An embedded submanifold inherits an intrinsic metric, but its bending in the ambient manifold is captured by the second fundamental form. The Gauss, Codazzi, and Ricci equations relate intrinsic curvature to this extrinsic data.
Tangential and normal decomposition
Let \(i:M^m\hookrightarrow(\overline M^{m+k},\overline g)\) be an isometric immersion. Along \(M\), \[ T\overline M|_M=TM\oplus NM. \] For tangent fields \(X,Y\), decompose the ambient derivative: \[ \overline\nabla_XY=\nabla_XY+\mathrm{II}(X,Y), \] where \(\nabla_XY\) is tangential and \(\mathrm{II}(X,Y)\in NM\) is the second fundamental form.
Proposition. The tangential component is the Levi-Civita connection of the induced metric, and \(\mathrm{II}\) is a symmetric \(NM\)-valued bilinear form.
Proof. Orthogonal projection preserves the connection rules. Ambient metric compatibility descends to the tangential component. Ambient torsion-freeness gives \[ 0=\overline\nabla_XY-\overline\nabla_YX-[X,Y]. \] Separating tangential and normal components proves both that \(\nabla\) is torsion free and that \(\mathrm{II}(X,Y)=\mathrm{II}(Y,X)\).
Shape operator and mean curvature
For a normal field \(\nu\), the Weingarten equation is \[ \overline\nabla_X\nu=-A_\nu X+\nabla_X^\perp\nu, \] and \[ g(A_\nu X,Y)=\overline g(\mathrm{II}(X,Y),\nu). \] For a hypersurface with unit normal \(\nu\), eigenvalues of \(A=A_\nu\) are the principal curvatures. The mean-curvature vector is \[ H=\frac1m\operatorname{tr}_g\mathrm{II}. \] The immersion is minimal exactly when \(H=0\).
Example (round sphere). For \(S^m(r)\subset\mathbb R^{m+1}\) with outward normal \(\nu=x/r\), \[ \overline\nabla_X\nu=\frac1rX. \] With the convention above, \(A=-r^{-1}I\) and \(\mathrm{II}(X,Y)=-r^{-1}g(X,Y)\nu\). Reversing the normal reverses these signs but not the intrinsic curvature.
Gauss and Codazzi equations
Theorem (Gauss equation). \[ \begin{aligned} g(R(X,Y)Z,W) ={}&\overline g(\overline R(X,Y)Z,W)\\ &+\overline g(\mathrm{II}(X,W),\mathrm{II}(Y,Z)) -\overline g(\mathrm{II}(X,Z),\mathrm{II}(Y,W)). \end{aligned} \]
Proof. Expand \(\overline R(X,Y)Z\) using \(\overline\nabla_XY=\nabla_XY+\mathrm{II}(X,Y)\), then take its tangential component and pair with \(W\). Derivatives of \(\mathrm{II}\) contribute only normal components; the tangential parts of differentiating a normal field are converted to shape operators by the Weingarten equation.
For a surface in \(\mathbb R^3\), the ambient curvature vanishes and the Gauss equation gives \[ K=\det A=k_1k_2. \] Thus Gaussian curvature, apparently defined through bending in space, is intrinsic. This is Gauss's Theorema Egregium.
The Codazzi equation controls covariant derivatives of \(\mathrm{II}\): \[ (\nabla_X\mathrm{II})(Y,Z)-(\nabla_Y\mathrm{II})(X,Z) =(\overline R(X,Y)Z)^\perp. \] Together with the normal Ricci equation, these are the compatibility conditions for reconstructing an immersion from first and second fundamental forms.
First variation of volume
Theorem. For a compactly supported normal variation with field \(V\), \[ \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_0 \operatorname{Vol}(M_t) =-m\int_M\overline g(H,V)\,\mathrm{d}V_g. \] Hence minimal submanifolds are precisely critical points of volume.