The Second Fundamental Form and the Gauss Equations
Let \((M,g)\) be a smooth Riemannian manifold, with Levi–Civita connection \(\nabla\). For an immersed submanifold, the tangent and normal parts of \(\nabla\) give the induced connection and second fundamental form. See the connection notes for the notation.
An immersed submanifold has two notions of differentiation: differentiation using its own metric and differentiation in the ambient manifold. Orthogonal projection relates them. The tangential part of the ambient derivative gives the induced connection, while the normal part measures how the submanifold bends.
After recalling covering metrics, we define this normal contribution and derive the Gauss equation. It expresses the intrinsic curvature in terms of ambient curvature and the second fundamental form.
Definition 1. Let \(M, \widetilde M\) be differentiable manifolds. Then, \(p : \widetilde M \to M\) is a covering map if \(p\) is smooth and surjective, and for all \(m \in M\), there exists a neighbourhood \(U = U_m \subset M\) with \(p^{-1} (U) = \bigsqcup_{i \in I} U_i\), such that \(p|_{U_i}: U_i \to U\) is a diffeomorphism.
A recap of basic facts from algebraic topology can be found in Appendix Covering Spaces and the Fundamental Group of the Circle.
Definition 2. Let \(M, \widetilde M\) be differentiable manifolds. Then, \(p : \widetilde M \to M\) is a Riemann covering map if \(p\) is a smooth covering map, and is additionally a local isometry.
Proposition 3. If \(p:N\to M\) is a smooth covering and \(g\) is a Riemannian metric on the base \(M\), there is a unique metric \(h=p^*g\) on \(N\) making \(p\) a Riemannian covering.
Proof. Set \(h_n(v,w)=g_{p(n)}(dp_n v,dp_n w)\). Smoothness follows in covering charts, and positive definiteness follows because \(dp_n\) is an isomorphism. The local-isometry condition is exactly this formula, so uniqueness follows. An arbitrary metric upstairs need not descend: that requires agreement of these inner products over all sheets. ◻
Proposition 4. Suppose a discrete group \(G\) acts freely and properly on a smooth Riemannian manifold \((N,h)\) by isometries. Then \(M=N/G\) has a unique quotient smooth structure and metric \(g\) for which the projection \(p:N\to M\) is a Riemannian covering. Here properness is for the discrete action; equivalently, only finitely many translates of one compact set can meet another.
Proof. The free proper discrete quotient construction gives smooth covering charts, as proved in the manifold examples. Define for \(n\in p^{-1}(m)\)
If \(n'=f(n)\) for \(f\in G\), differentiating \(p\circ f=p\) gives \(dp_{n'}\circ df_n=dp_n\), hence \((dp_{n'})^{-1}=df_n\circ(dp_n)^{-1}\). The isometry property of \(f\) therefore makes the displayed value independent of the lift \(n\). In a covering chart it is the pullback of \(h\) by a smooth inverse sheet, so it is smooth and positive definite. This formula is forced by local isometry and proves uniqueness. Discreteness is needed: a positive-dimensional free proper group action generally has positive-dimensional fibers and is not a covering. ◻
Example 5 (Flat tori). Let \(e_1,\ldots,e_n\) be a basis of Euclidean \(\mathbb R^n\), and let \(\Gamma=\{\sum_j k_je_j:k_j\in\mathbb Z\}\). Translation by \(\Gamma\) acts freely and properly: a compact difference set intersects the discrete lattice in only finitely many points. The quotient inherits a flat metric by Proposition 4.
Write \(x=\sum_j a_je_j\) with real coefficients. The map \(x\mapsto(e^{2\pi i a_1},\ldots,e^{2\pi i a_n})\) has precisely the lattice cosets as fibers, and induces a continuous bijection \(\mathbb R^n/\Gamma\to(S^1)^n\). The quotient is compact, since it is the image of the closed fundamental parallelepiped \(0\leq a_j\leq1\), so this is a homeomorphism. On a product of open intervals of length one in the \(a_j\), the map is a smooth parametrization of its open image, not a bijection onto the whole torus. Its inverse is a local chart, and chart changes add integer constants.
To see the descended metric directly, for \(y=p(x)\) define
For another lift \(x'=x+\gamma\), the translation \(\tau_\gamma\) satisfies \(dp_{x'}d\tau_\gamma=dp_x\) and is an isometry, proving lift independence. The differential, not a pointwise inverse of the covering map, acts on the vectors here. In the local coefficient coordinates,
Its coefficients are constant, so its Christoffel symbols and curvature vanish. This proves flatness for any lattice basis; if the basis is orthonormal the coefficients are \(\delta_{ij}\). The displayed product-of-circles identification transports this metric; it does not assert equality with the standard unit-circle product metric, which has a \(4\pi^2\) scale factor in these coordinates.
Definition 6. A submanifold \(N\subset M\), with the induced Riemannian metric, is totally geodesic if, for every \(p\in N\) and \(v\in T_pN\), the ambient geodesic with initial data \((p,v)\) remains in \(N\) for all sufficiently small times. Equivalently, each geodesic of \(N\) is an ambient geodesic on its interval of definition. The local condition is necessary when \(N\) is an open part of a larger totally geodesic submanifold.
Proposition 7. Let \((N,h)\) and \((M,g)\) denote differentiable manifolds. Let \(p: N \to M\) be a Riemann covering map. Then, the geodesics of \((M,g)\) are the projection of geodesics of \((N,h)\). On the other hand, geodesics of \((N,h)\) are liftings of geodesics of \((M,g)\).
Proof. A local isometry preserves the Levi–Civita connection by its metric and torsion characterization. Thus on each covering sheet
This proves that projections of geodesics are geodesics. Conversely a base geodesic on an interval has a unique lift after a point of its initial fiber is chosen. On each inverse sheet this lift is smooth and satisfies the geodesic equation, so it is a geodesic throughout its interval. No completeness assumption is needed for this interval-wise statement. ◻
Example 8 (Totally geodesic submanifolds).
Affine subspaces and their open sets are totally geodesic submanifolds of \(\mathbb R^d\).
Great circles of \(\mathbb S^2\).
Totally geodesic submanifolds can have any dimension allowed by the ambient manifold: affine planes in Euclidean space already give examples of dimension greater than one. Indeed, the straight line \(p+tv\), with \(p\) in an affine subspace and \(v\) tangent to it, remains in that subspace; for an open subset this holds for sufficiently small times.
For the unit sphere, a great circle parametrized by \(\gamma(t)=\cos t\,u+\sin t\,v\), with \(u,v\) orthonormal, satisfies \(\gamma''=-\gamma\). Its Euclidean acceleration is normal to the sphere, so its intrinsic acceleration vanishes by Proposition 9. Every circle-tangent initial vector produces a reparametrization of this curve, proving total geodesicity.
1. Second Fundamental Tensor and Form
We now split ambient derivatives into tangent and normal components. This will distinguish the connection of the induced metric from the bending of the immersion. In what follows, let \(N \subset M\) denote a submanifold of \(M\), and \(\langle\cdot,\cdot\rangle_M\) denote the metric on \(M\) (and likewise for other manifolds).
Proposition 9. Let \(\nabla^N\) be the Levi-Civita connection of the metric induced on \(N\subset M\). For tangent vector fields \(X,Y\in\Gamma(TN)\),
where \((\cdot)^\top\) denotes the orthogonal projection from \(T M\) to \(T N\).
The independence of extensions can also be checked in coordinates: if two extensions of \(Y\) differ by \(W\) vanishing along \(N\), then every tangent directional derivative of each coefficient of \(W\) vanishes on \(N\), and the Christoffel terms contain \(W(p)=0\). Thus \(\nabla_X^MW(p)=0\). This works locally on each branch of an immersion; for a general immersion the ambient tangent bundle is understood as pulled back to its domain.
Proof. Put \(D_XY=(\nabla_X^MY)^\top\), extending the fields locally to the ambient manifold. This is independent of the extensions: differentiation in a tangent direction depends only on the restriction of a field to \(N\). Projection preserves the connection rules. Since \([X,Y]\) is tangent to \(N\), the ambient torsion identity gives \(D_XY-D_YX=[X,Y]\). For a third tangent field \(Z\), metric compatibility gives
Thus \(D\) is torsion-free and compatible with the induced metric. Uniqueness of the Levi-Civita connection yields \(D=\nabla^N\). ◻
Remark 10. Let \(\eta(p)\) denote a vector field in a neighbourhood \(p \in N \subset M\), which is orthogonal to \(N\), i.e. \(\langle \eta(p), X\rangle = 0\) for all \(X \in T_p N\). Let \((T_p N)^\perp\) denote the orthogonal complement of \(T_p N\) in \(T_p M\), and the bundle \((TN)^\perp\) (with the usual construction, i.e. whose fibre at \(p\) is \((T_p N)^{\perp}\)) is called the normal bundle. In other words, \(\eta(p) \in (T_p N)^{\perp}\). We will need this to describe the Codazzi equation.
Proposition 11. For tangent \(X\) and normal \(\eta\) along \(N\), the value \((\nabla_X^M\eta)^\top(p)\) depends only on \(X(p)\) and \(\eta(p)\).
Proof. The connection is linear over smooth functions in \(X\). In the other variable,
Expand a normal field in a local frame of the normal bundle. This identity shows that its derivative's tangent projection at \(p\) depends only on the frame coefficients at \(p\), hence only on \(\eta(p)\). ◻
The preceding proposition defines a tensor on tangent and normal vectors. We retain the sign convention \(S(X,\eta)=(\nabla_X^M\eta)^\top\). With this convention, its scalar form is the negative of the pairing of the normal acceleration with \(\eta\).
Definition 12. At \(p\in N\subset M\), the second fundamental tensor in the convention used here is
Definition 13. The second fundamental form of \(N\) with respect to \(M\), in the normal direction \(\eta\), is defined as
Proposition 14. The second fundamental form \(\mathrm{II}_\eta(X,Y)\) is symmetric in \(X,Y\in T_pN\).
Proof. Extend \(X,Y\) as tangent fields and \(\eta\) as a normal field along \(N\). Metric compatibility and the ambient torsion identity give
The bracket term vanishes because \([X,Y]\) is tangent and \(\eta\) is normal. ◻
Remark 15 (Notions of Curvatures). Fix a normal field \(\eta\), and vary only the tangent vector \(X \in T_p N\). Define \(S_\eta : T_p N \to T_p N : X \mapsto S(X,\eta)\). Then, \(S_\eta\) is self-adjoint with respect to the metric. If \(\eta\) is a unit normal field, then the \(n=\dim N\) eigenvalues of \(S_\eta\) are all real, by self-adjointness. These eigenvalues are called the principal curvatures of \(N\) in the direction \(\eta\); the corresponding eigenvectors are called principal curvature vectors. Furthermore,
The mean curvature of \(N\) in the direction \(\eta\) is \(H_\eta \triangleq \frac{1}{n} \mathrm{tr} S_\eta\).
The Gauss-Kronecker curvature of \(N\) in the direction \(\eta\) is \(K_\eta \triangleq \det (S)_\eta\). This is often stated as Gauss curvature, in short. Equivalently, for an orthonormal basis \(e_1,\cdots,e_n \in T_p N\), \(K_\eta = \det(\langle S_\eta e_i,e_j\rangle)_{i,j=1}^n\).
As explored in Theorem 1, for a surface in Euclidean three-space, the sectional curvature of the induced metric equals the product of its two principal curvatures. In a curved ambient manifold the ambient sectional curvature also contributes, as the Gauss equation below shows.
The definitions use \(n\geq1\). Replacing the unit normal by its negative sends \(S_\eta\), every principal curvature, and \(H_\eta\) to their negatives, while \(K_\eta\) acquires \((-1)^n\). Thus for hypersurfaces of dimension two the determinant is normal-independent. In higher codimension a single normal direction's determinant is not the intrinsic sectional curvature.
The second fundamental form also tests whether intrinsic geodesics remain geodesics of the ambient manifold. For an intrinsic geodesic, the tangential acceleration vanishes; its ambient acceleration is therefore the normal contribution.
Theorem 16. Let \(N\subset(M,g)\) be a submanifold with the induced metric \(h\). The following statements are equivalent:
\(N\) is a totally geodesic submanifold of \(M\).
\(h\)-geodesic in \(N\) is a \(g\)-geodesic in \(M\).
All second fundamental forms vanish identically.
Proof. Write \(B(X,Y)=(\nabla_X^MY)^\perp\) for tangent vectors \(X,Y\). The connection rules make \(B\) bilinear, and the torsion identity makes it symmetric. Proposition 9 gives the Gauss formula
If \(N\) is totally geodesic, its geodesic with arbitrary initial tangent vector \(v\) is also an ambient geodesic. Both accelerations vanish, so \(B(v,v)=0\). Symmetry and polarization imply \(B(v,w)=0\) for all tangent \(v,w\). Conversely, if \(B=0\), the displayed formula says that each intrinsic geodesic has zero ambient acceleration. Uniqueness for the ambient geodesic equation gives the local condition in Definition 6.
Finally, metric compatibility gives \(\mathrm{II}_\eta(X,Y)=-\langle B(X,Y),\eta\rangle\) with the sign convention of Definition 13. All these scalar forms vanish exactly when \(B=0\), since a normal vector orthogonal to every normal vector is zero. This proves the equivalence of all three statements. ◻
2. Fundamental Equations
Apply the same tangent-normal decomposition to the curvature operators. The Gauss equation compares the Riemann curvature tensors on \(N\) and \(M\), where \(N\) is a submanifold of \(M\).
Theorem 17 (Gauss equation). Let \(N^n\subset M^m\), \(p\in N\), and let \(\eta_1,\ldots,\eta_{m-n}\) be an orthonormal basis of \((T_pN)^\perp\subset T_pM\). Write \(\mathrm{II}_\alpha=\mathrm{II}_{\eta_\alpha}\), \(S_\alpha=S_{\eta_\alpha}\). For \(X,Y,Z\in T_pN\), with the curvature convention \(R(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z\),
Proof. Extend the tangent vectors and orthonormal normal basis locally along \(N\); no global normal frame is required. From \(\mathrm{II}_\alpha(Y,Z)=-\langle\nabla_Y^MZ,\eta_\alpha\rangle\),
Differentiate with \(\nabla_X^M\) and project tangentially. Derivatives of the scalar coefficients multiply normal vectors and disappear, leaving
The analogous formula with \(X,Y\) interchanged and \((\nabla^M_{[X,Y]}Z)^\top=\nabla^N_{[X,Y]}Z\) give
Rearranging proves (4). This sign is tied to the explicitly stated convention for \(R\) and the retained convention \(S_\eta X=(\nabla_X\eta)^\top\). ◻
Corollary 18 (Metric Gauss equation). Under the hypotheses of Theorem 17, for \(X,Y,Z,W\in T_pN\),
For an orthonormal pair \(X,Y\), setting \(Z=X,W=Y\) gives
In a Euclidean hypersurface and a principal orthonormal pair this is the product of the corresponding principal curvatures.
Proof. Pair (4) with \(W\). Since \(W\) is tangent, the ambient curvature's normal projection pairs to zero, and \(\langle S_\alpha V,W\rangle=\mathrm{II}_\alpha(V,W)\). For the sectional formula, use \(\mathrm{II}_\alpha(U,V)=-\langle B(U,V),\eta_\alpha\rangle\) and sum over the orthonormal normal basis. The two minus signs in each product cancel. ◻
To clarify, many texts state Corollary 18 as the Gauss equation, i.e. the form expressed in terms of Riemann curvature tensor.
The promised Codazzi equation uses the normal connection \(\nabla_X^\perp\eta=(\nabla_X^M\eta)^\perp\). Define
With the same reversed curvature convention, its form is
Indeed the normal part of \(\nabla_Y^M\nabla_X^MZ\) is \(B(Y,\nabla_X^NZ)+\nabla_Y^\perp(B(X,Z))\). Subtract the analogous term in the other order and add \(B([X,Y],Z)\); inserting \([X,Y]=\nabla_X^NY-\nabla_Y^NX\) gives exactly the displayed covariant derivatives. In Euclidean ambient space the left side vanishes, so the covariant derivative of \(B\) is symmetric in its first two arguments.
Reading: Carmo (1992), Riemannian geometry.