Scalar Curvature Rigidity of Warped Product Metrics

. We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini–Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 with two antipodal points removed. This resolves a problem in Gromov’s “Four Lectures” in all dimensions. Our arguments are based on spin geometry.


Introduction
In this paper, we study rigidity results for metrics with lower scalar curvature bounds.One of the first results of this kind is the famous rigidity theorem of Llarull [14].Let g S n denote the standard round metric on S n with scalar curvature n(n − 1).Llarull showed that, if g is a metric on S n with g ≥ g S n and R g ≥ n(n − 1), then g = g S n .The proof of Llarull's theorem uses Dirac operator techniques in an ingenious way, and is inspired by the fundamental work of Gromov and Lawson [8,9].
In an important paper, Cecchini and Zeidler extended this line of thought and proved scalar and mean curvature rigidity results for odd-dimensional manifolds with boundary where the comparison metric is not the round metric, but a warped product metric (see [5,Section 10]).In the first part of the present paper we will remove the dimension parity assumption in some of their results.
Given a Riemannian manifold (M, g) with boundary, we denote by R g the scalar curvature of g.Moreover, we denote by ν g the outward unit normal with respect to g.We denote by H g the mean curvature of ∂M with respect to g, defined as the sum of the principal curvatures.The sign convention for H g is such that the mean curvature vector is given by −H g ν g .
Let n > 2, let θ − < θ + and let ρ : [θ − , θ + ] → R be a positive smooth function.We consider the warped product metric g 0 = dθ ⊗ dθ + ρ 2 (θ)g S n−1 (1.1) on S n−1 × [θ − , θ + ].The scalar curvature of g 0 is given by while the boundary mean curvature of g 0 is given by Our first result says that warped product metrics satisfy a scalar-mean curvature rigidity property, provided that the warping function is strictly logarithmically concave.
Then Φ is a Riemannian isometry.
We note that R g 0 in this theorem is not required to be non-negative.For n odd, Theorem A is implied by results of Cecchini-Zeidler, see [5,Theorem 10.2].
Applying this discussion to annuli in simply-connected space forms as in [5, Section 10], this removes the parity restriction in [5, Corollaries 10.4 and 10.5].
Example 1.2.The spatial Schwarzschild-de Sitter metrics on S n−1 × [θ − , θ + ] are rotationally symmetric and have scalar curvature equal to a positive constant.Similarly, the spatial Schwarzschild-anti-de Sitter metrics on S n−1 × [θ − , θ + ] are rotationally symmetric and have scalar curvature equal to a negative constant.These metrics can be expressed as warped products of the form g 0 = dθ ⊗ dθ + ρ 2 (θ)g S n−1 , see, e.g., [13, p. 64].If we restrict to an interval where log ρ is strictly concave, then we obtain the rigidity property in Theorem A.
The second theme of our paper is a rigidity result for metrics on the sphere S n with two antipodal points removed.This can be viewed as a limiting case of the band rigidity results treated in the first part of our paper.This is related to a conjecture of Gromov [7].He conjectured that Llarull's theorem holds for metrics that are defined on the sphere S n with finitely many points removed.In the special case of two antipodal punctures, Gromov sketched an argument based on µ-bubbles (see [7,Sections 5.5 and 5.7]).In the three-dimensional case, a detailed proof based on µ-bubbles was given by Hu, Liu, and Shi [11].An alternative proof in the three-dimensional case was given by Hirsch, Kazaras, Khuri, and Zhang [10].Using Dirac operator techniques, we generalize these results to all dimensions: Theorem B. Let n > 2. We consider the warped product metric g 0 = dθ ⊗ dθ + sin 2 θg S n−1 on S n−1 × (0, π).Let Ω be a non-compact, connected spin manifold of dimension n without boundary.Let g be a (possibly incomplete) Riemannian metric on Ω with scalar curvature R g ≥ n(n − 1).Suppose that Φ : (Ω, g) → S n−1 × (0, π), g 0 is a smooth map with the following properties: Then Φ is a Riemannian isometry.
Our argument relies on the spin geometric approach to scalar curvature rigidity as introduced in [14] and further developed in [5].A new feature of the present work is the construction of non-zero harmonic spinor fields for which the right-hand side of the integral Schrödinger-Lichnerowicz-Weitzenböck formula has a favorable sign, but which cannot be generated directly by index-theoretic arguments.This construction uses limits of sequences of non-zero harmonic spinor fields whose existence follows from index theory, cf.Corollaries 2.7 and 3.5.
In contrast to [5], our index calculations take place exclusively on compact manifolds.The corresponding "holographic" index theorem for compact manifolds with boundary is formulated and proved in Appendix B, which may be of independent interest.
After this paper was written, we learned of a preprint by Wang and Xie [15] announcing similar results.

Proof of Theorem A 2.1 Proof of Theorem A for n even
We first prove Theorem A for even n, which is not treated in [5].The necessary changes in the odd-dimensional case will be explained in the next section.
Fix an even integer n > 2 and a warping function ρ : [θ − , θ + ] → R such that (log ρ) ′′ < 0. Let g 0 denote the warped product metric in (1.1).Let M be a compact, connected spin manifold of dimension n with boundary ∂M .Let g be a Riemannian metric on M .Suppose that Φ : (M, g) → S n−1 × [θ − , θ + ], g 0 is a smooth map satisfying the assumptions of Theorem A.
Proof .Evaluating the inequality (2.1) at the vector ∇Θ gives From this, the assertion follows easily.■ We write ∂M = ∂ + M ∪ ∂ − M , where Proof .We can find a smooth function Θ : Since Φ is homotopic to Φ relative to ∂M , the assertion follows.■ For even n, the boundary ∂M is odd-dimensional which is inconvenient for the index calculations.As in [14], we remedy the situation by considering products with circles of large radius and sending the radius to infinity.Let r be a positive real number.We consider the product M = M × S 1 equipped with the product metric g = g +r 2 g S 1 .We write ∂ M = ∂ + M ∪∂ − M , where Lemma 2.3.There exists a smooth map , and β(t) = π for t ∈ 7π 8 , π .Moreover, let us fix a point a ∈ S n−1 .We consider the map This gives a map h : In the following, we assume that h : Choose a spin structure on M and let S denote the spinor bundle over M .Furthermore, let S denote the spinor bundle over M = M × S 1 , where S 1 is equipped with the trivial spin structure S 1 × Spin(1) → S 1 × SO (1).Note that with this choice, S is the pull-back of S, as a Clifford-module bundle, under the projection from M = M × S 1 to M .
Let E 0 denote the spinor bundle of the round sphere S n .The bundle E 0 is equipped with a preferred bundle metric and connection.Since n is even, we may decompose E 0 in the usual way as E 0 = E + 0 ⊕ E − 0 , where E + 0 and E − 0 are the ±1-eigenbundles of the complex volume form.Next we need an index computation.
It remains to show that max{ind 1 , ind 2 , ind 3 , ind 4 } > 0. Suppose that this is false.Then ind 1 = ind 2 = ind 3 = ind 4 = 0. We will apply the holographic index theorem in Appendix B and the Atiyah-Singer index theorem to show that the assumption ind 1 = ind 3 = 0 already leads to a contradiction.
The restriction S| ∂ − M can be identified with the spinor bundle on ∂ − M .We may write S| ∂ − M = S + ⊕ S − , where S + and S − denote the eigenbundles of the volume form on ∂ − M .Equivalently, S + and S − can be characterized as the eigenbundles of iν.This gives the splitting Since ind 1 = 0, Corollary B.3 tells us that the boundary Dirac operator which maps sections of Similarly, since ind 3 = 0, the boundary Dirac operator which maps sections of Working with E − 0 instead of E + 0 , we similarly obtain In the next step, we subtract (2.3) from (2.2).Using the fact that dim It follows from [12,Proposition 11.24, Chapter III] that ch By assumption, the map Φ : M → S n−1 × [θ − , θ + ] has non-zero degree.Hence, it follows from Lemma 2.2 that the map φ| ∂ − M : ∂ − M → S n−1 has non-zero degree.Consequently, the map φ| ∂ − M × id : ∂ − M × S 1 → S n−1 × S 1 has non-zero degree.By Lemma 2.3, the map h : has non-zero degree.This is a contradiction.■ By Proposition 2.4, we know that max{ind 1 , ind 2 , ind 3 , ind 4 } > 0. After switching the bundles E + 0 and E − 0 if necessary, we may assume that max{ind 1 , ind 2 } > 0. In the remainder of this section, we focus on the case ind 1 > 0. (The case ind 2 > 0 can be treated analogously.) Let Ẽ denote the pull-back of E + 0 under the map f .The bundle metric on E + 0 gives us a bundle metric on Ẽ.Moreover, the connection on E + 0 induces a connection on Ẽ.We denote by ∇ S⊗ Ẽ the tensor product connection on S ⊗ Ẽ.We denote by D S⊗ Ẽ the Dirac operator acting on sections of S ⊗ Ẽ, where {e 1 , . . ., e n+1 } is a local orthonormal frame on M .Finally, we define the boundary Dirac operator by Hu, where {e 1 , . . ., e n } is a local orthonormal frame on ∂ M .The boundary Dirac operator is selfadjoint and anti-commutes with Clifford multiplication by ν.
Recall the Weitzenböck formula (see [12,Theorem 8.17, Chapter II]), where R Ẽ is a section of the endomorphism bundle of S ⊗ Ẽ which depends on the curvature of the bundle Ẽ.We define a vector field T on M by T = 1 r ∂ ∂t , where t → (cos t, sin t) is the canonical local coordinate on S 1 .Note that T is parallel and has unit length with respect to the metric g.In the following, Ψ will denote a smooth function on M which will be specified later.We may extend Ψ to a smooth function on M satisfying T (Ψ) = 0.If u is a section of the bundle S ⊗ Ẽ and X is a vector field on M , we define Our argument is based on the following integral formula which links several geometric quantities on M and ∂ M .
Proof .Integrating the Weitzenböck formula and using the divergence theorem gives Hu.This gives Since D ∂ M is self-adjoint and anti-commutes with ν, we find Therefore, (2.5) Using the definition of P u and a local orthonormal frame e 1 , . . ., e n , T on M , we compute Using the divergence theorem, we find (2.7) We integrate (2.6) over M and insert (2.7) and obtain (2.8) Substituting (2.8) into (2.5),we obtain (2.9) Using the divergence theorem, we obtain (2.10) Adding (2.9) and (2.10) gives This completes the proof of Proposition 2.5.■ At this point, we specify our choice of the function Ψ.We define a function ψ Then we can find an element t 0 ∈ S 1 and a section u ∈ C ∞ M , S ⊗ Ẽ such that and Proof .Recall that we are assuming ind 1 > 0. In view of the deformation invariance of the index, we can find a section u ∈ C ∞ M , S ⊗ Ẽ such that Using Proposition 2.5, we obtain (2.12) By assumption and using (1.3), Consequently, the right-hand side in (2.12) is non-positive.
We next analyze the term R Ẽ .To that end, we fix a point (x, t) ∈ M .Let µ 1 , . . ., µ n+1 ≥ 0 denote the singular values of the differential d f(x,t) : T (x,t) M , g → (T f (x,t) S n , g S n ), arranged in decreasing order.Since the differential d f(x,t) has rank at most n, it follows that µ n+1 = 0.The eigenvalues of the symmetric bilinear form f * g S n with respect to the metric g = g + r 2 g S 1 are given by µ 2 1 , . . ., µ 2 n , 0. Using (2.1), we obtain, on the one hand, On the other hand, the inequality By assumption, r > 2ρ(Θ) at the point (x, t).Hence, the min-max characterization of the eigenvalues implies that µ 2 1 , . . ., µ 2 n−1 ≤ 1 ρ 2 (Θ) and µ 2 n ≤ 4 r 2 .Together with Proposition A.1, this implies Using (2.11), we obtain the pointwise estimate Putting these facts together, we conclude that Hence, we can find an element t 0 ∈ S 1 such that From this, the assertion follows.■ Corollary 2.7.There exists an element t 0 ∈ S 1 with the following property.Let f : M → S n be defined by Let E denote the pull-back of E + 0 under the map f .Then there exists a section s ∈ C ∞ (M, S⊗E) such that for every vector field X.
Proof .Let us consider a sequence r ℓ → ∞.For each ℓ, Proposition 2.6 implies the existence of an element t ℓ ∈ S 1 and a section u and After passing to a subsequence, we may assume that the sequence t ℓ converges to an element t 0 ∈ S 1 .We define maps f : for x ∈ M .Let E denote the pull-back of E + 0 under f , and let E (ℓ) denote the pull-back of For each ℓ, we define a bundle map σ (ℓ) : E (ℓ) → E as follows.For each point x ∈ M , the map σ (ℓ) x : E (ℓ) is defined as the parallel transport along the shortest geodesic from f (ℓ) (x) ∈ S n to f (x) ∈ S n .It is easy to see that σ (ℓ) is a bundle isometry for each ℓ.It follows that the map id ⊗ σ (ℓ) : S ⊗ E (ℓ) → S ⊗ E is a bundle isometry for each ℓ.
We may write where A (ℓ) is a 1-form taking values in the endomorphism bundle End S ⊗ E (ℓ) .Since t ℓ → t 0 , the maps f ℓ converge to f smoothly.From this, we deduce that A (ℓ) → 0 uniformly.By (2.13), ∇ S⊗E (ℓ) s (ℓ) is bounded in L 2 .Using (2.14), we conclude that After passing to a subsequence, the sequence id The inequality (2.13) implies that for every smooth vector field X on M , where (2.15) is understood in the sense of distributions.Since (2.15) holds for every smooth vector field X on M , it follows that s is a weak solution of an overdetermined elliptic system.By elliptic regularity, s is smooth and (2.15) holds classically.
Rescaling s concludes the proof.■ Definition 2.8.Let t 0 ∈ S 1 , f : M → S n , and E be defined as in Corollary 2.7.We define the modified connection ∇ Ψ on S ⊗ E by where ∇ S⊗E denotes the connection on S ⊗ E induced by the ones on S and E.
Lemma 2.9.The curvature tensor R Ψ of the connection ∇ Ψ defined in (2.16) satisfies: (2.17) Here R S⊗E denotes the curvature tensor of ∇ S⊗E .Moreover, the curvature term in the Weitzenböck formula satisfies where e 1 , . . ., e n is a local orthonormal frame.
Proof .We check (2.17) at a fixed point on M .Let X and Y be vector fields defined in a neighborhood of that point whose covariant derivatives vanish at the point.We compute at that point: Anti-symmetrizing with respect to X and Y yields (2.17).
As to (2.18), we use formula (8.8) in [12, Chapter II] and (2.17) and we find After these preparations, we now complete the proof of Theorem A for even n.Let t 0 ∈ S 1 , f : M → S n , and E be defined as in Corollary 2.7 and let ∇ Ψ denote the connection defined in (2.16).By Corollary 2.7, there exists a section s ∈ C ∞ (M, S ⊗ E) such that M |s| 2 = 1 and ∇ Ψ s = 0. Since M is connected and s is ∇ Ψ -parallel, we have that s ̸ = 0 at each point in M .Using (2.18) and the fact that R Ψ annihilates s, we find In the last step, we have again used (2.11).
Let us fix an arbitrary point x ∈ M .We consider the singular values µ 1 , . . ., µ n ≥ 0 of df x : (T x M, g) → (T f (x) S n , g S n ), arranged in decreasing order.Since f factors through S n−1 , the differential df x has rank at most n − 1, and we obtain µ n = 0.The eigenvalues of the symmetric bilinear form f * g S n with respect to the metric g are given by µ 2 1 , . . ., µ 2 n−1 , 0. Using (2.1) together with the inequality h(•, t 0 ) * g S n ≤ g S n−1 , we obtain at the point x.Having established the reverse inequality in (2.19), this inequality must be an equality.In particular all the inequalities in (2.19) are Since n > 2 and s ̸ = 0 at the point x and ψ ′ ̸ = 0, we can draw the following conclusion: • The singular values of df x : (T x M, g) → (T f (x) S n , g S n ) are given by 1 ρ(Θ) , . . ., 1 ρ(Θ) , 0. In other words, the eigenvalues of f * g S n with respect to g at the point x are given by Since |∇Θ| = 1 at the point x, Lemma 2.1 implies that dφ x (∇Θ) = 0, hence df x (∇Θ) = 0. Consequently, ∇Θ lies in the nullspace of f * g S n .Putting these facts together, we obtain at the point x.On the other hand, using (2.1) together with the inequality h(•, t 0 ) * g S n ≤ g S n−1 , we obtain at the point x.Combining (2.20) and (2.21), we conclude that at the point x.Since x is arbitrary, we conclude that g = Φ * (g 0 ).This means that Φ is a local isometry.Since the target of Φ is simply connected and the domain is connected, it follows that Φ is a global Riemannian isometry.The proof of Theorem A for even n is complete.

Proof of Theorem A for n odd
When n is odd, the proof of Theorem A is simpler, and we just indicate the necessary changes.Instead of working with M = M × S 1 , we work with M .Furthermore, instead of the map f = h • (φ × id) : M → S n , we directly work with φ : M → S n−1 .Let S denote the spinor bundle over M , and let E 0 denote the spinor bundle over the round sphere S n−1 .Since n − 1 is even, we may decompose E 0 = E + 0 ⊕ E − 0 , where E + 0 and E − 0 denote the ±1-eigenbundles of the complex volume form.Then ind 1 + ind 2 = 0, ind 3 + ind 4 = 0, and max{ind 1 , ind 2 , ind 3 , ind 4 } > 0.
The proof of Proposition 2.10 is analogous to the proof of Proposition 2.4 and uses the holographic index theorem.
After switching the bundles E + 0 and E − 0 if necessary, we may assume that max{ind 1 , ind 2 } > 0. As above, we focus on the case ind 1 > 0. (The case ind 2 > 0 can be handled analogously.) Let E denote the pull-back of E + 0 under φ.Let D S⊗E denote the Dirac operator on sections of S ⊗ E, and let D ∂M denote the boundary Dirac operator.
Let Ψ be a smooth function on M that will be specified later.If s is section of the bundle S ⊗ E and X is a vector field on M , we define a perturbed covariant derivative of s by the formula The proof of Proposition 2.11 is analogous to Proposition 2.5.
As above, we define a function ψ

.22)
We define Ψ : M → R by Ψ = ψ • Θ.At this point, we use the assumption that ind 1 > 0. In view of the deformation invariance of the index, we find a section s of S ⊗ E such that • s does not vanish identically, Using Proposition 2.11, we obtain (2.23) By assumption, H − (n − 1)Ψ ≥ 0 on ∂ + M and H + (n − 1)Ψ ≥ 0 on ∂ − M .Therefore, the right-hand side in (2.23) is non-positive.Fix a point x ∈ M , and let µ 1 , . . ., µ n ≥ 0 denote the singular values of the differential dφ x : (T x M, g) → T f (x) S n−1 , g S n−1 , arranged in decreasing order.Since the differential dφ x has rank at most n − 1, it follows that µ n = 0.
Since φ is 1-Lipschitz by assumption, we obtain Since |∇Θ| ≤ 1 and |∇Ψ| ≤ −ψ ′ (Θ), this gives the pointwise estimate In the last step, we have used (2.22).Putting these facts together, we conclude that for every vector field X.This is the analogue of Corollary 2.7.From here on, the proof of Theorem A in the odd-dimensional case proceeds in the same way as in the even-dimensional case.
3 Proof of Theorem B

Proof of Theorem B for n even
We first prove Theorem B for even n.The necessary adaptations in the odd-dimensional case will be explained at the end of this section.
Proof .The degree of Φ is obtained by counting the elements of the set Φ −1 ({z}) with suitable signs.Since Φ −1 ({z}) is contained in M δ , the degree of Φ| M δ coincides with the degree of Φ. ■ We consider the product Ω = Ω × S 1 equipped with the product metric g = g + r 2 g S 1 .Let Mδ = M δ × S 1 ⊂ Ω.We write ∂ Mδ = ∂ + Mδ ∪ ∂ − Mδ , where By Lemma 2.3, we can find a smooth map h : Choose a spin structure on Ω and let S denote the spinor bundle over Ω.Let S denote the spinor bundle over the product Ω.Note that S is the pull-back of S under the canonical projection from Ω = Ω × S 1 to Ω.
Let E 0 denote the spinor bundle of the round sphere S n .The bundle E 0 is equipped with a natural bundle metric and connection.Since n is even, we may decompose E 0 in the usual way as E 0 = E + 0 ⊕ E − 0 , where E + 0 and E − 0 are the eigenbundles of the complex volume form.For each δ ∈ ∆, we define ind 1 , ind 2 , ind 3 , ind 4 as in Proposition 2.4, working on Mδ instead of M .Let By Proposition 2.4, we know that In particular, at least one of the sets ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 must contain 0 in its closure.After switching the bundles E + 0 and E − 0 if necessary, we may assume that one of the sets ∆ 1 , ∆ 2 must contain 0 in its closure.In the remainder of this section, we assume that the set ∆ 1 contains 0 in its closure.(The case when the set ∆ 2 contains 0 in its closure can be handled analogously.) In the following, we consider a real number δ ∈ ∆ 1 .In other words, ind 1 is positive.Let Ẽ denote the pull-back of E + 0 under the map f .The bundle metric on E + 0 gives us a bundle metric on Ẽ.Moreover, the connection on E + 0 induces a connection on Ẽ.As above, we denote by ∇ S⊗ Ẽ the connection on S ⊗ Ẽ.We denote by D S⊗ Ẽ the Dirac operator acting on sections of S ⊗ Ẽ.
By assumption, r > 2 sin Θ. Arguing as in Section 2, we can bound Moreover, using the inequality |∇Θ| ≤ 1, we obtain 3 , π − δ , and . Using these facts together with the inequality R ≥ n(n − 1), we conclude that denote the pull-back of E + 0 under the map f .Then there exists a section s ∈ H Here, ∇ S⊗E denotes the connection on S ⊗ E.
Proof .Let us consider a sequence r ℓ → ∞.For each ℓ, we can find an element t ℓ ∈ S 1 and a section u and After passing to a subsequence, we may assume that the sequence t ℓ converges to an element t 0 ∈ S 1 .As in Section 2, we define maps f : Ω → S n and f (ℓ) : Ω → S n by for x ∈ Ω.Let E denote the pull-back of E + 0 under f , and let E (ℓ) denote the pull-back of E + 0 under the map f (ℓ) .The restriction of u (ℓ) to M δ × {t ℓ } gives a section s Since M δ is connected, we may estimate (see [2]).This implies M δ s (ℓ) 2 ≤ C(δ, ε).Analogously to Section 2, we consider a sequence of bundle maps σ (ℓ) : E (ℓ) → E. After passing to a subsequence, the sequence id ⊗ σ (ℓ) s (ℓ) ∈ C ∞ (M δ , S ⊗ E) converges, in the weak topology of H 1 (M δ , S ⊗ E), to a section s.The section s ∈ H 1 (M δ , S ⊗ E) has all the desired properties.■ Corollary 3.5.There exists an element t 0 ∈ S 1 with the following property.Let f : Ω → S n be defined by f (x) := f (x, t 0 ) = h(φ(x), t 0 ) for x ∈ Ω.Let E denote the pull-back of E + 0 under the map f .Then there exists a section s ∈ C ∞ (Ω, S ⊗ E) such that for every vector field X.Here, ∇ S⊗E denotes the connection on S ⊗ E.
Proof .Recall that we are assuming that the set ∆ 1 contains 0 in its closure.In other words, we can find a sequence of real numbers δ ℓ ∈ ∆ 1 converging to 0. After passing to a subsequence, we may assume that the sequence δ ℓ is monotonically decreasing.Consequently, M δ l is an increasing sequence of compact domains in Ω, and l M δ l = Ω.We choose a sequence ε ℓ ∈ (0, δ ℓ ) such that H ≥ −(n − 1) cot ε ℓ at each point on ∂M δ ℓ .By Corollary 3.4, we can find a sequence of elements t ℓ ∈ S 1 with the following property.Let f (ℓ) : Ω → S n be defined by f (ℓ) (x) := f (x, t ℓ ) = h(φ(x), t ℓ ) for x ∈ Ω.Let E (ℓ) denote the pull-back of E + 0 under the map f (ℓ) .Then there exists a section s Since Ω is connected, results in [2] imply that the sequence s (ℓ) is bounded in H 1 loc .After passing to a subsequence, we may assume that the sequence t ℓ converges to an element t 0 ∈ S 1 .We define a map f : Ω → S n by f (x) := f (x, t 0 ) = h(φ(x), t 0 ) for x ∈ Ω.Let E denote the pull-back of E + 0 under f .As in Section 2, we consider a sequence of bundle maps σ (ℓ) : E (ℓ) → E. After passing to a subsequence, the sequence id ⊗ σ (ℓ) s (ℓ) ∈ H 1 (M δ ℓ , S ⊗ E) converges weakly in H 1 loc to a section s ∈ H 1 loc (Ω, S ⊗ E).The section s satisfies and for every vector field X, where (3.3) is understood in the sense of distributions.Since s is a weak solution of an overdetermined elliptic system, we conclude that s is smooth and (3.3) holds in the classical sense.■ Having established Corollary 3.5, the proof of Theorem B now proceeds as in Section 2, with the choice ρ(θ) = sin θ and ψ(θ) = cot θ.As in Section 2, we conclude that g = Φ * (g 0 ).In other words, Φ is a local Riemannian isometry.Since Φ is proper, the domain is connected, and the target is simply connected, Φ is a global Riemannian isometry.This completes the proof of Theorem B for n even.

Proof of Theorem B for n odd
When n is odd, the proof of Theorem B is simpler, as we can work with Ω and M δ directly, and we do not need to consider the Cartesian product with S 1 .We omit the details.

A The curvature term in the Weitzenböck formula
In this section, we recall a well-known estimate for the curvature term in the Weitzenböck formula.Let us fix integers n, N ≥ 2. Let (M, g) be a Riemannian spin manifold of dimension n, and let f : (M, g) → S N , g S N be a smooth map to the round unit sphere of dimension N .Let S → M denote the spinor bundle of M , let E 0 → S N denote the spinor bundle of S N and set E = f * E 0 .Let R E denote the curvature term appearing in the Weitzenböck formula for the square of the twisted Dirac operator on S ⊗ E, so that The following estimate for the curvature term R E is well known.
Proposition A.1 (cf.Llarull [14]).Let x ∈ M and let µ 1 , . . ., µ n ≥ 0 denote the singular values of the differential df x : (T x M, g x ) → T f (x) S N , g S N .Then Proof .Let m denote the rank of the differential df x .Clearly, m ≤ min{n, N }.We assume that the singular values are arranged so that µ k > 0 for 1 ≤ k ≤ m and µ k = 0 for m+1 ≤ k ≤ N .We can find an orthonormal basis {e 1 , . . ., e n } of (T x M, g x ) and an orthonormal basis {ε 1 , . . ., ε N } of T f (x) S N , g S N such that df x (e k ) = µ k ε k for 1 ≤ k ≤ m and df x (e k ) = 0 for m + 1 ≤ k ≤ n.
Let F E 0 ∈ Ω 2 (S N , End(E 0 )) denote the curvature of E 0 , and let F E ∈ Ω 2 (M, End(E)) denote the curvature of E. For each s ∈ S x ⊗ E x = S x ⊗ (E 0 ) f (x) , formula (8.22) in [12, Chapter II] gives Since the curvature operator of S N acts as the identity on 2-forms, we obtain by formula (4.37) in [12, Chapter II] (also compare [14, Lemma 4 Putting everything together, it follows that For each pair j ̸ = k, Clifford multiplication by (e We prove a theorem which relates the index on a manifold with boundary with that of the boundary.We only need it for twisted spinorial Dirac operators but since it may be of independent interest, we show it for the larger class of self-adjoint Dirac-type operators. Let M be a compact Riemannian manifold with boundary ∂M with outward unit normal vector field ν.Let S → M be a Hermitian vector bundle.Let D be a differential operator of first order.Its principal symbol is characterized by D(f s) = f Ds + σ D (df )s.We say that D is of Dirac type if its principal symbol satisfies the Clifford relations for all ξ, η ∈ T * x M and x ∈ M .In particular, if D is of Dirac type, then D is elliptic.All generalized Dirac operators in the sense of Gromov and Lawson are of Dirac type.
We assume that the restriction of S to the boundary ∂M splits into two orthogonal subbundles, for all ξ ∈ T * ∂M .Here ν ♭ is the 1-form metrically related to ν.
follows from Corollary 7.23, Proposition 7.24, and Theorem 7.17 combined with Corollary 8.6 in [1].Note that the completeness and coercivity at infinity required in [1] is automatic in our situation as M is compact.
Since σ D ν ♭ preserves the bundles S + and S − , the boundary conditions s ∈ S + and s ∈ S − are adjoint to each other.Corollary 8.6 in [1] In particular, this proves (B.2).Elliptic operators on compact manifolds without boundary are always Fredholm.Since D ∂ is elliptic and formally self-adjoint, its L 2 -closure is self-adjoint.Since the L 2 -closures of the restrictions C ∞ ∂M, S ± → C ∞ ∂M, S ∓ are adjoints of each other, we obtain (B.4).It remains to prove (B.3).
Let V > and V < denote the subspaces of the Sobolev space H 1 2 (∂M, S) spanned by the eigenspaces of D ∂ corresponding to the positive or negative eigenvalues, respectively.Let H ⊂ C ∞ (∂M, S) denote the kernel of D ∂ .Since D ∂ interchanges S + and S − , we may decompose H = H + ⊕ H − , where H ± := H ∩ C ∞ ∂M, S ± .This gives an L 2 -orthogonal decomposition Let σ denote the self-adjoint bundle involution on S with the property that S ± are the eigenspaces to the eigenvalues ±1.Since D ∂ interchanges S + and S − , it anti-commutes with σ.Thus, σ maps the eigenspace of D ∂ for the eigenvalue λ isomorphically onto that of −λ.Now any s ∈ V > ⊕ V < can be expanded into eigensections, s = λ̸ =0 s λ .If furthermore s ∈ H and therefore σs λ = s −λ .Thus, (V > ⊕ V < ) ∩ H 1 2 ∂M, S + is the graph of the map σ : V > → V < .We introduce a deformation parameter t ∈ [0, 1] and consider the graph of tσ.More precisely, we put B t := H + ⊕{s+tσs : s ∈ V > }.Each B t is an ∞-regular elliptic boundary condition for D in the sense of [1].By deformation invariance of the index, ind(D, B 1 ) = ind(D, B 0 ).In other words, ind(D : C ∞ + (M, S) → C ∞ (M, S)) coincides with the index of D subject to the boundary condition H + ⊕ V > .
We observe that H + ⊕ V > is a finite-dimensional modification of the Atiyah-Patodi-Singer boundary condition V > .Since σ ν ♭ anti-commutes with D ∂ , the adjoint boundary condition of V > is σ ν ♭ V > ⊥ = V ⊥ < = H ⊕ V > where ⊥ denotes the L 2 -orthogonal complement in H This concludes the proof.■ The following consequence is known as cobordism invariance of the index: Hence, s| ∂M = 0.
We show that this implies s = 0 on all of M .By adding a small collar neighborhood to M along ∂M we embed M into an open manifold M .We extend the bundle S and the Dirac-type operator D to M .We extend s by zero to M and obtain a continuous section s.Let ϕ be a compactly support test section on M .Then We define a bundle S + over ∂M so that the fiber of S + at a point x ∈ N j is the eigenspace of iσ D ν ♭ with eigenvalue ε j .Similarly, we define a bundle S − over ∂M so that the fiber of S − at a point x ∈ ∂M is the eigenspace of iσ The result now follows from Theorem B.1.■

•
Let ind 1 denote the index of the Dirac operator on S ⊗ f * E + 0 with boundary conditions u = −iν • u on ∂ + M and u = iν • u on ∂ − M .• Let ind 2 denote the index of the Dirac operator on S ⊗ f * E + 0 with boundary conditions u = iν • u on ∂ + M and u = −iν • u on ∂ − M .• Let ind 3 denote the index of the Dirac operator on S ⊗ f * E − 0 with boundary conditions u = −iν • u on ∂ + M and u = iν • u on ∂ − M .• Let ind 4 denote the index of the Dirac operator on S ⊗ f * E − 0 with boundary conditions u = iν • u on ∂ + M and u = −iν • u on ∂ − M .
index 0. To obtain a contradiction, we compute the index of the boundary Dirac operators using the Atiyah-Singer index theorem.Denote the total Â-class of ∂ − M by Â ∂ − M .The Chern character of the bundle f | ∂ − M * E + 0 is given by the pull-back of ch E + 0 under f | ∂ − M .In particular, the Chern character of the bundle f | ∂ − M * E + 0 only contains terms in the 0-th and n-th cohomology groups.Since the boundary Dirac operator which maps sections of S

Proposition 2 .
10 (cf.Cecchini-Zeidler[5]). Consider the indices of the following operators:• Let ind 1 denotethe index of the Dirac operator on S ⊗ φ * E + 0 with boundary conditions s = −iν • s on ∂ + M and s = iν • s on ∂ − M .• Let ind 2 denote the index of the Dirac operator on S ⊗ φ * E + 0 with boundary conditions s = iν • s on ∂ + M and s = −iν • s on ∂ − M .• Let ind 3 denote the index of the Dirac operator on S ⊗ φ * E − 0 with boundary conditions s −iν • s on ∂ + M and s = iν • s on ∂ − M .• Let ind 4 denote the index of the Dirac operator on S ⊗ φ * E − 0 with boundary conditions s = iν • s on ∂ + M and s = −iν • s on ∂ − M .

1
If D is of Dirac type then so is D ∂ .If D ∂ interchanges the subbundles, i.e., D ∂ : C ∞ ∂M, S ± → C ∞ ∂M, S ∓ , then we call D ∂ an odd operator.We denote by C ∞ ± (M, S) the space of all sections s of S which are smooth up to the boundary and satisfy s(x) ∈ S ± x for all x ∈ ∂M .Theorem B.1.Let D be a formally self-adjoint Dirac-type operator and let D ∂ be a formally self-adjoint odd operator adapted to D. Assume that σ D ν ♭ preserves the bundles S + and S − and anti-commutes with D ∂ .Then the operators D : C ∞ ± (M, S) → C ∞ (M, S) and D ∂ : C ∞ ∂M, S ± → C ∞ ∂M, S ∓ are Fredholm and their indices satisfy ind(D : C

Corollary B. 2 .
In addition to the assumptions in Theorem B.1 assume that S ± are the eigensubbundles of S| ∂M of the involution iσ D ν ♭ for the eigenvalues ±1.Then S| ∂M = S + ⊕ S − and the indices occurring in Theorem B.1 vanish.Proof .Without loss of generality we can assume that M is connected.If ∂M = ∅, then the assertion is obvious.Therefore, we assume ∂M ̸ = ∅.We claim that the operators D :C ∞ ± (M, S) → C ∞ (M,S) have trivial kernel.To see this, suppose that s ∈ C ∞ ± (M, S) → C ∞ (M, S) satisfies Ds = 0. Since D is formally self-adjoint, we obtain 0 = M ⟨Ds, s⟩ − M ⟨s, Ds⟩ = ∂M σ D ν ♭ s, s = ∓i ∂M |s| 2 .