Global Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity

2.1 Notations: Manifolds and Vector Bundles

Throughout this paper, we denote (M, g) as an n-dimensional Riemannian manifold. By definition, M is a second-countable, Hausdorff topological space with an atlas of local charts \(\mathcal {A}=\{(U_\alpha , \phi _\alpha )\,: \,\alpha \in \mathcal {I}\}\) such that each \(U_\alpha \subset M\) is an open subset, \(\phi _\alpha : U_\alpha \mapsto \phi _\alpha (U_\alpha ) \subset \mathbb {R}^n\) is a homeomorphism, and the transition functions \(\phi _\alpha \circ \phi _\beta ^{-1}\) between the overlapping charts \(U_\alpha \) and \(U_\beta \) have required regularity.

Given a manifold M, we say that (E, M, F) is a vector bundle of degree \(k\in \mathbb {N}\) over M if there is a surjection \(\pi : E \mapsto M\) such that, for any \(P \in M\), there exists a local neighborhood \(U \subset M\) containing P so that there is a diffeomorphism \(\psi _U:\pi ^{-1}(U) \mapsto U \times F\) with \(\mathrm{pr}_1\circ \psi _U = \pi \) on \(\pi ^{-1}(U)\), where map \(\mathrm{pr}_1\) is the projection map onto the first coordinate, E and F are also differentiable manifolds, and \(F \simeq \mathbb {R}^k\) (vector space isomorphism). In this bundle, E is called the total space, F is the fiber, M is the base manifold, \(\pi \) is the projection of the bundle, and \(\psi _U\) is termed as a local trivialization. For simplicity, we also say that E is a vector bundle over M.

If \(E_1\) and \(E_2\) are both vector bundles over M, we can take the direct sum and the quotient of the bundles, by taking the vector space direct sum and quotient of the fibers. Also, for a vector bundle \((E,M,F=\mathbb {R}^k)\) with projection \(\pi \), the space of smooth sections is defined by \(\Gamma (E):=\{s\in C^\infty (M;E)\,: \,\pi \circ s=\mathrm{id}_M\}\). We can define the affine connection \(\nabla ^E: \Gamma (TM)\times \Gamma (E) \mapsto \Gamma (E)\), by linearity and the Leibniz rule. For our purpose, \(\nabla ^E\) is required to restrict to the Levi–Civita connection on M.

As a primary example, consider \(E=TM\), the tangent bundle over M. Then \(\pi \) is the projection onto the base point in M, and \(\Gamma (TM)\) is the space of smooth vector fields, agreeing with the previous notation. Moreover, \(\nabla ^{TM}\) is precisely the Levi–Civita connection. Another example is the cotangent bundle \(T^*M\) over M, whose fibers are the dual vector spaces of the fibers of TM.

Our next example is crucial to this paper. Consider \(E_1=T^*M\otimes T^*M\otimes \cdots \otimes T^*M\), the tensor product of q-copies of \(T^*M\), for \(q=0,1,2,\ldots \). This is the (covariant) q-tensor algebra over M, which can be viewed as q-linear maps on TM. Now let \(E_2\subset E_1\) be the subspace of all the alternating q-tensors on TM, i.e., the q-linear maps that change sign when switching any pair of its indices \(\{i,j\}\subset \{1,\ldots ,q\}\). By convention, we write \(E_2=:\bigwedge ^q T^*M\), known as the alternating q-algebra. Moreover, the sections are \(\Omega ^q(M):=\Gamma (\bigwedge ^q T^*M)\), known as the differential q-forms. A general element \(\alpha \in \Omega ^q(M)\) is written as a linear combination of alternating forms \(\xi _1 \wedge \cdots \wedge \xi _q\), where \(\{\xi _j\}_{j=1}^q\) is a q-tuple of linearly independent differential 1-form on M. If \(\dim (M)=n\), \(\Omega ^q(M)= \{0\}\) for \(q \ge n+1\). Thus, we always restrict to \(0 \le q \le n\).

There is a natural isomorphism between TM and \(T^*M\), by identifying canonically each fiber of TM with its dual. It induces a canonical isomorphism \(\sharp : \Omega ^1(M) \mapsto \Gamma (TM)\), for which we write \(\sharp ^{-1}=:\flat \). Note that, if a vector field and its corresponding 1-form in local coordinates are written by the Einstein summation convention, \(\sharp \) (or \(\flat \)) amounts to raising (or lowering) the indices of the coefficients. Clearly, they extend to the isomorphisms between \(T^*M \otimes \cdots \otimes T^*M\) (i.e., covariant tensors) and \(TM \otimes \cdots \otimes TM\) (i.e., contravariant tensors).

For instance, consider the covariant derivative \(\nabla : \Omega ^q(M)\mapsto \Omega ^{q+1}(M)\) defined above. For \(q=1\) and \(\alpha \in \Omega ^1(M)\), we set \(X:=\alpha ^\sharp \in \Gamma (TM)\). Since \(\nabla _YX \in \Gamma (TM)\) for any \(Y \in \Gamma (TM)\), we can view \(\nabla X = \nabla \alpha ^\sharp \in \Gamma (TM \otimes TM)\), i.e., \((\nabla \alpha ^\sharp )^\flat \in \Omega ^2(M)\). This example shows that, via the identifications \(\sharp \) and \(\flat \), the covariant derivative \(\nabla \) on \(\Omega ^q(M)\) generalizes the definition of the Levi–Civita connection.

We now define the Sobolev spaces \(W^{k,p}(M; \bigwedge ^q T^*M)\) on an n-dimensional Riemannian manifold (M, g), which generalize \(W^{k,p}(\mathbb {R}^n)\) for \(k\in \mathbb {Z}\) and \(1\le p \le \infty \). First, for differential q-forms \(\alpha ,\beta \in \Omega ^q(M)\), g on M defines an inner product \(g(\alpha ,\beta ) =\langle \alpha ,\beta \rangle \) by

$$\begin{aligned} \langle \alpha ,\beta \rangle \, dV_g := \alpha \wedge *\beta , \end{aligned}$$

(2.3)

which is an equality of n-forms on M. Then, for alternating contravariant q-tensor fields T and S, set \(\langle T,S\rangle := \langle T^\flat , S^\flat \rangle \), which is consistent with the previous notations. Next, the \(L^p\)-norm of \(\alpha \in \Omega ^q(M)\) is defined as

$$\begin{aligned} \Vert \alpha \Vert _{L^p} := \Big (\int _M \big [*(\alpha \wedge *\alpha )\big ]^{\frac{p}{2}} \,\mathrm{d}V_g\Big )^{\frac{1}{p}} = \Big (\int _M \langle \alpha ,\alpha \rangle ^{\frac{p}{2}}\,\mathrm{d}V_g\Big )^{\frac{1}{p}}. \end{aligned}$$

(2.4)

Moreover, for \(\alpha \in \Omega ^q(M)\), set

$$\begin{aligned} \Vert \alpha \Vert _{W^{k,p}}:=\left( \sum _{j=0}^k \Vert \nabla ^j \alpha \Vert ^p_{L^p} \right) ^{\frac{1}{p}} = \left( \sum _{j=0}^k \Vert \underbrace{\nabla \circ \cdots \circ \nabla }_{j \text { times}} \alpha \Vert ^p_{L^p} \right) ^{\frac{1}{p}}. \end{aligned}$$

(2.5)

We denote by \(W^{k,p}(M; \bigwedge ^qT^*M)\) the completion of the space of compactly supported q-forms with respect to \(\Vert \cdot \Vert _{W^{k,p}}\). Notice that, for any contravariant tensor field X, the following holds:

$$\begin{aligned} \Vert X\Vert _{W^{k,p}}:=\Vert X^\flat \Vert _{W^{k,p}}. \end{aligned}$$

In fact, \(W^{k,p}(M;E)\) can be defined for an arbitrary bundle E. Furthermore, we can define the \(W^{k,p}\) connection \(\nabla ^E\) on bundle E. This can be done since the moduli space of connections is an affine space modeled over the tensor algebra Open image in new window . We refer the reader to Jost [26] for the detailed construction. A key feature for this definition lies in its intrinsic nature, since the local coordinates on M are not needed to define \(W^{k,p}(M;\bigwedge ^q T^*M)\). In particular, when \(p=2\), we denote \(H^{k}(M;\bigwedge ^q T^*M):=W^{k,2}(M;\bigwedge ^q T^*M)\) which are Hilbert spaces.

2.2 Isometric Immersions

On the other hand, some progress has been made on the existence and regularity of immersions/embeddings of two-dimensional Riemannian manifolds \(M^2\) into \(\mathbb {R}^3\), with the minimal target dimension 3, the Janet dimension. In this setting, the problem can be reduced to a fully nonlinear Monge–Ampère equation, whose type is determined by the Gauss curvature K of M. For \(K>0, K=0\), and \(K<0\), the corresponding equation is elliptic, parabolic, and hyperbolic, respectively. The case of \(K>0\) has a solution in the large due to Nirenberg [39], while the other two cases are more delicate, and are still widely open in the general setting; see Han–Hong [24].

2.3 The Gauss–Codazzi–Ricci Equations

The isometric immersion problem can also be approached via the GCR equations as the compatibility conditions, instead of directly tackling Eq. (2.7) (cf. do Carmo [13]).

The GCR equations are derived from the orthogonal splitting of the tangent spaces along the isometric immersion \(f: (M,g) \hookrightarrow (\mathbb {R}^{n+k}, g_0)\). Indeed, the tangent spaces satisfy \(T_P\mathbb {R}^{n+k}\cong \mathbb {R}^{n+k}\cong T_PM \bigoplus T_PM^\perp \) for each \(P \in M\), where \(T_PM^\perp \) is the fiber of the normal bundle \(TM^\perp \) at point P, and \(TM^\perp \) is defined as the quotient vector bundle \(T\mathbb {R}^{n+k}/TM\). Here and hereafter, we obey the widely adopted convention to identify TM with T(fM), that is, we view f as the inclusion map.

From now on, we will always use Latin letters \(X,Y,Z,\ldots \) to denote tangential vector fields, i.e., elements in \(\Gamma (TM)\), and Greek letters \(\xi ,\eta ,\zeta ,\ldots \) to denote normal vector fields, i.e., elements in \(\Gamma (TM^\perp )\).

In Theorem 2.1, we have expressed the GCR equations in the most compact form. Nevertheless, to analyze the weak rigidity, it is helpful to rewrite the Codazzi and Ricci equations in a less concise manner.

In the GCR equations in the form of either Theorem 2.1 or Theorem 2.2, we view \((B, \nabla ^\perp )\) as unknowns and g (hence R) as given. The GCR equations constitute a necessary condition for the existence of isometric immersions. Tenenblat [47] proved that, if everything is smooth, this is also sufficient for the local existence of isometric immersions.

From the point of view of PDEs, the three equations in Theorem 2.2 form a system of first-order nonlinear equations. The left-hand sides of Eqs. (2.19)–(2.20) can be regarded as the principal parts, while the nonlinear terms on the right-hand sides are of zero-th order. The nonlinear terms are quadratic in the form of \(B\otimes B\), \(\nabla ^\perp \otimes \nabla ^\perp \), and \(B\otimes \nabla ^\perp \).

3.1 General Compensated Compactness Theorem on Banach Spaces

Throughout this section, for a normed vector space X with dual space \(X^*\), we use \(\langle \cdot ,\cdot \rangle _X\) to denote the duality pairing of \((X,X^*)\). Let \(\mathcal {H}\) be a Hilbert space over field \(\mathbb {K}=\mathbb {R}\text { or } \mathbb {C}\) such that \(\mathcal {H}=\mathcal {H}^*\). Let Y and Z be two Banach spaces over \(\mathbb {K}\) with their dual spaces \(Y^*\) and \(Z^*\), respectively. In what follows, we consider two bounded linear operators \(S:\mathcal {H}\mapsto Y\), \(T:\mathcal {H}\mapsto Z\), and their adjoint operators \(S^\dagger : Y^*\mapsto \mathcal {H}\) and \(T^\dagger : Z^* \mapsto \mathcal {H}\), respectively.

Furthermore, the following conventional notations are adopted: For any normed vector spaces X, \(X_1\), and \(X_2\), we write \(\{s^\epsilon \}\subset X\) for a sequence \(\{s^\epsilon \}\) in X as a subset, and \(X_1 \Subset X_2\) for a compact embedding between the normed vector spaces. We use \(\Vert \cdot \Vert _X\) to denote the norm in X. Also, we use \(\rightarrow \) to denote the strong convergence of sequences and \(\rightharpoonup \) for the weak convergence. Furthermore, we denote the closed unit ball of space X by \(\bar{B}_X:=\{x\in X: \Vert x\Vert \le 1\}\), the open unit ball by \(B_X:=\{x\in X: \Vert x\Vert <1\}\). Moreover, for a linear operator \(L: X_1\mapsto X_2\), its kernel is written as \(\ker (L)\subset X_1\), and its range is denoted by \(\text {ran}(L)\subset X_2\). Finally, for \(X_1\subset X\) as a vector subspace, its annihilator is defined as \(X_1^\perp :=\{f\in X^*:f(x)=0 \text { for all } x\in X_1\}\).

In the setting above, we are concerned with the following question:

The goal of this subsection is to provide a sufficient condition for the convergence \(\langle u^\epsilon , v^\epsilon \rangle _\mathcal {H}\rightarrow \langle \bar{u},\bar{v}\rangle _\mathcal {H}\) as \(\epsilon \rightarrow 0\). Roughly speaking, it requires the existence of a “nice” pair of bounded linear operators \(S:\mathcal {H}\mapsto Y\) and \(T:\mathcal {H}\mapsto Z\) such that S and T are orthogonal to each other, and \(S\oplus T\) gains certain compactness/regularity. More precisely, we prove

Proof

We divide the proof into eight steps.

1. In order to show that Open image in new window has a finite-dimensional kernel, we consider the following subset of \(\mathcal {H}\):

Open image in new window

where Open image in new window is a compact embedding between the Hilbert spaces.

Suppose that Open image in new window is compact. First notice that j(E) is the closed unit ball of \(j[\ker (S\oplus T)]\), which is a Banach space, due to the closedness of the kernel. It then follows from the classical Riesz lemma that \(j[\ker (S\oplus T)]\) is finite-dimensional. Since j is an embedding, we can conclude that \(\dim \ker (S\oplus T)=\dim (j[\ker (S\oplus T)])<\infty \).

To prove the compactness of j(E), take any \(h\in E\) and consider the following estimate deduced from condition (Op 2):

Open image in new window

(3.3)

Hence, the boundedness of E in \(\mathcal {H}\) and the compactness of Open image in new window imply that j(E) is compact. Thus, the first step is complete.

2. We now show that \(S\oplus T\) is a closed-ranged operator, i.e., \(\text {ran}(S\oplus T)\subset Y\bigoplus Z\) is a closed subspace. To this end, we first prove the existence of a constant \(\epsilon _0>0\) such that, for all \(h\in \mathcal {H}\),

Open image in new window

(3.4)

In fact, if the inequality were false, then, for any \(\mu \in (0,1)\), we could find \(\{h^\mu \}\subset [\ker (S\oplus T)]^\perp \) and Open image in new window such that

$$\begin{aligned} \Vert (S\oplus T)h^\mu \Vert _{Y\bigoplus Z}\le \mu \end{aligned}$$

and

$$\begin{aligned} \mathrm{dist}\big (h^\mu , \ker (S\oplus T)\big )\ge \hat{\epsilon }_0 \end{aligned}$$

for some \(\hat{\epsilon }_0>0\). Such a choice of \(\{h^\mu \}\) is possible, owing to the finite-dimensionality of \(S\oplus T\). Now, plugging \(\{h^\mu \}\) into condition (Op 2), we have

Open image in new window

It follows from the compactness of j that \(\{j(h^\mu )\}\) is pre-compact in Open image in new window . Let j(h) be a limit point of \(\{j(h^\mu )\}\) in Open image in new window . Then one must have \(h\in \ker (S\oplus T)\), in view of \(\Vert (S\oplus T)h^\mu \Vert _{Y\bigoplus Z}\le \mu \). However, this contradicts the fact that \(\{h^\mu \}\) were chosen to have a distance at least \(\hat{\epsilon }_0\) from \(\ker (S\oplus T)\). Therefore, we have established estimate (3.4).

To proceed, consider a sequence \(\{h^\mu \}\subset \mathcal {H}\) such that \((S\oplus T)h^\mu \rightarrow w\) for some \(w\in Y\bigoplus Z\). Our goal is to show that \(w\in \text {ran}(S\oplus T)\). Indeed, by the projection theorem for Hilbert spaces, we can decompose \(h^\mu =k^\mu +r^\mu \) with \(k^\mu \in \ker (S\oplus T)\) and \(r^\mu =[\ker (S\oplus T)]^\perp \) (which is a closed subspace of \(\mathcal {H}\)). Then estimate (3.4) gives

Open image in new window

(3.5)

As a consequence, \(\{j(r^\mu )\}\) is a Cauchy sequence in \(j(\mathcal {H})\), which implies that there exists \(j(r)\in j(\mathcal {H})\) such that \(j(r^\mu )\rightarrow j(r)\) in \(j(\mathcal {H})\) as \(\mu \rightarrow 0\). Since j is an embedding, it follows that \(r^\mu \rightarrow r\) in \(\mathcal {H}\). Therefore, by the closed graph theorem of Banach spaces, \((S\oplus T)h^\mu =(S\oplus T)r^\mu \rightarrow (S\oplus T)r\). It now follows that \(w=(S\oplus T)r\) so that the range of \(S\oplus T\) is closed.

3. As an immediate corollary, we can obtain the following decomposition of \(\mathcal {H}\):

$$\begin{aligned} \mathcal {H}= \ker (S\oplus T) \bigoplus \text {ran}(S^\dagger \vee T^\dagger ). \end{aligned}$$

(3.6)

Here, \(\bigoplus \) denotes the topological direct sum of Banach spaces, and the direct summands are orthogonal as Hilbert spaces.

Indeed, by the projection theorem, \(\mathcal {H}=\ker (S\oplus T)\bigoplus [\ker (S\oplus T)]^\perp \). On the other hand,

$$\begin{aligned}{}[\ker (S\oplus T)]^\perp =\overline{\text {ran}[(S\oplus T)^\dagger ]}=\overline{\text {ran}(S^\dagger \vee T^\dagger )}. \end{aligned}$$

We recall the closed range theorem in Banach spaces, which states that a bounded linear operator is closed-ranged if and only if its adjoint operator is closed-ranged (cf. [20]). Hence, we may deduce from the preceding equalities that

$$\begin{aligned}{}[\ker (S\oplus T)]^\perp =\text {ran}(S^\dagger \vee T^\dagger ). \end{aligned}$$

The decomposition in Eq. (3.6) now follows immediately.

With this decomposition, it now suffices to prove Theorem 3.1 for surjective operators S and T. Indeed, all the conditions in (Op 1)–(Op 2) and (Seq 1)–(Seq 2) continue to hold when \(Y\bigoplus Z\) is replaced by \(\text {ran}(S\oplus T)\), which has been proved to be a closed subspace of \(Y\bigoplus Z\). Here we have used the fact that closed subspaces of reflexive Banach spaces are still reflexive. Therefore, in the sequel, we always assume \(\text {ran}(S\oplus T)=Y\bigoplus Z\) without loss of generality.

4. Now, we introduce the following operator Open image in new window :

Open image in new window

(3.7)

For this generalized Laplacian, we also prove that it has a finite-dimensional kernel, and its range is closed (in fact, surjective, in view of the reduction at the end of Step 3).

Indeed, we can find the range of Open image in new window as follows:

Open image in new window

(3.8)

where the decomposition in Eq. (3.6) has been used in the second equality.

On the other hand, notice that

Open image in new window

(3.9)

In this expression, the first direct summand equals \(\ker (S\oplus T)\), by a similar argument as in Eq. (3.8). For the second summand, a standard result in functional analysis gives \(\ker (S^\dagger \vee T^\dagger ) = [\text {ran}(S\oplus T)]^\perp \), which is assumed to be \(\{0\}\) at the end of Step 3. It follows that Open image in new window , which is finite-dimensional, by Step 1.

To summarize, Open image in new window is a closed-ranged operator with a finite-dimensional kernel; without loss of generality, we may assume Open image in new window to be surjective.

5. We now show a crucial result concerning Open image in new window :

Open image in new window

First, applying the open mapping theorem in Banach spaces to operator Open image in new window , we can find a constant \(\delta >0\) and an element Open image in new window such that Open image in new window .

From this inclusion, we can prove

Open image in new window

(3.10)

In fact, for any \(v_0 \in \delta \bar{B}_{Y\bigoplus Z}\), we can write \(v_0\) as a convex combination of elements in \(w_0+\delta B_{Y\bigoplus Z}\), e.g., \(v_0=\frac{1}{2}\big ((v_0+w_0)+(v_0-w_0)\big )\). Observe that Open image in new window is a convex set, as Open image in new window is a bounded linear operator and \(B_{Y^*\bigoplus Z^*}\) is convex. Since \(w_0+\delta B_{Y\bigoplus Z}\) lies in Open image in new window , we conclude that Open image in new window . Thus, (3.10) now follows.

As a consequence, given any Open image in new window , there exists \(\eta \in \bar{B}_{Y^*\bigoplus Z^*}\) such that Open image in new window . Define \(\xi :=\frac{\Vert w\Vert }{\delta }\eta \), which yields

Open image in new window

Now, the proof for \((\clubsuit )\) is completed by choosing \(M=\delta ^{-1}\).

6. Now, we employ (3.6) to decompose sequences \(\{u^\epsilon \}\) and \(\{v^\epsilon \}\), and the weak limits \(\bar{u}\) and \(\bar{v}\). In the sequel, we denote the canonical projection of \(\mathcal {H}\) onto the first factor by \(\pi _1:\mathcal {H}=\ker (S\oplus T)\bigoplus \text {ran}(S^\dagger \vee T^\dagger ) \mapsto \ker (S\oplus T)\). By Step 1, \(\pi _1\) is a finite-rank (hence compact) operator.

Employing such a decomposition, we can write

Open image in new window

(3.11)

for some \(a,\tilde{a}, a^\epsilon ,\tilde{a}^\epsilon \in Y^*\) and \(b,\tilde{b}, b^\epsilon , \tilde{b}^\epsilon \in Z^*\). Moreover, applying the orthogonality condition (Op 1), we have

Open image in new window

Since \(\{u^\epsilon \}\) and \(\{v^\epsilon \}\) are weakly convergent by assumption (Seq 1), and \(\pi _1\) is a compact operator, we obtain that \(\langle \pi _1u^\epsilon , \pi _1v^\epsilon \rangle \rightarrow \langle \pi _1\bar{u}, \pi _1\bar{v}\rangle \). It thus remains to establish

Open image in new window

(3.12)

In the next two steps, we prove (3.12).

7. The starting point is to observe that the left-hand side of (3.12) can be rewritten as follows:

Open image in new window

(3.13)

On the other hand, let us apply S to \(u^\epsilon \) and T to \(v^\epsilon \) in (3.11). Using the definition of \(\pi _1\) and the orthogonality condition (Op 1), we immediately find that

Open image in new window

(3.14)

i.e., Open image in new window . As \(\{Su^\epsilon \}\subset Y\) and \(\{Tv^\epsilon \}\subset Z\) are pre-compact by (Seq 2), it suffices to show the boundedness of \(\{(\tilde{a}^\epsilon , b^\epsilon )\}\subset Y^*\bigoplus Z^*\) to conclude (3.12). To see this point, assuming the boundedness, one can deduce that

Open image in new window

For \(\text {I}^\epsilon \), since \(\Vert (\tilde{a}^\epsilon ,b^\epsilon )\Vert _{Y^*\bigoplus Z^*}\le M\), we can use the pre-compactness of \(\{Su^\epsilon \}\subset Y\) and \(\{Tv^\epsilon \}\subset Z\) in assumption (Seq 1) to conclude that

$$\begin{aligned} \text {I}^\epsilon \le M \big \Vert \big (S(u^\epsilon -\bar{u}), T(v^\epsilon - \bar{v})\big ) \big \Vert _{Y\bigoplus Z} \rightarrow 0. \end{aligned}$$

For \(\text {II}^\epsilon \), we need to invoke the reflexivity of Y and Z. As a Banach space is reflexive if and only if its dual space is reflexive, we know that \(Y^*\bigoplus Z^*\) is reflexive. Then the bounded sequence \(\{(\tilde{a}^\epsilon , b^\epsilon )\}\) is weakly pre-compact, by Theorem 3.31 in [20]. Moreover, Theorem 4.47 (Eberlein-S̆mulian theorem) in [20] implies the weak convergence of \(\{(\tilde{a}^\epsilon ,b^\epsilon )\}\). Thus, viewing Open image in new window as an element in \((Y\bigoplus Z)^{**}\), we conclude that \(\text {II}^\epsilon \rightarrow 0\).

8. From the above arguments, it remains to prove the boundedness of \(\{(\tilde{a}^\epsilon , b^\epsilon )\}\subset Y^*\bigoplus Z^*\). We further remark that \((\tilde{a}^\epsilon , b^\epsilon )\) can be chosen modulo Open image in new window . More precisely, it is enough to exhibit one particular representative \((\tilde{a}^\epsilon , b^\epsilon )\) in Open image in new window such that \(\Vert (\tilde{a}^\epsilon , b^\epsilon )\Vert _{Y^*\bigoplus Z^*}\le C\), where \(C<\infty \) is independent of \(\epsilon \). To see this, notice that

Open image in new window

analogous to Eqs. (3.13)–(3.14). Therefore, we have the freedom to choose a representative \((\tilde{a}^\epsilon ,b^\epsilon )\) in the co-set Open image in new window , without changing the expression on the left-hand side of (3.12).

For this purpose, let us invoke the key estimate \((\clubsuit )\) proved in Step 5 to find a constant \(M\in (0, \infty )\) satisfying

$$\begin{aligned} \Vert (\tilde{a}^\epsilon , b^\epsilon )\Vert _{Y^*\bigoplus Z^*} \le M\Vert (Sv^\epsilon , Tu^\epsilon )\Vert _{Y\bigoplus Z}. \end{aligned}$$

Then it suffices to prove the uniform boundedness of \(\{(Sv^\epsilon , Tu^\epsilon )\}\) in \(Y\bigoplus Z\).

Indeed, as \(u^\epsilon \rightharpoonup \bar{u}\) and \(v^\epsilon \rightharpoonup \bar{v}\) according to assumption (Seq 1), we know that \(\{u^\epsilon \}\) and \(\{v^\epsilon \}\) are bounded in \(\mathcal {H}\), by using the weak lower semi-continuity of \(\Vert \cdot \Vert _\mathcal {H}\). By the reflexivity of Hilbert spaces (due to the Riesz representation theorem), \(\{u^\epsilon \}\) and \(\{v^\epsilon \}\) are weakly pre-compact in \(\mathcal {H}\). Next, by standard results in functional analysis, the continuous linear operator \(S\oplus T:\mathcal {H}\mapsto Y\bigoplus Z\) with respect to the strong topologies is also continuous when both \(\mathcal {H}\) and \(Y\bigoplus Z\) are endowed with the weak topologies (see [20] for details). As continuous mappings take pre-compact sets to pre-compact sets, \(\{(Sv^\epsilon , Tu^\epsilon )\}\) is weakly pre-compact in \(Y\bigoplus Z\); equivalently, we have established the uniform boundedness of \(\{(Sv^\epsilon , Tu^\epsilon )\}\) in the strong topology of \(Y\bigoplus Z\), because the Banach spaces Y and Z are reflexive.

Therefore, we have proved that

$$\begin{aligned} \Vert (\tilde{a}^\epsilon , b^\epsilon )\Vert _{Y^*\bigoplus Z^*} \le M \sup _{\epsilon >0}\Vert (Sv^\epsilon , Tu^\epsilon )\Vert _{Y\bigoplus Z}\le M'<\infty , \end{aligned}$$

(3.15)

from which the convergence in (3.12) follows. This completes the proof. \(\square \)

Let us close this subsection with two remarks on Theorem 3.1. First, in terms of the applications, Y and Z are usually Hilbert spaces \(H^s=W^{s,2}\) for \(s\in \mathbb {Z}\), or S and T are known to be Fredholm operators. In such cases, our arguments can be essentially simplified. Second, the reflexivity of Y and Z is necessary for Theorem 3.1 to hold.

Remark 3.1

If Y and Z are Hilbert spaces, then Open image in new window is self-adjoint: Open image in new window . Thus, we can decompose

Open image in new window

(3.16)

(cf. Proposition 7.32 in [20]) so that the closed-rangedness of Open image in new window and the crucial estimate \((\clubsuit )\) are automatically verified. As a consequence, after establishing the finite-dimensionality of Open image in new window as in the first half of Step 1, we can proceed to Step 5 in the proof of Theorem 3.1. On the other hand, if S and T are given a priori to have finite-dimensional kernels and co-kernels, then \(S\oplus T\) is a Fredholm operator (whose range is automatically closed). Again, we can directly proceed to Step 5.

3.2 Intrinsic Div-Curl Lemma on Riemannian Manifolds

Now we adapt the general functional-analytic theorem, Theorem 3.1, to the geometric settings. Our aim is to obtain a global intrinsic div-curl lemma on Riemannian manifolds (Theorem 3.3), independent of local coordinates, which will be applied to analyze the global weak rigidity of isometric immersions of Riemannian manifolds in the subsequent development. Let us begin with the notions of several geometric quantities.

One fundamental result concerning the Laplace–Beltrami operator is the Hodge decomposition theorem (cf. Warner [48]):

To conclude this section, we point out that some connections between the Hodge decomposition theorem and compensated compactness have been observed by Robbin–Rogers–Temple in [40] (also cf. Tartar [45]), and some generalized versions of the div-curl lemma have been obtained by Kozono–Yanagisawa [27]–[28], among others. Our general functional-analytic compensated compactness theorem, Theorem 3.1, further provides such a connection in the abstract form. In particular, our proof is essentially based upon the analysis of operator Open image in new window , which is an analogue of the Laplace–Beltrami operator \(\Delta \).

In fact, a global intrinsic div-curl lemma, more general than Theorem 3.3, on Riemannian manifolds can also be established, which applies for the two sequences \(\{\omega ^\epsilon \}\) and \(\{\tau ^\epsilon \}\) lying in \(L^r_{\mathrm{loc}}\) and \(L^s_{\mathrm{loc}}\), where \(1<r,s<\infty \) and \(\frac{1}{r}+\frac{1}{s}=1\).

To make this paper self-contained, we will present its proof in the appendix.

4.1 Global Weak Rigidity Theorem

Our main result in this section is the following:

This result can be regarded as a global version on Riemannian manifolds of Theorem 3.3 in Chen–Slemrod–Wang [8]. In this paper, both the statement and the proof for this weak rigidity theorem are global, intrinsic, independent of the local coordinates of the Riemannian manifolds, which offers further geometric insights into the GCR equations.

4.2 First Formulation: Identification of the Tensor Fields with Special Div-Curl Structure on Riemannian Manifolds

Now we begin the proof of Theorem 4.1. First we seek tensor fields with special div-curl structure for the GCR equations on manifolds.

Recall our previous convention: \(X,Y,Z,\ldots \in \Gamma (TM)\) and \(\xi ,\eta ,\beta ,\ldots \in \Gamma (TM^\perp )\). For each fixed \((Z,\eta ,\xi )\), define the tensor fields \(V^{(B)}_{Z,\eta }, V^{(\nabla ^\perp )}_{\xi ,\eta }: \Gamma (TM)\times \Gamma (TM)\mapsto \Gamma (TM)\) and 1-forms \(\Omega ^{(B)}_{Z,\eta }, \Omega ^{(\nabla ^\perp )}_{\xi ,\eta } \in \Omega ^1(M)=\Gamma (T^*M)\) by

$$\begin{aligned}&V^{(B)}_{Z,\eta }(X,Y):= B(X,Z,\eta )Y-B(Y,Z,\eta )X, \end{aligned}$$

(4.8)

$$\begin{aligned}&V^{(\nabla ^\perp )}_{\xi ,\eta }(X,Y):=\langle \nabla ^\perp _Y \xi , \eta \rangle X - \langle \nabla ^\perp _X \xi , \eta \rangle Y, \end{aligned}$$

(4.9)

Open image in new window

(4.10)

Open image in new window

(4.11)

To avoid further notations, we denote the vector fields canonically isomorphic to \(\Omega ^{(B)}_{Z,\eta }\) and \(\Omega ^{(\nabla ^\perp )}_{\xi ,\eta }\) (via \(\sharp \) and \(\flat \)) by the same symbols.

Our geometric picture is as follows: The V-tensors take two tangential vector fields (X, Y) to a vector field spanned by X and Y, which are anti-symmetric in (X, Y). Thus, \(V^{(B)}_{Z,\eta }(X,Y)\) and \(V^{(\nabla ^\perp )}_{\xi ,\eta }(X,Y)\) are precisely the rate of rotations of \(\Omega ^{(B)}_{Z,\eta }\) and \(\Omega ^{(\nabla ^\perp )}_{\xi ,\eta }\) in the 2-planes generated by (X, Y).

The 1-forms \((\Omega ^{(B)},\Omega ^{(\nabla ^\perp )})\) are simply contractions of \((B,\nabla ^\perp )\), and the tensors \((V^{(B)}, V^{(\nabla ^\perp )})\) can be obtained by applying the \(\Omega \)-tensors to the 2-Grassmannian of TM.

Our first formulation concerns the divergence of V in Eqs. (4.8)–(4.9) and the curl (as 2-forms) of \(\Omega \) in Eqs. (4.10)–(4.11):

As a remark, it is crucial to recognize that \(\delta \big (V^{(B)}_{Z,\eta }(X,Y)\big )\) and \(d\big (\Omega ^{(B)}_{Z,\eta }\big )(X,Y)\), as well as \(\delta \big (V^{(\nabla ^\perp )}_{\xi ,\eta }(X,Y)\big )\) and \(d\big ( \Omega ^{(\nabla ^\perp )}_{\xi ,\eta }\big )(X,Y)\), are essentially the same. They only differ by a zero-th order term involving the Lie bracket [X, Y]. This observation turns out to be crucial in the proof of Theorem 4.1.

4.3 Second Formulation: The Div-Curl Structure of the GCR Equations

We now express the GCR equations in another form, which is suitable for applying the intrinsic div-curl lemma, i.e., Theorem 3.3. To achieve this, we employ the geometric quantities \(V^{(B)}, \Omega ^{(B)},V^{(\nabla ^\perp )}\), and \(\Omega ^{(\nabla ^\perp )}\), in the reduced GCR equations (2.18)–(2.20) in Theorem 2.2 to obtain

4.4 Proof of Theorem 4.1

We can now prove the global weak rigidity theorem, Theorem 4.1, for the GCR equations on Riemannian manifolds. In fact, the main ingredients of the proof have been provided in the previous two formulations, which express the GCR equations in the form suitable for employing Theorem 3.3.

To conclude this section, we remark that Eqs. (2.1)–(2.3) in [8], which are the Gauss, Codazzi, and Ricci equations in local coordinates, can be seen directly from our global formulations in Theorems 2.1–2.2.

5.1 From PDEs to Geometry: An Equivalence Theorem

We now address the central question raised above. First, it should be noticed that the results in Sect. 4 are essentially PDE-theoretic. Despite the geometric—global and intrinsic—nature of the formulation and proof of Theorem 4.1, we have only analyzed the GCR equations per se, but have not referred to their geometric origin, i.e., the isometric immersion problem. One would expect that, providing that our formulation is natural, the weak rigidity of isometric immersions and the weak rigidity of the GCR equations should be essentially the same problem. We now formalize this observation and prove it in mathematical rigor.

We point out that the realization problem, i.e., the construction of isometric immersions from the GCR equations, has been investigated in the recent years; see Ciarlet–Gratie–Mardare [9], Mardare [30]–[32], Szopos [44], and the references cited therein. These previous results, which solve the realization problem locally, can be summarized in the following theorem.

Here and in the sequel, we write \(\mathfrak {gl}(q;\mathbb {R})\) for the space of \(q\times q\) matrices with real entries, \(O(q) \subset \mathfrak {gl}(q;\mathbb {R})\) for the space of symmetric \(q \times q\) matrices, and \(\mathfrak {so}(q)\subset \mathfrak {gl}(q;\mathbb {R})\) the space of \(q\times q\) anti-symmetric matrices.

The strategy for the proof in [30]–[32] and [44] can be briefly sketched as follows: First, the GCR equations can be transformed into two types of first-order matrix-valued PDE systems with \(W^{2,p}\) coefficients, known as the Pfaff and Poincaré systems; then, applying various analytic results established in [30]–[32], one can construct explicitly the local isometric immersions by solving the Pfaff and Poincaré systems with the rough coefficients. Nevertheless, despite the successful solution to the realization problem (at least locally), the transformations from the GCR equations to the Pfaff and Poincaré systems in [30]–[32] and [44] appear quite delicate, which involve many different types of geometric quantities in local coordinates (e.g., metrics, connections, and curvatures) as the entries of the same matrix of enormous size.

In this section, we give an alternative global geometric proof of Theorem 5.1. In addition, we solve the problems listed in goals (i) and (ii) at one stroke. Our perspective is essentially different from those in [30]–[32] and [44]: We aim at establishing the equivalence of the GCR equations and the existence of isometric immersions on Riemannian manifolds with \(W^{1,p}_\mathrm{loc}\) metric, \(p>n\), via the Cartan formalism for exterior differential calculus (see [43]).

We first state the main theorem of this section, which concerns the equivalence of three formulations of the GCR equations in disguise. Roughly speaking, we view the Cartan formalism as the bridge between the GCR equations and isometric immersions.

In Theorem 5.2, we require \(p>n\) (instead of \(p>2\) for the weak rigidity of the GCR equations, as in Sect. 4) to guarantee that the immersion is \(C^1\), which agrees with the classical notions from differential geometry. In Sect. 5.2, the Cartan formalism is introduced. This clarifies the precise meaning for the second item in the above theorem. Then, in Sect. 5.3, we give a proof of Theorem 5.2.

Finally, the implication: \(\mathrm{(i)} \Rightarrow \mathrm{(iii)}\) leads to the following global realization theorem on Riemannian manifolds, independent of the local coordinates:

5.2 Cartan Formalism

On a local chart \(U \subset M\) where the vector bundle E is trivialized (i.e., \(E|_U\) is diffeomorphic to \(U\times \mathbb {R}^k\)), let \(\{\omega ^i\} \subset \Omega ^1(U)\) be an orthonormal co-frame dual to the orthonormal frame \(\{\partial _i\} \subset \Gamma (TU)\). The latter is called a moving frame adapted to U. The Cartan formalism is thus also known as the method of moving frames.

We remark that the structure equations, as well as various other Lie group-valued equations we will introduce below, are intrinsic. That is, these equations are independent of the moving frames, and hence are natural in coordinate-free notations. This ensures the global and intrinsic nature of the proof of Theorem 5.2, which is the content in Sect. 5.3. For more details on the Cartan formalism, see [43].

5.3 Proof of Theorem 5.2

With the Cartan formalism introduced, we are now in the position to prove the main result of this section, i.e., Theorem 5.2. Our proof is intrinsic and global in nature, i.e., covariant with respect to the change of local frames. We divide the proof in seven steps.

1. We first notice the equivalence between the GCR equations and the Cartan formalism (cf. [43]). Also, it is well known that the existence of a local isometric immersion implies the GCR equations (cf. [13]). Although the above classical results are established in the \(C^\infty \) category in [13, 43], it is easy to check that the proofs remain unaltered for the lower regularity case, since \(g\in W^{1,p}_\mathrm{loc}\) ensures that all the calculations involved make sense in the distributional sense. Thus, it remains to prove \(\mathrm{(ii)}\Rightarrow \mathrm{(iii)}\), i.e., the Cartan formalism implies the existence of an isometric immersion.

We follow closely the arguments in Tenenblat [47] for the \(C^\infty \) case. We first prove the local version of the theorem and then extend to the global version on simply connected manifolds via topological arguments.

2. Recall that \(W=\{\omega ^a_b\}\in \Omega ^1(U; \mathfrak {so}(n+k))\) and \(w=(\omega ^1, \ldots , \omega ^n, 0, \ldots , 0)\in \Omega ^1(U; \mathbb {R}^{n+k})\). In order to find an isometric immersion based upon the structural equations (5.1)–(5.2), we first solve for an affine map A (taking \(\partial _j\) to \(\eta _\alpha \)), which essentially consists of the components of the normal affine connection, and then the isometric immersion f is constructed from A.

3. In the \(C^\infty \) category, to establish the solvability of PDEs in form (5.9)–(5.10), which can be viewed as complete integrability conditions, we only need to check the involutiveness, in view of the Frobenius theorem. This constitutes the arguments in [47]. In the lower regularity case, we seek for the correct analogue to the Frobenius theorem. The following lemma can serve for this purpose:

We also remark that the uniqueness in (ii) is proved by Schwartz [42]. Thanks to Lemma 5.1, solving for systems (5.9) and (5.10) can be reduced to checking the compatibility conditions in the form of Eqs. (5.12) and (5.14), provided that the regularity assumptions are satisfied.

5. Next, we solve for immersion f from the Poincaré system (5.10). Since \(w\cdot A \in W^{1,p}_\mathrm{loc}\) with \(p>n\), the regularity assumption in Lemma 5.1(ii) is satisfied, both for \(w\cdot A\) and its first derivatives. Thus, if we verify the compatibility condition in the form of Eq. (5.14), we can establish the existence of \(f\in W^{2,p}_\mathrm{loc}\), thanks to Lemma 5.1.

Moreover, observe that \(df= w\,\cdot A\) from Eq. (5.10), \(A \in O(n+k)\) is a field of nonsingular matrices, and \(\{\omega ^1, \ldots , \omega ^n\}\) are linearly independent, so that \(df \ne 0\) in \(W^{1,p}_\mathrm{loc}\). Since \(p > n\), the Sobolev embedding yields that \(df \ne 0\) almost everywhere, which verifies that f is indeed an immersion. The almost everywhere uniqueness of f follows from Lemma 5.1 and the \(\mathbb {R}^{n+k}\rtimes O(n+k)\)-symmetry of the Euclidean space.

7. It remains to globalize our arguments for simply connected manifolds. This follows from a standard argument in topology.

Given any two points \(x \ne y \in M\), we connect them by a continuous curve (again since \(g \in W^{1,p}_\mathrm{loc}\hookrightarrow C^0_\mathrm{loc}\) for \(p>n\)), denoted by \(\gamma : [0,1]\mapsto M\) with \(\gamma (0)=x\) and \(\gamma (1)=y\). Let f be the \(W^{2,p}_\mathrm{loc}\) isometric immersion in a neighborhood of x, whose existence is guaranteed by the preceding steps of the same proof. Cover curve \(\gamma ([0,1])\) by finitely many charts \(\{V_1,\ldots ,V_N\}\). By the uniqueness statements in Lemma 5.1, we can extend the isometric immersion f to \(\bigcup _{i=1}^N V_i\), especially including a neighborhood of y.

Thus, it suffices to verify that the extension of f is independent of the choice of \(\gamma \). Indeed, if \(\eta :[0,1]\mapsto M\) is another continuous curve connecting x and y, by concatenating \(\gamma \) with \(\eta \), we can obtain a loop \(L\subset M\). As M is simply connected, the restriction \(f|_L\) is homotopic to a constant map so that \((f\circ \gamma )(1)=(f\circ \eta )(1)\). In this way, we have verified that f can be extended to a global isometric immersion of M into \(\mathbb {R}^{n+k}\), provided that M is simply connected.

This completes the proof of Theorem 5.2.

As a corollary of Theorems 4.1 and 5.2, we can deduce the weak rigidity of isometric immersions.

One natural question arises at this stage is about the criteria for the existence of global isometric immersions for a non-simply connected manifold M with \(W^{1,p}_\mathrm{loc}\) metrics, especially for the case of the minimal target dimension (i.e., the Janet dimension). However, as far as we have known, it is still open, primarily owing to some topological obstructions to the GCR equations. Suppose that there is a loop \(\gamma \) generating nontrivial homotopy on M, it is far from being clear if one can find a well-defined immersion along the entire loop. The problems on global isometric immersions/embeddings for general manifolds remain vastly open in the large. See Schoen–Yau [41] and Bryant–Griffith–Yang [4] for further discussions.

Therefore, the best we may say for non-simply-connected manifolds are the local versions of Theorem 5.2 and Corollaries 5.1 and 5.2.

7.1 Weak Rigidity of the GCR Equations and Isometric Immersions Revisited

We now provide a proof of the weak rigidity of the Cartan formalism, which leads to the weak rigidity of the GCR equations and isometric immersions, in view of the equivalence between the Cartan formalism, the GCR equations, and isometric immersions established in Sect. 5. Therefore, we also provide an alternative proof of Theorem 4.1 and Corollary 5.2. The arguments are summarized as follows:

We remark that the above proof lies in the same spirit as in Chen–Slemrod–Wang [8] for the GCR equations. Related arguments are also present in [1, 11, 18, 19, 33, 34, 35, 45, 46] and the references cited therein.

7.2 Weak Rigidity of Isometric Immersions with Different Metrics

We now develop an extension of the weak rigidity theory of the GCR equations and isometric immersions established above. In the earlier sections, we have analyze the isometric immersions of a fixed Riemannian manifold. In particular, despite the change of second fundamental forms and normal connections as we shift between distinctive immersions, metric \(g\in W^{1,p}_{\text {loc}}\) is always fixed. In what follows, we generalize the weak rigidity results (cf. Theorem 4.1 and Corollary 5.2) to the Riemannian manifolds with unfixed metrics \(\{g^\epsilon \}\).

More precisely, we establish the following theorem.

We also refer the reader to some recent results on the compactness of \(W^{2,p}\) immersions of n-dimensional manifolds for \(p>n\) with appropriate gauges for the case \(n=2\) in Langer [29] and the higher dimensional case in Breuning [3].