The Second Fundamental Form
Let \((\tilde{M},\tilde{g})\) be a Riemannian manifold and \(M\subset \tilde{M}\) be a submanifold. By restricting \(\tilde{g}\) on \(M\) the submanifold is assigned with a natural Riemannian metric. At each point \(p\) we have the orthogonal decomposition with respect to \(\tilde{g}_p\), \[ T_p\tilde{M}=T_pM\oplus T_p^\perp M \] where \(T_p^\perp M\), called the normal space at \(p\), denotes the orthogonal complement of \(T_pM\) in \(T_p\tilde{M}\). Let \(T^\perp M=\cup_{p\in M}T_p^\perp M\). It can be verified that \(T^\perp M\) is a vector bundle on \(M\), called the normal bundle. The idea to study Riemannian submanifolds is straightforward: we differentiate the vector fields along \(M\) and look at its tangent components and normal components respectively. Let \(\tilde{\nabla}\) be the metric connection on \(\tilde{M}\) and \(X\in\Gamma(TM)\) be a tangent vector field on \(M\). On the one hand, for any vector fields \(X,Y\in \Gamma(TM)\), we have \[ \tilde{\nabla}_XY=(\tilde{\nabla}_XY)^\top+(\tilde{\nabla}_XY)^\perp \] The first term is nothing but the covariant derivative of \(Y\) on \(M\). Let \(\nabla\) be the metric connection on \(M\) with respect to the induced Riemannian metric. Then \((\tilde{\nabla}_XY)^\top=\nabla_XY\). Define \[ h(X,Y)=(\tilde{\nabla}_XY)^\perp=\tilde{\nabla}_XY-\nabla_XY \] It can be verified that \(h\) is a normal-bundle-valued symmetric tensor field on \(M\), called the second fundamental form. Equation (3) is called the Gauss formula.
Remark. Given \(\xi\in \Gamma(T^\perp M)\), we can define a tensor field \(\Pi_\xi(X,Y)=\tilde{g}(h(X,Y),\xi)\). In literature \(\Pi\) is also called the second fundamental form.
On the other hand, for a vector field \(\xi\in \Gamma(T^\perp M)\), we have \[ \tilde{\nabla}_X\xi=(\tilde{\nabla}_X\xi)^\top+(\tilde{\nabla}_X\xi)^\perp \] Define the connection \(\nabla^\perp:\Gamma(T^\perp M)\times\Gamma(TM)\to \Gamma(T^\perp M)\) by \((\nabla^\perp)_X\xi=(\tilde{\nabla}_X\xi)^\perp\). It can be verified that \(\nabla^\perp\) is a connection on the normal bundle \(T^\perp M\) which is compatible with the bundle metric, called the normal connection. Define \[ A_\xi(X)=-(\tilde{\nabla}_X\xi)^\top \] The operator \(A_\xi:T_pM\to T_pM\) is called the shape operator or the Weingarten map. The minus sign is chosen so that the following equation holds \[ \tilde{g}(A_\xi(X),Y)=\tilde{g}(h(X,Y),\xi) \] The equation \[ \tilde{\nabla}_X\xi=-A_\xi(X)+\nabla^\perp_X\xi \] is called the Weingarten formula.
Example. Let \(M^{d-1}\subseteq \mathbb{E}^d\) be a hypersurface, \(\xi\) be a unit normal vector field on \(M\). The Gauss map \(g:M\to \mathbb{S}^{d-1}\) is defined by \(g(p)=\xi_p\). For any \(X\in T_pM\), let \(\gamma\) be a smooth curve such that \(\gamma'(0)=X\). Then \[ A_\xi X=-(\xi(\gamma)'(0))^\top=-g(\gamma)'(0)=-g_*(X) \] that is, \(-A_\xi=g_*\) is the differential of Gauss map. It is clear from this example that for hypersurfaces the normal connection is trivial.
Let \(e_1,\cdots,e_m\) be a basis of the tangent space and \(\xi_1,\cdots,\xi_{d-m}\) be a basis of the normal space. Assume \[ A_{\xi_\alpha}e_i=\sum_{j=1}^mA_{\alpha i}^je_j \] Easy computation shows that \[ \begin{aligned} &\Pi_{\xi_\alpha}(e_i,e_j)=A_{\alpha i}^j\\ &h(e_i,e_j)=\sum_{\alpha=1}^{d-m}A_{\alpha i}^j\xi_\alpha \end{aligned} \] Define the mean curvature vector field on \(M\) by \(H=\frac{1}{m}trace(h)\) where \(m\) is the dimension of \(M\). By definition \(H\) is independent of the choice of basis. Under the above notation we have \[ H=\frac{1}{m}\sum_{i=1}^m h(e_i,e_i)=\frac{1}{m}\sum_{i=1}^m\sum_{\alpha=1}^{d-m}A_{\alpha i}^i\xi_\alpha=\sum_{\alpha=1}^{d-m}(\frac{1}{m}\text{trace}(A_{\alpha})\xi_{\alpha})=\sum_{\alpha=1}^{d-m}H^\alpha\xi_\alpha \] where \(H^\xi=\tilde{g}(H,\xi)\) is called the mean curvature along \(\xi\). The value \(\|H\|=(\sum_\alpha\|H^\alpha\|^2)^{1/2}=\frac{1}{m}(\sum_{\alpha=1}^{d-m}\text{trace}(A_\alpha)^2)^{1/2}\) is called the mean curvature. If \(M\) is a hypersurface, \(\|H\|=\frac{1}{m}|\text{trace}(A_\alpha)|\). For a surface in 3-dimensional Euclidean space, this coincides with the absolute mean curvature.
Fundamental Equations
To involve the curvature tensor, we need second order derivative. Thus, we differentiate the Gauss formula and the Weingarten formula. For Gauss formula we have \[ \begin{aligned} \tilde{\nabla}_X\tilde{\nabla}_YZ&=\tilde{\nabla}_X\nabla_YZ+\tilde{\nabla}_Xh(Y,Z)\\ &=\nabla_X\nabla_Y+h(X,\nabla_YZ)-A_{h(Y,Z)}(X)+\nabla^\perp_Xh(Y,Z) \end{aligned} \] If we define the curvature tensor to be \(\tilde{R}(X,Y)=[\tilde{\nabla}_X,\tilde{\nabla}_Y]-\tilde{\nabla}_{[X,Y]}\), we have \[ \begin{aligned} \tilde{R}(X,Y)Z=&R(X,Y)Z+h(X,\nabla_YZ)-h(Y,\nabla_XZ)-h([X,Y],Z)\\ &-A_{h(Y,Z)}(X)+A_{h(X,Z)}(Y)+\nabla^\perp_Xh(Y,Z)-\nabla^\perp_Yh(X,Z) \end{aligned} \] Define the covariant differentiation of \(h\) to be \[ (\nabla_Xh)(Y,Z)=\nabla^\perp_Xh(Y,Z)-h(\nabla_XY,Z)-h(\nabla_XZ,Y) \] Then the tangent component of \(\tilde{R}(X,Y)Z\) is \[ (\tilde{R}(X,Y)Z)^\top=R(X,Y)Z+A_{h(X,Z)}(Y)-A_{h(Y,Z)}(X) \] while the normal component is \[ (\tilde{R}(X,Y)Z)^\perp=(\nabla_Xh)(Y,Z)-(\nabla_Yh)(X,Z) \] Equation (12) is called the Gauss equation and equation (13) is called the Codazzi equation. We can rewrite the Gauss equation as \[ \tilde{R}(X,Y,Z,W)=R(X,Y,Z,W)+\langle h(X,Z),h(Y,W)\rangle-\langle h(Y,Z),h(X,W)\rangle \] Especially the sectional curvature can be expressed as \[ \tilde{K}(X,Y)=K(X,Y)-\langle h(X,X),h(Y,Y)\rangle+\|h(X,Y)\|^2 \] If the ambient space is Euclidean, then \(\overline{R}\) vanishes identically. Let \(e_i,e_j\) span a two-plane \(\pi_{ij}\), then the sectional curvature of \(\pi_{ij}\) is \[ K(\pi_{ij})=-\|h(e_i,e_j)\|^2+\langle h(e_i,e_i),h(e_j,e_j)\rangle=\sum_{\alpha=1}^{d-m}(-(A_{\alpha i}^j)^2+A_{\alpha i}^iA_{\alpha j}^j) \] From \(A_\alpha\) we extract the \(2\times 2\) submatrix \(A_\alpha|_{\pi_{ij}}\) with \(i\)th and \(j\)th row and column. Then \[ K(\pi_{ij})=\sum_{\alpha=1}^{d-m}\det(A_\alpha|_{\pi_{ij}}) \] Example. If \(M\) is a hypersurface of \(\mathbb{E}^d\) with a unit normal vector field \(\xi\), the sectional curvature is \(K(\pi_{ij})=\det(A_\xi|_{\pi_{ij}})\). If \(e_1,\cdots,e_{d-1}\) is a basis that diagonalizes the shape operator with eigenvalues \(\lambda_1,\cdots,\lambda_{d-1}\), then \(K(\pi_{ij})=\lambda_i\lambda_j\). The eigenvalues are called principal curvature and eigenvectors are called principal directions. The determinant of \(A_\xi\) is called the Gauss-Kronecker curvature.
Similarly we can differentiate the Weingarten formula, which yields \[ \begin{aligned} \tilde{\nabla}_X\tilde{\nabla}_Y\xi&=-\tilde{\nabla}_XA_\xi(Y)+\tilde{\nabla}_X\nabla^\perp_Y\xi\\ &=-\nabla_X A_\xi(Y)-h(X,A_\xi(Y))-A_{\nabla^\perp_Y\xi}(X)+\nabla^\perp_X\nabla^\perp_Y\xi \end{aligned} \] Define the curvature tensor on the normal bundle by \(R^\perp(X,Y)\xi=[\nabla^\perp_X,\nabla^\perp_Y]\xi-\nabla^\perp_{[X,Y]}\xi\), and define the covariant differentiation of the shape operator by \((\nabla_XA)_\xi(Y)=\nabla_X(A_\xi Y)-A_{\nabla^\perp_X\xi}(Y)-A_\xi(\nabla_XY)\). We obtain that \[ \begin{aligned} \tilde{R}(X,Y)\xi=&R^\perp(X,Y)\xi-(\nabla_X A)_\xi(Y)+(\nabla_Y A)_\xi(X)\\ &-h(X,A_\xi(Y))+h(Y,A_\xi(X)) \end{aligned} \] The tangent component is \[ (\tilde{R}(X,Y)\xi)^\top=-(\nabla_X A)_\xi(Y)+(\nabla_Y A)_\xi(X) \] while the normal component is \[ (\tilde{R}(X,Y)\xi)^\perp=R^\perp(X,Y)\xi+h(X,A_\xi(Y))-h(Y,A_\xi(X)) \] Equation (19) is called the Ricci equation. Note that equation (18) is equivalent to the Codazzi equation as follows \[ \begin{aligned} \langle\tilde{R}(X,Y)\xi,Z\rangle=&\langle-(\nabla_X A)_\xi(Y),Z\rangle+\langle(\nabla_Y A)_\xi(X),Z\rangle\\ =&\langle\nabla_X(A_\xi Y)-A_{\nabla^\perp_X\xi}(Y)-A_\xi(\nabla_XY),Z\rangle\\ &+\langle\nabla_Y(A_\xi X)-A_{\nabla^\perp_Y\xi}(X)-A_\xi(\nabla_YX),Z\rangle\\ =&-\langle\nabla^\perp_X\xi,h(Y,Z)\rangle-\langle\nabla^\perp_Y\xi,h(X,Z)\rangle-\langle h(\nabla_XY,Z),\xi\rangle\\ &-\langle h(\nabla_YX,Z),\xi\rangle+X\langle h(Y,Z),\xi\rangle-\langle h(\nabla_XZ,Y),\xi \rangle\\ &+Y\langle h(X,Z),\xi\rangle-\langle h(\nabla_YZ,X),\xi\rangle\\ =&\langle\nabla_X^\perp h(Y,Z),\xi\rangle+\langle\nabla_Y^\perp h(X,Z),\xi\rangle-\langle h(\nabla_XY,Z),\xi\rangle\\ &-\langle h(\nabla_YX,Z),\xi\rangle-\langle h(\nabla_XZ,Y),\xi \rangle-\langle h(\nabla_YZ,X),\xi\rangle\\ =&\langle(\nabla_X h)(Y,Z)-(\nabla_Y h)(X,Z),\xi\rangle\\ =&\langle\tilde{R}(X,Y)Z,\xi\rangle \end{aligned} \] The three equations (Gauss, Codazzi, Ricci) are called fundamental equations for a submanifold \(M\hookrightarrow\tilde{M}\).
Examples
Planar Curves
A planar curve is a 1-dimensional manifold embedded in the 2-plane. Let \(\mathbf{t}\) be the tangent vector field and \(\mathbf{n}\) be the normal vector field. We have \[ A_{\mathbf{n}}\mathbf{t}=-(\overline{D}_{\mathbf{n}}\mathbf{t})^\top=\kappa \mathbf{t} \] where \(\kappa\) is the curvature. Thus the estimation can be simply given by \[ \hat{A}=\frac{\Delta\mathbf{n}\cdot\mathbf{t}}{\Delta p\cdot\mathbf{t}}\approx \kappa \]
Space Curves
A space curve is a 1-dimensional manifold embedded in 3-space. Let \(\mathbf{t}\) be the tangent vector field, \(\mathbf{n}\) be the normal vector field, and \(\mathbf{b}\) be the binormal vector field. We have the Frenet formula \[ \frac{d}{ds}\left(\begin{array}{c}\mathbf{t}\\ \mathbf{n}\\ \mathbf{b}\end{array}\right)=\left(\begin{array}{ccc}0&\kappa&0\\ -\kappa&0&\tau\\ 0&-\tau&0\end{array}\right)\left(\begin{array}{c}\mathbf{t}\\ \mathbf{n}\\ \mathbf{b}\end{array}\right) \] where \(\kappa\) is curvature and \(\tau\) is torsion. Let \(\xi=\cos(\theta)\mathbf{n}+\sin(\theta)\mathbf{b}\) be a unit normal vector field. We have \[ A_\xi\mathbf{t}=\cos(\theta)A_\mathbf{n}\mathbf{t}+\sin(\theta)A_\mathbf{b}\mathbf{t}=\cos(\theta)\kappa\mathbf{t} \] Similarly, let \(\xi_\perp=-\sin(\theta)\mathbf{n}+\cos(\theta)\mathbf{b}\) be the unit normal vector perpendicular to \(\xi\). Then \[ A_{\xi_\perp}\mathbf{t}=-\sin(\theta)\kappa\mathbf{t} \] By definition the mean vector field is \(H=\cos(\theta)\kappa\xi-\sin(\theta)\kappa\xi_\perp=\kappa\mathbf{n}\). The mean curvature is \(\|H\|=\kappa\).
Surfaces
A surface is a 2-dimensional manifold embedded in 3-space. The shape operator coincides with the differential of Gauss map (with an additional minus sign). Thus the norm of mean curvature vector \(\|H\|\) is in fact the absolute value of mean curvature for surfaces. The sectional curvature is Gaussian curvature for surfaces.
Clifford Torus
Let \(f:\mathbb{R}^2\to\mathbb{R}^4\) be defined by \[ f(\theta,\phi)=\frac{1}{\sqrt{2}}(\cos(\sqrt{2}\theta),\sin(\sqrt{2}\theta),\cos(\sqrt{2}\phi),\sin(\sqrt{2}\phi)) \] The image of \(f\) is \(\mathbb{S}^1(\sqrt{1/2})\times\mathbb{S}^1(\sqrt{1/2})\), hence a torus. By computation the tangent vector fields are \[ \begin{aligned} & f_\theta = (-\sin(\sqrt{2}\theta),\cos(\sqrt{2}\theta),0,0)\\ & f_\phi = (0,0,-\sin(\sqrt{2}\phi),\cos(\sqrt{2}\phi)) \end{aligned} \] The Riemannian metric is given by \[ g = d\theta^2+d\phi^2 \] which is a flat metric, implying the sectional curvature is identically zero. The normal vector fields are given by \[ \begin{aligned} & \xi = \frac{1}{\sqrt{2}}(-\cos(\sqrt{2}\theta),-\sin(\sqrt{2}\theta),\cos(\sqrt{2}\phi),\sin(\sqrt{2}\phi))\\ & \nu =\frac{1}{\sqrt{2}}(\cos(\sqrt{2}\theta),\sin(\sqrt{2}\theta),\cos(\sqrt{2}\phi),\sin(\sqrt{2}\phi)) \end{aligned} \] By definition the shape operators are given by \[ \begin{aligned} &A_\xi[f_\theta,f_\phi]=[f_\theta,f_\phi]\left[\begin{array}{cc}1&0\\0&-1\end{array}\right]\\ &A_\nu[f_\theta,f_\phi]=[f_\theta,f_\phi]\left[\begin{array}{cc}-1&0\\0&-1\end{array}\right] \end{aligned} \] Therefore, the mean curvature vector field is \(H=-\nu\) and the mean curvature is \(1\).
Rotation Group
Let \(SO(2)\) be the rotation group which consists of matrices in the form \[ \left[\begin{array}{cc}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)& \cos(\theta)\end{array}\right] \] which can be viewed as a curve in the 4-space. Consider the parametrization \[ f(\theta)=(\cos(\theta),\sin(\theta),-\sin(\theta),\cos(\theta)) \] Direct computation shows that the tangent vector field is \[ e_\theta=1/\sqrt{2}(-\sin(\theta),\cos(\theta),-\cos(\theta),-\sin(\theta)) \] and consider the following basis of normal space \[ \begin{aligned} & \xi_1=(\cos(\theta),\sin(\theta),0,0)\\ & \xi_2=(0,0,\sin(\theta),-\cos(\theta))\\ & \xi_3=1/\sqrt{2}(-\sin(\theta),\cos(\theta),\cos(\theta),\sin(\theta)) \end{aligned} \] Then the shape operators are \[ A_{\xi_1}e_\theta=-1/\sqrt{2}e_\theta,A_{\xi_2}e_\theta=1/\sqrt{2}e_\theta,A_{\xi_3}e_\theta=0 \] Hence the mean vector field is \(H(\theta)=-1/\sqrt{2}\xi_1+1/\sqrt{2}\xi_2=-1/\sqrt{2}f(\theta)\).
Ellipsoid
Let \(F:\mathbb{R}^{n+1}\to\mathbb{R}\) be the function \[ F(x^1,\cdots,x^{n+1})=\sum_{i=1}^{n+1}(\frac{x^i}{a^i})^2-1 \] Since \(\nabla F(x)\neq 0\) for all \(x\) such that \(F(x)=0\), the level set \(M=F^{-1}(0)\) is a \(n\)-dimensional submanifold. \(\nabla F/\|\nabla F\|\) serves as the unit normal vector field on \(M\). The mean curvature is given by the following formula \[ H=-\frac{1}{n}div(\frac{\nabla F}{\|\nabla F\|}) \]