection and the superscript denotes projection onto the normal bundle. For a hypersurface in Euclidean space with unit normal $N$, the scalar form is $II(X,Y)=⟨A(X,Y),N⟩=⟨-dN(X),Y⟩$. In a surface parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.

For major radius $a>b>0$ and minor radius $b$, let $R=a+b\cos u$. With the inward normal determined by the usual ordered $(u,v)$ parametrization, the first fundamental form and second fundamental form areReversing the normal reverses $II$. The Gaussian curvature of a ring torus is independent of this orientation.

For a ring torus of radii $a>b>0$, the Gaussian curvature isIt is positive on the outer side, negative on the inner side and zero on the top and bottom circles. The sign contrast expresses two principal curvatures of the same sign outside and opposite signs inside.

The Gauss–Codazzi equations relate the intrinsic curvature and connection of an embedded submanifold to its second fundamental form and the ambient curvature. The Gauss equation controls tangential curvature; the Codazzi equation controls the covariant derivative of the second fundamental form. The Gauss formula is the tangential-normal splitting used to derive these relations.

For a submanifold with induced Levi-Civita connection $\nabla$ and second fundamental form $A$, the ambient derivative decomposes asFor a parametrized surface this reads $X_{ij}={\Gamma }_{ij}^k{X}_k+h_{ij}N$.

For a Euclidean embedded submanifold,It expresses intrinsic curvature in terms of the second fundamental form.

For an embedded submanifold of a Riemannian manifold, use $R(v,w)y={\nabla }_v{\nabla }_{w}y-{\nabla }_w{\nabla }_{v}y-{\nabla }_{[v,w]}y$. The Gauss formula and tangential derivative of a normal field giveSubtract the expression with $v,w$ interchanged and the bracket derivative, whose normal part pairs to zero. This proves the displayed curvature identity. For flat Euclidean ambient space, its ambient curvature term vanishes and one obtains the ordinary Gauss equation. Declaring the four-slot convention $R(x,y,v,w)=⟨R(v,w)y,x⟩$ avoids sign ambiguities when permuting arguments.

For a two-plane $\sigma =\mathrm{span}(X,Y)$ in a Riemannian tangent space,

Ricci curvature is the trace of the Riemann curvature tensor in its first and third arguments. For a unit tangent vector $X$ and an orthonormal basis $X,e_2,\dots ,e_n$,

An Einstein manifold has Ricci curvature equal to a constant multiple of its metric. In the Riemannian setting, the round unit sphere has $\Lambda =n-1$. This condition is weaker than constant sectional curvature in higher dimensions. The definition can also be applied to pseudo-Riemannian metrics, but positivity is assumed in the completeness theorems discussed here.

For an orthonormal triple and distinct $i,j,k$, tracing sectional curvature gives ${\mathrm{Ric}}_{ii}=K_{ij}+K_{ik}$. Solving the three equations gives the displayed formula. Equivalently $K_{ij}={\mathrm{Ric}}_{ii}+{\mathrm{Ric}}_{jj}-\mathrm{Scal}/2$. This pointwise assertion does not require the triple to diagonalize the Ricci curvature.

Scalar curvature is the complete trace of the Riemann curvature tensor:An $n$-manifold of constant sectional curvature $K$ has $\mathrm{Scal}=n(n-1)K$.

A Ricci-flat Riemannian manifold has identically zero Ricci tensor. Every flat Riemannian manifold is Ricci-flat, while the converse can fail in dimension at least four.

For a connected Riemannian manifold of dimension $n\ge 2$, assume geodesic completeness and $\mathrm{Ric}\ge (n-1)\kappa g$ for a constant $\kappa >0$. The Riemannian distance then has diameter at most $\pi /\sqrt{\kappa }$; the manifold is compact and its fundamental group is finite. The dimension restriction excludes the vacuous one-dimensional Ricci bound.

If a connected complete Riemannian manifold has $\mathrm{Ric}\ge (n-1)\kappa g$ with $\kappa >0$, its universal cover with the lifted metric is complete and satisfies the same bound. The Bonnet-Myers theorem makes that cover compact. A covering fiber is closed and discrete and hence finite; the number of its elements equals the order of the fundamental group. Applying the diameter theorem only to the base would not prove this conclusion: the positive lower bound must also be used on the cover.

This metric pulls back the round spherical metric by $(u,v)\mapsto (\cos u\cos v,\cos u\sin v,\sin u)$. It has unit Gaussian curvature and Ricci curvature equal to its metric, yet the meridian reaches a missing boundary in finite time. Every curve between $(0,0)$ and $(0,L)$ has length at least $|L|/\sqrt{2}$, so its diameter is infinite. It shows that the positive Ricci bound alone does not replace geodesic completeness in the Bonnet-Myers theorem.

A complete connected Riemannian manifold with nonnegative Ricci curvature that contains a line splits isometrically as $N\times \mathbb{R}$.

A Riemannian manifold is flat when every sectional curvature is zero. A complete simply connected flat $n$-manifold is isometric to Euclidean space ${\mathbb{R}}^n$.

A flat torus is ${\mathbb{R}}^d/\Lambda$ with its quotient Euclidean metric, where $\Lambda$ is a full-rank Euclidean lattice. Its volume is the lattice covolume. The dual lattice indexes its complete Fourier series of Laplacian eigenfunctions.

With the nonnegative Laplace-Beltrami operator, a frequency $w$ in the dual lattice gives the eigenfunction $e^{2\pi i⟨w,x⟩}$ and eigenvalue $4{\pi }^2|w{|}^2$. Its multiplicity is the number of dual vectors with that norm. Fourier series prove completeness. In dimension two the shortest vector, shortest independent vector and covolume determine a reduced Gram matrix, proving spectral rigidity.

The vector-length multiset of a rank-two Euclidean lattice determines it up to an orthogonal map. Let $a$ be the shortest nonzero length. Removing two copies of each positive integer multiple of $a$ removes one primitive lattice line; the shortest remaining length $b$ is the shortest independent-vector length. Those two vectors form a lattice basis: otherwise a nonzero point in the centred basis parallelogram would be an independent lattice vector of length less than $b$. Lattice-point density determines the covolume $A$. The basis Gram matrix then has diagonal entries $a^2,b^2$ and absolute off-diagonal entry $\sqrt{a^2{b}^2-A^2}$. This proves two-dimensional spectral rigidity for a flat torus through its dual lattice.

On a complete connected Riemannian manifold of nonpositive sectional curvature, the exponential map at every point is a covering map. If the manifold is simply connected, each exponential map is a diffeomorphism from a tangent space onto the manifold.

With outward normal $N(x)=x$, the unit sphere has $A(X,Y)=-⟨X,Y⟩N$. The Gauss equation therefore gives sectional curvature one on every tangent two-plane.

For a Euclidean embedded submanifold, the covariant derivative of the second fundamental form satisfies

For the ambient Levi-Civita connection, a normal field $\eta$, and tangent fields $X,Y$, differentiate $⟨\eta ,Y⟩=0$ using metric compatibility. The tangential part of ${\nabla }_{X}Y$ is orthogonal to $\eta$, giving the displayed identity. It controls how an ambient derivative of a normal vector can develop a tangential component, and is the vector-valued version of the shape operator relation.

For a nondegenerate induced metric tensor, a totally geodesic submanifold has vanishing second fundamental form; equivalently, every ambient geodesic initially tangent to it remains in it. This extends the notion of a totally geodesic hypersurface to arbitrary codimension.

A smooth nondegenerate component of the fixed-point set of an isometry is a totally geodesic submanifold. If a geodesic starts tangent to the fixed component, its image under the isometry has identical position and tangent. Uniqueness of the geodesic equation makes the two geodesics coincide throughout their common domain, so the original geodesic stays in the fixed set.

The normal curvature of an oriented surface in a unit tangent direction $v$ isIt equals the normal component of the acceleration of any unit-speed surface curve tangent to $v$.

If a unit tangent makes angle $\theta$ with the first principal direction, the normal curvature is ${\kappa }_n={\kappa }_1{\cos }^2\theta +{\kappa }_2{\sin }^2\theta$. This follows by diagonalizing the second fundamental form relative to the first fundamental form. When the Gaussian curvature is nonnegative, $|{\kappa }_n|\le 2|H|$.

The second fundamental form of an embedded submanifold of a Riemannian manifold is symmetric. Its ambient Levi-Civita connection is a torsion-free connection, soThe Lie bracket of vector fields tangent to the submanifold is tangent and therefore has zero normal component. This proof applies to a curved ambient manifold as well as Euclidean space.

For an oriented surface with Gauss map $N$, the shape operator at $p$ is the self-adjoint linear map

The principal curvatures are the two eigenvalues $k_1,k_2$ of the shape operator. Thus $K=k_1{k}_2$ and $H=(k_1+k_2)/2$.

A principal direction is an eigenvector of the shape operator. The normal curvature in a unit principal direction is the corresponding principal curvature.

An umbilical point of a surface is a point where the two principal curvatures are equal, equivalently where the shape operator is a scalar multiple of the identity.

For a proper Euclidean motion of Euclidean three-space $E(x)=Ax+b$, transport the unit normal by $\tilde{N}(E(p))=AN(p)$. The corresponding shape operators are orthogonally conjugate:Their determinant and trace, hence Gaussian curvature and mean curvature, are unchanged.

Gaussian curvature is the determinant of the shape operator:

A supporting sphere for an embedded surface at $p$ is a sphere tangent at $p$ such that the surface lies locally on one side of the sphere. Comparing second fundamental forms at the tangency controls the surface's normal curvature.

If a compact regular surface lies in a closed Euclidean ball of radius $R$, maximize distance from the ball's centre. At a maximizing point, comparison of the second fundamental form with the tangent supporting sphere makes both principal curvatures have magnitude at least $R^{-1}$ and the same sign. The Gaussian curvature there is therefore at least $R^{-2}$.

An elliptic point of a surface is a point at which the Gaussian curvature is positive, equivalently where the two principal curvatures have the same nonzero sign.

Every regular cone parametrized by $X(r,t)=rc(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, whileis tangent, so two coefficients of the second fundamental form vanish and $\det II=0$.

The mean curvature vector of an $n$-dimensional embedded submanifold is the trace $\mathbf{H}={\sum }_{i}A(e_i,e_i)$. For a surface in ${\mathbb{R}}^3$, the scalar mean curvature is half the trace of the shape operator:

A minimal submanifold has vanishing mean curvature vector. For a surface in ${\mathbb{R}}^3$, this says that the scalar mean curvature is zero, equivalently that its two principal curvatures sum to zero.

An outer area-minimizing surface is a closed surface in a Riemannian initial-data slice whose area is no greater than that of any enclosing competitor. This is a variational property, not a consequence of containment alone. It is important when choosing the horizon-area quantity in the Riemannian Penrose inequality and when examining an apparent-horizon area comparison.

The isothermal coordinates $(u,v)$ give the immersion $(u-u^3/3+u{v}^2,v-v^3/3+u^{2}v,u^2-v^2)$. Its coordinate functions are harmonic and its metric is $(1+u^2+v^2{)}^2(d{u}^2+d{v}^2)$, so it is a complete immersed minimal surface with self-intersections.

With $A(X,Y)=({\overline{\nabla }}_{X}Y{)}^{\perp }$ and $\mathbf{H}=\mathrm{tr}A$, a variation with velocity $V$ satisfiesThus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.

If $M\subseteq {\mathbb{R}}^{n+m}$ and $f$ is smooth in the ambient space, then

For the squared distance $r^2(x)=|x{|}^2$, a minimal $n$-submanifold satisfies ${\Delta }_M{r}^2=2n$. On a compact boundaryless manifold, $r^2$ attains a maximum where the Laplacian is nonpositive, a contradiction for $n>0$.

Replacing the chosen unit normal $N$ by $-N$ replaces the shape operator $S$ by $-S$. On a surface, Gaussian curvature $K=\det S$ is unchanged, while mean curvature $H=\mathrm{tr}(S)/2$ changes sign.

For $X(x,y)=(x,y,h(x,y))$ and $W=\sqrt{1+h_x^2+h_y^2}$,The upward unit normal is $(-h_x,-h_y,1)/W$.

For the graph of $h$,

Suppose two graph surfaces $z=f(x,y)$ and $z=g(x,y)$ are tangent at the origin, are oriented upward, and $g\ge f$ nearby. Then $g-f$ has a local minimum, so its Hessian matrix is positive semidefinite. At the common horizontal tangent plane,

The same contact does not order Gaussian curvature, because determinant is not monotone under addition of a positive-semidefinite matrix when the original Hessian is indefinite or negative definite. For example,satisfy $g\ge f$ and are tangent at zero, but $K_g(0)=1<4=K_f(0)$.

After a rigid motion, write the plane as $z=0$ and the surface locally as $z=h(x,y)$. Along the curve of tangency, $h=0$ and $\nabla h=0$. Differentiating $\nabla h(\gamma (s))=0$ shows that the Hessian annihilates the nonzero tangent ${\gamma }^{\prime }(s)$, so its determinant and therefore the Gaussian curvature vanish.

The graph of $h$ is minimal exactly whenEquivalently, it is the Euler--Lagrange equationfor the area functional $\int \sqrt{1+|\nabla h{|}^2}$.

The flux $F(p)=p/\sqrt{1+|p{|}^2}$ of the minimal surface equation for a graph has a positive definite symmetric matrix derivative. Its eigenvalues are $(1+|p{|}^2{)}^{-3/2}$ in the direction of $p$ and $(1+|p{|}^2{)}^{-1/2}$ orthogonally; at $p=0$ every eigenvalue is one. On $|p|\le M$ it satisfiesThus an averaged linearization of a nonlinear divergence-form equation is uniformly elliptic where the relevant gradients are bounded. Pointwise positivity alone does not supply a uniform bound on an unbounded gradient range.

For a $C^2$ solution $u$ of the minimal surface equation for a graph on the closure of a bounded domain,Write $a^{ij}(Du)u_{ij}=0$ with $a^{ij}(p)={\delta }_{ij}-p_i{p}_j/(1+|p{|}^2)$. Its differentiated operator $\mathcal{L}=a^{ij}{D}_{ij}+b^{\ell }{D}_{\ell }$, where $b^{\ell }=(\partial a^{ij}/\partial p_{\ell })u_{ij}$, satisfiesThe weak maximum principle for elliptic operators gives the result. Interior Schauder estimates justify differentiating a solution initially assumed only $C^2$.

If a smooth minimal surface equation for a graph solution on ${\mathbb{R}}^n\setminus \{0\}$ is homogeneous of degree one, with $n\ge 2$, then its graph extends to a plane through zero. Its gradient is homogeneous of degree zero, so $|Du{|}^2$ attains a global maximum on the unit sphere. The strong maximum principle for elliptic operators makes it constant. The differentiated equation in the gradient maximum principle for a minimal graph then forces $D^{2}u=0$, and degree-one homogeneity removes the affine constant. This argument applies to cones that are graphs of smooth functions away from their vertex; it is not a claim that every minimal surface cone is flat.