Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the Hodge Laplacian on differential forms, and discuss similarities and differences of this new ``magnetic-type'' operator.


Introduction and statement of results
The classical magnetic Laplacian on a Riemannian manifold (M n , g) associated to a smooth real 1-form α ∈ Ω 1 (M ) acts on the space of smooth complex-valued functions C ∞ (M, C) and is given by where d α := d M + iα and δ α := δ M − i⟨α ♯ , •⟩ (note that δ M is the L 2 -adjoint of d M ).Here α ♯ ∈ X (M ) is the vector field corresponding to the 1-form α via the musical isomorphism ⟨α ♯ , X⟩ = α(X).The 1-form α is called the magnetic potential and d M α is the magnetic field.The magnetic Laplacian ∆ α can be viewed as a first order perturbation of the usual Laplacian ∆ M = δ M d M , namely for any f ∈ C ∞ (M, C), In the case of a closed manifold or a compact manifold with boundary, both operators ∆ M and ∆ α (with suitable boundary conditions when ∂M ̸ = ∅) have a discrete spectrum with non-decreasing eigenvalues with multiplicity denoted by (λ k (M )) k∈N and (λ α k (M )) k∈N , respectively.There are very few Riemannian manifolds where the complete set of eigenvalues can be given explicitly.Amongst them is the unit round sphere S n with the standard metric g, whose eigenfunctions can be described as spherical harmonics.In Appendix A, we give an explicit derivation of the spectrum of a magnetic Laplacian on (S 3 , g) with a special magnetic potential α.This derivation is based on the Hopf fibration S 1 → S 3 → S 2 , and α is a constant magnetic field along the S 1 -fibers.
In analogy with the generalization of the usual Laplacian ∆ M on functions to the Hodge Laplacian δ M d M + d M δ M on differential forms, it is natural to generalize the magnetic Laplacian on functions to complex differential forms as follows.On the set of complex-valued differential p-forms Ω p (M, C), we define where d α := d M + iα∧ and δ α := δ M − iα ♯ ⌟ is its formal adjoint.Both d α and δ α can also be expressed via the magnetic covariant derivative ∇ α X Y := ∇ M X Y + iα(X)Y for any X, Y ∈ C ∞ (T M ⊗ C) (see formula (3.1)).We refer to this operator ∆ α acting on Ω p (M, C) as the magnetic Hodge Laplacian on complex p-forms.
We establish the following results for the magnetic Hodge Laplacian on an oriented Riemannian manifold (M n , g): (a) We show that the magnetic Hodge Laplacian commutes with the Hodge star operator (see Corollary 3.2).
(b) We derive a magnetic analogue of the classical Bochner-Weitzenböck formula (see Theorem 3.4).
(c) We prove gauge invariance of the magnetic Laplacian on forms ∆ α (see Corollary 3.8).
(d) We obtain a Shigekawa-type result (see Theorem 3.9) for the magnetic Hodge Laplacian ∆ α on a closed Riemannian manifold M in the case where M has a parallel p-form and α is a Killing 1-form (for the original statement, see [29]).
(e) Following a result by Gallot-Meyer [13] for the Hodge Laplacian, we derive a lower bound for the first eigenvalue of the magnetic Hodge Laplacian for closed manifolds (see Theorem 4.2).
(f) Following a result by Colbois-El Soufi-Ilias-Savo [6] for the magnetic Laplacian on functions, we derive an upper bound for the first eigenvalue of the magnetic Hodge Laplacian for closed manifolds (see Theorem 4.3).
(g) We show that in general the diamagnetic inequality does not hold for magnetic Hodge Laplacians (Corollary 4.6).In fact, we give a counterexample which is based on the calculations in Appendix A. In addition, we give an explicit characterization which determines when the diamagnetic inequality holds for ∆ tξ with ξ a Killing vector field (see Corollary 4.5).
(h) Following the work of Raulot-Savo in [26], we derive a Reilly formula for the magnetic Hodge Laplacian on Riemannian manifolds with boundary (see Theorem 5.1) and use it to derive a lower bound for the first eigenvalue of the magnetic Hodge Laplacian on an embedded hypersurface of a Riemannian manifold (see Theorem 6.2).
(i) Following the work of Guerini-Savo in [14], we derive a "gap" estimate between the first eigenvalues of consecutive p-values of the magnetic Hodge Laplacians on Ω p (M, C) for isometrically immersed manifolds (M n , g) in Euclidean space R n+m (see Theorem 6.3).
Acknowledgment: The third named author thanks Durham University for its hospitality during his stay.He also thanks the Alexander von Humboldt foundation and the Alfried Krupp Wissenschaftskolleg in Greifswald.

Review of the magnetic Laplacian for functions
Before we introduce the magnetic Hodge Laplacian in the next section, we first recall some results for the classical magnetic Laplacian on functions.Let (M n , g) be a Riemannian manifold and α ∈ Ω 1 (M ).The magnetic Laplacian ∆ α acting on complex-valued smooth functions defined by formula (1.1) has the property of gauge invariance, that is ∆ α (e if ) = e if ∆ α+d M f for any smooth real-valued function f .When M is compact (with or without boundary), the spectrum of ∆ α (or with suitable boundary conditions when ∂M ̸ = ∅) is discrete.Therefore, by the gauge invariance, the spectrum of ∆ α is equal to the spectrum of ∆ α+d M f .Thus, when α is exact, the spectrum of ∆ α reduces to that of the usual Laplace-Beltrami operator.In [8,Prop. 3], it is proven that one can always assume that α is a co-closed 1-form (and tangential, i.e. ν⌟α = 0, when M has a boundary) without changing the spectrum of ∆ α .Moreover, by using the Hodge decomposition on compact manifolds, the authors show in [6, Prop.1] that one can further consider α to be of the form where ψ is a 2-form on M (with ν⌟ψ = 0 when ∂M ̸ = ∅), and h is a harmonic 1-form on M , that is, d M h = δ M h = 0 (with ν⌟h = 0 when ∂M ̸ = ∅), and again the spectrum does not change.Here, we point out that the first eigenvalue λ α 1 (M ) of ∆ α is not necessarily zero like for the usual Laplacian ∆ M as shown in [29,Ex. 1].This interesting property of the magnetic Laplacian was characterized by Shigekawa (see [29,Prop. 3.1 and Thm. 4.2]) as follows.
Theorem 2.1 (Shigekawa).Let (M n , g) be a closed Riemannian manifold and Then the following are equivalent: Hence, when α cannot be gauged away, meaning that α does not belong to the set B M , the first eigenvalue is necessarily positive.This gauge invariance can be described by the following: If α τ ∈ B M for some τ ∈ C ∞ (M, S 1 ), the Laplacians ∆ α and ∆ α+ατ are unitarily equivalent, that is Thus ∆ α and ∆ α+ατ have the same spectrum as stated before.Now, the diamagnetic inequality compares the first eigenvalue of ∆ α to the one for the Laplacian ∆ M and says that λ α 1 (M ) ≥ λ 1 (M ), with equality if and only if the magnetic potential α can be gauged away.When M has no boundary, the diamagnetic inequality provides no information since λ 1 (M ) = 0.However, when we consider manifolds with boundary and the magnetic Laplacian is associated to the Dirichlet or Robin boundary conditions, the diamagnetic inequality still holds and tells us that the first eigenvalue λ α 1 (M ) is always positive.
A simple estimate for the first eigenvalue of the magnetic Laplacian can be deduced straightforwardly from the min-max principle.Indeed, when applying the Rayleigh quotient to a constant function, we get, after choosing δ M α = 0, that Several papers have been devoted to estimating the first eigenvalue of the magnetic Laplacian, see, for example, [2,5,12,16,19,20,8,9,10,7,11,6].Among these results, we quote two of them [11], [6] on closed Riemannian manifolds.
The first result gives magnetic Lichnerowicz-type estimates for the first two eigenvalues: then we have where .
The technique used to obtain this result is an integral Bochner-type formula which involves the magnetic Hessian that is associated to the magnetic covariant derivative ∇ α .A related result to Theorem 2.2 for the magnetic Laplacian with Robin boundary conditions on compact Riemannian manifolds (M, g) with smooth boundary was proved in [15].In the setup of the above theorem, it is natural to ask whether the estimates are sharp for some α that is not gauged away.For this, we employ the example of the round sphere S 3 where the magnetic field α is collinear to the Killing vector field that defines the Hopf fibration.We refer to Appendix A for more details on the computation.
Example (Unit sphere S 3 with α = tY 2 ).Let (S 3 , g) be the unit sphere in R 4 with standard metric g of curvature 1.We use the notation introduced in Appendix A. Let α = tY 2 where Y 2 is the unit Killing vector field on S 3 .Using (A.6), we obtain On the other hand, we conclude from (A.9) that λ α 1 (S 3 ) = t 2 and λ α 2 (S 3 ) = 3 − 2t + t 2 for small t ∈ [0, t max ].The relations between these two smallest eigenvalues and their estimates for small t > 0 are illustrated in Figure 1.As we can see from Figure 1, sharpness of the upper estimate of λ α 1 (S 3 ) is lost (see the discussion after Lemma 4.1).
The second result was given in [6] in the general setting of magnetic Schrödinger operators ∆ α + q with Neumann boundary conditions.For simplicity, we formulate it in the special case of a closed Riemannian manifold (M n , g) with vanishing potential q = 0. We will return to this estimate later in Subsection 4.2.
Theorem 2.3 ([6, Thm.2]).Let (M n , g) be a closed Riemannian manifold and let α ∈ Ω 1 (M ) be of the form α = δ M ψ + h with ψ ∈ Ω 2 (M ) and h a harmonic 1-form.Then, , where λ ′′ 1,1 (M ) is the first eigenvalue of the Hodge Laplacian ∆ M on co-exact 1-forms, L Z is the lattice of integer harmonic 1-forms in Ω 1 (M ), and In order to check the sharpness of this inequality, we consider again the case of the round sphere with the magnetic field given by the Killing vector field.
Finally, as we mention in the introduction, examples of closed Riemannian manifolds (M n , g) with non-trivial magnetic potential α ∈ Ω 1 (M ) (that is, magnetic potential which cannot be gauged away) for which the full spectrum of the magnetic Laplacian ∆ α can be explicitly given, are very scarce (see, for example, [8,7] for such computations).

The magnetic Hodge Laplacian for differential forms
In this section, we introduce the magnetic Hodge Laplacian for differential forms, prove a magnetic Bochner formula, and discuss its gauge invariance.Henceforth (M n , g) will denote an oriented n-dimensional Riemannian manifold and Ω p (M ) and Ω p (M, C) will denote the spaces of real and complex differential p-forms for 0 ≤ p ≤ n.The spaces of real and complex vector fields on M are denoted by X (M ) and X C (M ).To simplify notation, we will often identify real and complex vector fields with real and complex 1-forms via the (complex-linear) musical isomorphisms.That is, Ω 1 (M, C) → X C (M ); ω → ω ♯ given by ω(X) = ⟨X, ω ♯ ⟩, where ⟨•, •⟩ stands for the Hermitian scalar product extended from the Riemannian metric g to X C (M ).

The magnetic Hodge Laplacian
Fix a smooth 1-form α ∈ Ω 1 (M ) (a magnetic potential) and consider the magnetic differential on Ω p (M, C), given by It is not difficult to check that the L 2 -adjoint of d α acting on complex differential forms (when M is without boundary) w.r.t. the Hermitian inner product where δ M = (−1) n(p+1)+1 * d M * is the formal adjoint of d M on p-forms (both extended complex linearly to complex differential forms) and the Hodge star operator is extended to a complex linear operator * : Ω p (M, C) → Ω n−p (M, C).
Recall here that the interior product "⌟" is the pointwise adjoint of the wedge product "∧".Both d α and δ α are the differential and co-differential associated to the magnetic connection on differential forms ∇ α X := ∇ M X + iα(X) on Ω p (M, C).That means we have where {e 1 , . . ., e n } is a local orthonormal frame of T M .Now, we define the magnetic Hodge Laplacian acting on Ω p (M, C) as follows: We first have the following observation: Proof.The proof is straightforward from the fact that * d M = (−1) p+1 δ M * and * (α∧) = (−1) p α ♯ ⌟ * on p-forms.Also, we have that * δ M = (−1) p d M * and The following is an immediate consequence of Lemma 3.1 above.
Proof.Indeed, on p-forms, we have The magnetic Laplacian ∆ α has the same principal symbol as the Hodge Laplacian ∆ M (see Equation (3.10) in the next section), since it differs by lower order terms.Therefore, it is an elliptic, essentially self-adjoint operator acting on smooth complex forms on a closed oriented Riemannian manifold or acting on smooth complex forms with Dirichlet boundary condition on an oriented Riemannian manifold with boundary (see Subsection 5.1 below).Therefore, ∆ α has a discrete spectrum consisting of nonnegative eigenvalues (λ α j,p (M )) j∈N , denoted in ascending order with multiplicities.Moreover, as for the usual Hodge Laplacian, its spectrum on p-forms is the same as the one on (n − p)-forms and the first eigenvalue is characterized by where ω runs over all smooth p-forms with ω| ∂M = 0, if ∂M ̸ = ∅.We also note that the differential d α does not satisfy the crucial property d α • d α = 0 to introduce cohomology groups.In fact, we have where d M α ∈ Ω 2 (M ) is the magnetic field.We could, however, still define magnetic Betti numbers via . Moreover, we have b α 0 (M ) = b α n (M ) = 0 for any magnetic potential α that cannot be gauged away, that is α / ∈ B M , by the diamagnetic inequality.In Theorem 3.9, we investigate the existence of closed Riemannian manifolds (M n , g) with a magnetic potential α that cannot be gauged away, for which some of the corresponding magnetic Betti numbers b α k (M ), 1 ≤ k ≤ n − 1, are non-zero.

A magnetic Bochner formula
Recall that the Hodge Laplacian ∆ M := d M δ M +δ M d M is related to the Bochner Laplacian on M via a curvature term by the Bochner-Weitzenböck formula.Namely, we have (see, e.g, [24,Thm. 7.4.5]or [31, p. 14]) where B [p] , called the Bochner operator, is a symmetric endomorphism on Ω p (M ) given by B [p] = n j,k=1 e * k ∧ e j ⌟R M (e j , e k ).Here R M is the curvature operator associated to the Levi-Civita connection ∇ M which is given by R for all X, Y ∈ X (M ) and {e 1 , . . ., e n } is a local orthonormal frame of T M .The Bochner Laplacian ∇ * ∇ is given by In the following, we derive a similar magnetic Bochner-Weitzenböck formula for ∆ α , which will provide a relation between the Hodge Laplacians ∆ α and ∆ M .For this, we recall the following definition.Given a Euclidean vector space V of dimension n and an endomorphism A : V → V , there exists a canonical extension A [p] of A on the set of differential p-forms (p ≥ 1) given by for v 1 , . . ., v p ∈ V .By convention, we take A [0] = 0.One can easily show from the definition that the endomorphism A [p] can be written in terms of A as where {e 1 , . . ., e n } is an orthonormal frame of V .If A is a symmetric (resp.skew-symmetric) endomorphsim on V , then so is A [p] on Λ p (V * ).In this case, if we denote the eigenvalues of A by η 1 ≤ . . .≤ η n , then we have the following estimates.For any ω ∈ Λ p (V * ) where σ p := η 1 + . . .+ η p are called the p-eigenvalues of A [p] and ∥A∥ is the operator norm of A. In order to state the magnetic Bochner-Weitzenböck formula, we introduce the following magnetic Bochner operator on Ω p (M, C): where as before {e i } i=1,...,n is a local orthonormal frame of T M .Here R α is the curvature operator associated to the magnetic covariant derivative ∇ α , that is . Now, we express the magnetic Bochner operator in terms of the usual one by the following lemma.
Lemma 3.3.On the set of complex differential p-forms, the magnetic Bochner operator B [p],α is equal to where A [p],α is the canonical extension to complex p-forms of the skew-symmetric endomorphism A α on T M given by A α (X) = (X⌟d M α) ♯ for any vector field X on M .
Proof.An easy computation shows that, for any X, Y ∈ X (M ) and ω ∈ Ω p (M, C), The proof can then be deduced from the definition of B [p],α and the fact that A α is skew-symmetric.
We make the following observation.Using the identity on p-forms * (X ♭ ∧) = (−1) p X⌟ * valid for any vector field X, one can easily show that where * is the Hodge star operator on M and ⟨•, •⟩ is the pointwise Hermitian product on Ω p (M, C).In the same way, and since the endomorphism A α is skew-symmetric, one can also show that A on complex p-forms.Notice here that iA [p],α is a symmetric endomorphism on Ω p (M, C).Now we formulate the magnetic Bochner-Weitzenböck formula.
Theorem 3.4 (Magnetic Bochner-Weitzenböck formula).Let (M n , g) be a Riemannian manifold and α ∈ Ω 1 (M ).Then we have where ej .Moreover, we have Proof.The proof follows the same computations as for the Hodge Laplacian ∆ M .For this, we use the expressions of d α and δ α in (3.1) on an orthonormal frame {e j } n j=1 on T M chosen in a way that ∇ M e j = 0 at some point x ∈ M .By the fact that, for all X, Y ∈ X C (M ), we have •, which can be proven by a straightforward computation (the same relation holds for the interior product), we can write at x ∈ M : e * k ∧ (e j ⌟R α (e j , e k )), where in the fourth equality we used the relation for any differential form β. This shows that (3.9) holds.To obtain (3.10), we just combine Lemma 3.3 with the Bochner-Weitzenböck formula (3.4) and the fact that at x ∈ M Remark.Formula (3.10) is a generalization of the formula for the magnetic Laplacian for functions, given by Now, we will consider a particular case for the magnetic field α.We will assume that it is a Killing 1-form, that is its corresponding vector field α ♯ by the musical isomorphism is a Killing vector field.In this case, the standard Hodge Laplacian ∆ M commutes with L α since it commutes with all isometries.Indeed, we will show that, when α is of constant norm, the exterior differential d M and codifferential δ M both commute with the magnetic Laplacian.Notice here that, in general, d α and δ α do not commute with ∆ α as a consequence of (3.3) and even when α ♯ is Killing.We now show that Equation (3.10) has the simpler expression (3.11) in this case.We also recall that for simplicity α and α ♯ are identified throughout the paper.Proposition 3.5.Let (M n , g) be a Riemannian manifold and let α be a Killing where L α is the Lie derivative in the direction of α.In particular, and, therefore, the magnetic Laplacian preserves the set of exact and co-exact forms.
Proof.The fact that α is Killing gives A α (X) = X⌟d M α = 2∇ M X α for any vector field X ∈ T M .Therefore, we get by (3.6) that where T [p],X is the canonical extension of the endomorphism T X = ∇ M X, for any X, given by the expression in (3.6).Now, the identity L X = ∇ M X + T [p],X valid on p-forms for any vector field X on T M [27, Lem.2.1] allows us to deduce that Hence, Equation (3.10) and the fact that δ M α = 0 since α is Killing gives the desired identity (3.11).In order to prove that L α commutes with ∆ α , we first use α(|α| 2 ) = 2g(∇ M α α, α) = 0 which is a consequence of the fact that α is Killing.Now, we compute, for any p-form ω, Thus, by the fact that L α commutes with the Laplacian ∆ M , we get that L α ∆ α = ∆ α L α .Now we assume |α| is constant.It follows from Cartan's formula L X ω = X⌟d M ω + d M (X⌟ω) that L X commutes with d M for any vector field X.Since d M commutes with ∆ M and with L α as well as with multiplication by the constant |α| 2 , we deduce that d M commutes with ∆ α .That the codifferential δ M commutes with ∆ α comes from the fact that δ M commutes with ∆ M and with L α , which is a consequence of δ M = ± * d M * and L α * = * L α by Equation (3.12).(Recall here that A [p],α * = * A [n−p],α ).This finishes the proof.
Remark.The relation (3.12) shows that for any complex differential forms ω and ω ′ on M , the following relation holds pointwise when α is a Killing vector field (not necessarily of constant norm), since A [p],α is skew-symmetric.
When the magnetic potential α is Killing of constant norm on (M n , g), we have seen that the magnetic Laplacian ∆ α preserves the set of exact and co-exact forms on M .In the following, we will assume M to be compact and will let λ α 1,p (M ) be the first non-negative eigenvalue of ∆ α on differential p-forms and λ α 1,p (M ) ′ (resp.λ α 1,p (M ) ′′ ) be the first non-negative eigenvalue restricted to exact (resp.co-exact) p-forms.As in the standard case [26], we can prove by Hodge duality that Recall here that the magnetic Laplacian commutes with the Hodge star operator.However, we will see in the next proposition, that the relation λ α 1,p (M ) = min(λ α 1,p (M ) ′ , λ α 1,p (M ) ′′ ) that usually holds for the Laplacian ∆ M is not always true for ∆ α .
For the next proposition, we need the following well known result, which we present for completeness.Lemma 3.6.Let (M n , g) be a compact manifold and let X be a Killing vector field on M .For any harmonic form ω ∈ Ω(M ) we have Proof.Let ω ∈ Ω(M ) be harmonic.Using Cartan's formula, we see that L X ω is exact.Moreover, since the Lie derivative of a Killing vector field commutes both with d M and δ M , the Lie derivative L X ω is both exact and harmonic.Therefore, by Hodge decomposition, L X ω = 0. Proposition 3.7.Let (M n , g) be a compact Riemannian manifold and let α be a Killing 1-form.The first non-negative eigenvalue Proof.Let ω be a complex p-eigenform of the magnetic Hodge Laplacian associated to the first eigenvalue λ α 1,p (M ).By the Hodge decomposition, we write where From the equation ∆ α ω = λ α 1,p (M )ω, by uniqueness of the decomposition and the fact that both d M and δ M commute with ∆ α , we obtain the relation ∆ α ω 2 = λ α 1,p (M )ω 2 .Now, if ω 2 does not vanish, then by the fact that α is Killing and ω 2 is harmonic, we have by Lemma 3.6 that L α ω 2 = 0. Thus, by Equation (3.11), we get that ∆ α ω 2 = |α| 2 ω 2 and, therefore, . When H p (M ) ̸ = 0 then there is a non-vanishing p-harmonic form ω on M and thus, as before, ∆ α ω = |α| 2 ω.Thus, by the min-max principle we deduce the required estimate.This finishes the proof.
Example.As in the previous examples, consider the manifold M = S 3 equipped with the standard metric of curvatue 1.Let Y 2 be the unit Killing vector field as in Appendix B. It follows that the 1-forms d M u, d M v and α = tY 2 are all simultaneous eigenforms of the operators ∆ α such that Moreover, d M u, d M v are exact eigenforms associated to the smallest eigenvalue λ ′ 1,1 (M ) = 3 and α is a co-exact eigenform associated to the smallest eigenvalue λ ′′ 1,1 (M ) = 4 (see [23]).Therefore, we have for small t > 0, since H 1 (M ) = 0. On the other hand, we get by Equation (A.9) that for small t > 0, λ α 1,0 (M ) = t 2 = |α| 2 .However, we have that

Gauge invariance of the magnetic Hodge Laplacian
Another consequence of the magnetic Bochner-Weitzenböck formula (3.10) is the following result.
In particular, ∆ α and ∆ α+ατ have the same spectrum on a closed oriented Riemannian manifold.
Proof.The proof relies mainly on the following identity.For any f ∈ C ∞ (M, C) and ω ∈ Ω p (M, C), we have Hence, for f = τ ∈ C ∞ (M, S 1 ), we use Equation (3.10) to compute Taking the divergence of d M τ = iτ α τ , we get that Hence, Equation (3.14) reduces to In the second equality, we used the fact that A α = A α+ατ since α τ is a closed form.This allows us to deduce the result.
The gauge invariance of the magnetic Laplacian allows us to state a Shikegawa type result for differential forms.Theorem 3.9.Let (M n , g) be a compact Riemannian manifold and let α be a one-form on M .Assume that M carries a non-zero parallel p-form ω 0 on M .Then we have the following: (a) If α ∈ B M , then λ α 1,p (M ) = 0 and there exists an eigenform ω of ∆ α associated with the eigenvalue λ α 1,p (M ) such that f := ⟨ω, ω 0 ⟩ is nowhere vanishing.
(b) Conversely, assume that α is Killing.If λ α 1,p (M ) = 0 and there exists an eigenform ω of ∆ α associated with the eigenvalue λ α 1,p (M ) such that f := ⟨ω, ω 0 ⟩ is not vanishing, then α ∈ B M and, in this case, it is a parallel form.
Proof.We first prove (a).Since α = d M τ iτ ∈ B M for some τ ∈ C ∞ (M, S 1 ), we deduce from Corollary 3.8 that the magnetic Laplacian has the same spectrum as the Hodge Laplacian ∆ M .Hence the first eigenvalue λ α 1,p (M ) is equal to 0 due to the existence of a parallel form ω 0 which gives that d M ω 0 = δ M ω 0 = 0.Moreover, one can easily check that the form ω := τ ω 0 satisfies In the same way, we prove that δ α ω = 0. Therefore, we have ∆ α ω = 0. Hence the function f = ⟨ω, ω 0 ⟩ = τ |ω 0 | 2 is nowhere zero since τ ∈ S 1 and the parallel form ω 0 is of constant norm.Now, we prove (b).For this, we assume that α is Killing and we compute the Laplacian of the function f .We choose a local orthonormal frame {e i } of T M such that ∇ M e i | x = 0 at some point x.Since the form ω 0 is parallel, we write In this computation, we used the fact that B [p] ω 0 = 0 since ω 0 is parallel, and also that L α ω 0 = 0 by Lemma 3.6.Therefore, Equation (1.2) and divα ♯ = 0 (since α is Killing) allows us to deduce that ∆ α f = 0 and, therefore λ α 1 (M ) = 0. Now, the classical Shikegawa's result (Theorem 2.1) allows us to get that α ∈ B M which is also equivalent to the fact that d M α = 0 and c α ∈ 2πZ for all closed curves c in M .Now, the condition d M α = 0 means that ∇ M α is a symmetric two-tensor which is also skew-symmetric by the fact that α is Killing.Hence, the form α is parallel.
Remark.We know from Lemma 3.6 that, on a compact manifold (M n , g), for any harmonic form ω and a Killing one-form α, we have that L α ω = 0.However, there are ∆ α -harmonic forms for which this fact no longer holds.Indeed, assume that M carries a Killing one-form α which is also in B M , that is α = d M τ iτ (for instance, such forms exist on the flat torus) and hence parallel by the same arguments as in the above proof.Assume also that a non-zero parallel p-form ω 0 exists on M .We have seen from the proof of Theorem 3.9 that ω = τ ω 0 is a ∆ α -harmonic form.Now, we compute since α is parallel and, hence, is of constant norm.
We illustrate Theorem 3.9 with two examples.

Examples.
(a) The flat torus T n is trivialized by parallel p-forms for any p.Hence, one can always find, for any non-trivial differential form ω, a parallel form ω 0 such that f = ⟨ω, ω 0 ⟩ is not vanishing.Let α be any Killing one-form, we get (b) Let us consider the product manifold M = S 1 × S 3 with the product metric.
For A ∈ R, we let α = Aω 0 be the one-form on M , where ω 0 := dθ is the parallel unit one-form on S 1 .It is not difficult to check that α ∈ B S 1 ×S 3 if and only if A ∈ Z.We show When A ∈ Z, the spectrum of ∆ α is the same as the spectrum of ∆ M , and hence λ α 1,1 (M ) = 0 due to the existence of a parallel one-form.For the converse, assume that λ α 1,1 (S 1 × S 3 ) = 0 and that A / ∈ Z.Hence, α / ∈ B S 1 ×S 3 and by Theorem 3.9, we obtain that f = ⟨ω, ω 0 ⟩ = 0 for any eigenform ω associated to λ α 1,1 (M ).Therefore, if we consider an orthonormal frame on {ξ, e 1 , e 2 } on T S 3 such that ξ is the unit Killing vector field that defines the Hopf fibration with ∇ S 3 e1 ξ = e 2 and ∇ S 3 e2 ξ = −e 1 (since the complex structure on S 2 is given by J(X) = ∇ S 3 X ξ), we write where f 0 , f 1 , f 2 are smooth functions on S 1 ×S 3 .Now, the condition d α ω = 0 allows us to get that ∂f k ∂θ = −iAf k for k = 0, 1, 2, which gives that f k = g k e −iAθ with functions g k which are constant on S 1 .However, the functions f k are only periodic functions on S 1 when A ∈ Z, which is a contradiction.

Eigenvalue estimates for the magnetic Hodge Laplacian on closed manifolds
In this section, we establish several eigenvalue estimates for the magnetic Hodge Laplacian on a closed oriented Riemannian manifold (M n , g).In particular, we show that the diamagnetic inequality cannot hold in general.

A magnetic Gallot-Meyer estimate
The aim of this subsection is to derive a lower bound for the first eigenvalue of the magnetic Hodge Laplacian on p-forms that is analogous to that of Gallot-Meyer.We begin with the following lemma similar to [13, Lem.6.8], relating the magnetic connection to the magnetic differential and co-differential.
Lemma 4.1.Let (M n , g) be a Riemannian manifold and let α be a magnetic potential.For any complex differential p-form ω with p ≥ 1, we have Proof.The proof relies on defining the magnetic twistor form as in the usual case: For any complex p-form ω and vector field X ∈ X C (M ), we define Using Equation (3.1), the norm of P α is equal to Here we use the fact that any complex p-form β on M can be written as β = Applying Inequality (4.1) to the 1-form ω := d α f , where f is a smooth complex-valued function, we get that If the equality is attained, then (d α )2 f = 0 which, by (3.3), is equivalent to [11, p. 1147], we should have equality in the above inequality which means that necessarily d M α = 0.This explains why sharpness of the upper bound for λ α 1 (M ) in (2.2) is lost.The next result now reads as a "magnetic version" of the Gallot-Meyer estimate [13, Thm.6.13].
Theorem 4.2.Let (M n , g) be a closed oriented Riemannian manifold, and let α be a smooth 1-form on M .Assume that B [p],α ≥ K for some K > 0 and p ≥ 1.Then, we have where C = max(p + 1, n − p + 1).
Proof.Let ω be a p-eigenform of ∆ α associated to the first eigenvalue λ α 1,p (M ).We apply the magnetic Bochner formula to ω, integrate it over M and use inequality (4.1) to obtain from which we deduce the desired inequality.
Remark.In view of Equality (3.8) and since the Hodge star operator commutes with the magnetic Laplacian ∆ α by Corollary 3.2, it is enough to consider p ≤ n Example.In order to check whether the condition B [p],α ≥ K required in the previous theorem can be satisfied for some K > 0, we will employ the example of the round sphere S n for some odd n = 2m + 1 where the magnetic field α is given by α = tξ, for t > 0, and ξ is the unit Killing vector field on S n that defines the Hopf fibration.Indeed, since on the round sphere B [p] = p(n − p), we get that B [p],α = p(n − p) − tiA [p],ξ .Now, as A ξ X = X⌟d M ξ = 2∇ M X ξ for any vector field X, we can always find an orthonormal basis of T S n such that the matrix of A ξ consists of the eigenvalue 0 and block matrices of type 0 ±2 ∓2 0 .The eigenvalue 0 corresponds to the eigenvector ξ and the block matrices come from the fact that ∇ M ξ is the complex structure on ξ ⊥ .Hence, in this basis, the eigenvalues of the symmetric matrix iA ξ are −2, 0, 2 with multiplicities n−1 2 , 1, n−1 2 respectively.An easy computation shows that the p-eigenvalues of the matrix iA ξ are equal to Recall here that n is odd.Hence the second inequality in (3.7) allows us to deduce that 2 .Thus, for t > 0, we deduce that Clearly, for any parameter t ≤ n−p 2 or p 2 , the number K is positive.Hence, Theorem 4.2 yields the following estimates for the first eigenvalue of the magnetic Laplacian ∆ α on S n with α = tξ,

A differential form analogue of a Colbois-El Soufi-Ilias-Savo estimate
In [6, Thm.2], the authors give an upper bound for the first Neumann eigenvalue of ∆ α defined on complex functions in terms of some distance function of harmonic 1-forms to a specific lattice and the norm of the magnetic field d M α for Riemannian manifolds with boundary.In the following, we prove a similar result in the setting of differential forms for closed oriented Riemannian manifolds (M n , g).Before we state the result, let us first introduce some relevant notation: We denote by m = b 1 (M ) the first Betti number and let c 1 , . . ., c m be a basis of H 1 (M, Z) and A 1 , . . ., A m ∈ H 1 (M ) be its dual basis, that is Let L Z be the lattice If H 1 (M ) = 0 we set L Z = 0. Note that, by Hodge Theory, we can think of L Z as a discrete subset of all real harmonic 1-forms.We now introduce the following distance functions for any real 1-form β ∈ Ω 1 (M ): When L Z = 0, the above distances reduce to ||β|| 2 or ||β|| ∞ .Now, we state the main result of this section.Theorem 4.3.Let (M n , g) be a closed Riemannian manifold and α ∈ Ω 1 (M ) be a magnetic potential of the form α = δ M ψ + h with h a harmonic 1-form and ψ a 2-form.Then we have the following eigenvalue estimate for the magnetic Hodge Laplacian on complex p-forms: where ω 0 is a real eigenform of the Hodge Laplacian ∆ M associated to the first eigenvalue λ 1,p (M ), and λ ′′ 1,1 (M ) denotes the first eigenvalue of the Hodge Laplacian on co-exact 1-forms.
Proof.The proof mainly follows the same lines as in [6].Firstly, we choose ω 0 to be a real p-form.Let η ∈ L Z , that is for some integers n 1 , . . ., n m ∈ Z.We fix x 0 ∈ M and define The r.h.s. is well defined and independent of the path from x 0 to x chosen, since x x0 η coincides for any pair of homotopic curves from x 0 and x and agrees up to a multiple of 2π for any arbitrary pair of paths from x 0 to x as η ∈ L Z .Then we have d M u = iuη.Therefore, for the p-form ω := uω 0 , we compute Similarly, Now we take the norms and use orthogonality of its real and imaginary parts to obtain and similarly Using the fact that |X ∧ ω| 2 + |X ♯ ⌟ω| 2 = |X| 2 • |ω| 2 for any vector field X, we add the above two equations and choose ω 0 to be an eigenform of the Hodge Laplacian to estimate Since η ∈ L Z was arbitrary, this proves Inequality (4.2).
For the proof of Inequality (4.3), recall that we have α = δ M ψ + h.Since harmonic 1-forms are L 2 -orthogonal to the forms in δ M (Ω 2 (M )), we have Since δ M ψ is co-exact, we have and therefore, This finishes the proof of the theorem.
Remark.The factor requires knowledge of the p-eigenform of the smallest eigenvalue.Under certain curvature conditions, it can be estimated from above as explained in [22].

The diamagnetic inequality does not hold for the magnetic Hodge Laplacian
A natural question is whether the diamagnetic inequality also holds for the magnetic Hodge Laplacian.That is, whether the inequality holds or not for some p ≥ 1.An example where the diamagnetic inequality holds is the flat n-dimensional torus M = T n .Clearly, the first eigenvalue λ 1,p (M ) = 0 for any p due to the existence of a parallel p-form.Hence the inequality λ α 1,p (M ) ≥ 0 = λ 1,p (M ) is satisfied.However, according to (3.15), the first eigenvalue λ α 1,p (M ) can be positive.In this subsection, we provide an example to show that the diamagnetic inequality does not hold in general.While this inequality is true for p = 0, we provide a counterexample for p = 1.We also give an explicit characterisation which determines whether this inequality holds for ∆ tξ where ξ is a Killing vector field.We start with the following estimate: Theorem 4.4.Let (M n , g) be a closed oriented Riemannian manifold and ξ ∈ Ω 1 (M ).Then, for any t ∈ R, we have, for α = tξ, where ω ∈ Ω p (M, C) is an eigenform of the Hodge Laplacian ∆ M (linearly extended to complex p-forms) associated with the eigenvalue λ 1,p (M ), and L X is the Lie derivative in the direction of the vector field X ∈ X (M ).In particular, if Im M ⟨L ξ ω, ω⟩dµ g is negative for some complex eigenform ω, then we get for small positive t that λ α 1,p (M ) < λ 1,p (M ), which means that the diamagnetic inequality does not hold.
Proof.Let ω be any p-form in Ω p (M, C).By the characterization of the first eigenvalue, we have for α = tξ eigenform, we deduce that Im M ⟨L ξ ω, ω⟩dµ g = β M |ω| 2 dµ g < 0. Therefore by Theorem 4.4 the diamagnetic inequality does not hold for small positive t.
To prove the second statement, we have from Relation (3.11) that ∆ α = ∆ M − 2itL ξ + t 2 |ξ| 2 holds for α = tξ.Also, we know from Proposition 3.5 that L ξ ∆ α = ∆ α L ξ .Therefore, the operator ∆ α preserves the space V as well as its orthogonal complement, by the fact that it is a self-adjoint operator.As ∆ M is perturbed analytically, the family (∆ tξ ) t is an analytic family of self-adjoint operators with compact resolvent and therefore the Hilbert basis of p-eigenforms of ∆ M and their corresponding eigenvalues can be extended analytically in the perturbation parameter t to a Hilbert basis of p-eigenforms of ∆ tξ and their corresponding eigenvalues (see [18,Thm. VII.3.9]).Since by assumption Ker(∆ M − λ 1,p (M )Id) ⊂ V and the fact that the spectrum is discrete (with finite dimensional eigenspaces), we deduce that λ 0 1,p (V ) = λ 1,p (M ), where λ 0 1,p (V ) denotes the lowest eigenvalue of ∆ M on p-forms, restricted to the invariant subspace V .Therefore, by choosing the analytic perturbation ω t of any basis element ω in the ∆ M -eigenspace corresponding to the eigenvalue λ 1,p (M ) and using the fact that ω t is of unit L 2 -norm, we obtain the estimate In the last inequality, we use the min-max principle for ∆ M .This implies that λ tξ 1,p (V ) ≥ λ 1,p (M ) for all t.The continuity of the maps t → λ tξ j,p (V ) and t → λ tξ j,p (M ) along with the fact that the eigenvalues are discrete and λ 0 1,p (V ) = λ 1,p (M ) imply that λ tξ 1,p (V ) = λ tξ 1,p (M ) for small t.Hence, we deduce that λ tξ 1,p (M ) ≥ λ 1,p (M ) for small |t|.
Below we consider the 3-dimensional round sphere and show that the diamagnetic inequality is not satisfied for a suitable choice of magnetic potential.For more details on the computation, we refer to Appendix A. Corollary 4.6.Let (M = S 3 , g) be the 3-dimensional unit sphere (centered at the origin) equipped with the canonical Riemannian metric g of curvature 1.Let ξ = Y 2 be the unit Killing vector field on S 3 as in Appendix A. Then, for small t > 0, we have, for α = tY 2 , which means that the diamagnetic inequality does not hold in general for differential 1-forms.
Proof.Hence, from Corollary 4.5, we just need to find a 1-eigenform of the Laplacian ∆ M which is not in the kernel of L ξ .For this, we use the computations done in Appendix B. Let (a, b), (z 1 , z 2 ) ∈ C 2 \(0, 0) and set Recall that ∆ M v = 3v and that 3 is the smallest eigenvalue of ∆ M associated to the 1-form ω := d M v. Hence, we compute In the last equality, we use the following consequence of the identity (A.2): Hence the result follows from Corollary 4.5.
5 The magnetic Hodge Laplacian on manifolds with boundary 5.1 A magnetic Green's formula for differential forms Let (M n , g) be a compact oriented Riemannian manifold with smooth boundary ∂M and let α ∈ Ω 1 (M ).We denote by ν the unit inward normal vector field to ∂M and by ι : ∂M → M the canonical injection.For any pair of complex differential forms ω 1 and ω 2 , the magnetic Stokes formula Here ι * is the pull-back of differential forms on M to the boundary.Indeed, it can be deduced from the usual Stokes formula and the expression of d α and δ α .As a consequence, we get Hence, we deduce that the magnetic Laplacian on smooth differential forms with Dirichlet boundary condition is self-adjoint and, being elliptic, it has a discrete spectrum that consists of real nonnegative eigenvalues.

A magnetic Reilly formula
In the following, we establish a Reilly formula for the magnetic Hodge Laplacian on a compact oriented Riemannian manifold (M n , g) with smooth boundary ∂M as in [26,Thm. 3].(Note that the dimension of the manifold in [26] is n + 1 in contrast to our setting).
Theorem 5.1.Let (M n , g) be a compact oriented Riemannian manifold with smooth boundary ∂M and let α ∈ Ω 1 (M ).Then we have for any ω ∈ Ω p (M, C), p ≥ 1, the magnetic Reilly formula Weingarten tensor of the boundary and Here II [p] is the extension of II as defined in (3.5).
Note that when p = 1, by taking ω = d α f for any smooth complex-valued function f and using the fact that B [1],α = Ric M + iA α (here A [1],α = −A α since A α is skew-symmetric), the Reilly formula in Theorem 5.1 reduces to the one stated in [11,Cor. 4.2] for manifolds without boundary and to [15, Thm.1.2] for manifolds with boundary.

Eigenvalue estimates for the magnetic Hodge
Laplacian on manifolds with boundary 6.1 A magnetic Raulot-Savo estimate In the following, we will estimate the first eigenvalue of the magnetic Laplacian on the boundary of an oriented Riemannian manifold in terms of the so-called p-curvatures as in [26,Thm. 1].We mainly follow and refer to [26] for further details.We consider a Riemannnian manifold (M n , g) with smooth boundary ∂M , and denote by η 1 (x) ≤ . . .≤ η n−1 (x) the eigenvalues of the Weingarten tensor II = −∇ M ν at any point x ∈ ∂M .Here, as before, ν is the inward unit normal vector field to the boundary.For any p ∈ {1, . . ., n−1}, the p-curvatures σ p (x) are defined as σ p (x) := η 1 (x) + . . .η p (x) and we set From Inequality (3.7), we have the following estimates ⟨II [p] ω, ω⟩ ≥ σ p (∂M )|ω| 2 and ⟨II [p] ω, ω⟩ ≤ (σ n−1 (∂M ) − σ n−1−p (∂M ))|ω| 2 , (6.1) for any ω ∈ Ω p (∂M ).Recall here that II [p] is the canonical extension of II to differential p-forms as in Equation (3.5).Also, it is not difficult to check the following inequality q , for p ≤ q, at any point x on the boundary with equality if and only if η 1 (x) = η 2 (x) = . . .= η q (x).
On manifolds with boundary, there are two notions of cohomology groups.We briefly recall them: The absolute cohomology group H p A (M ) which is defined as the set of harmonic forms on M satisfying the absolute boundary conditions, that is for any p ∈ {1, . . ., n}, In [26,Thm. 4], the authors provide geometric obstructions to the vanishing of these cohomologies using the Reilly formula.Namely, these conditions are related to the Bochner operator on M and to the p-curvatures of the boundary.Following the same idea, we will use the magnetic Reilly formula to deduce a similar vanishing result on the absolute cohomology groups by requiring a condition on the magnetic Bochner operator B [p],α .We have the following result.
In the following, we will consider a magnetic 1-form α on M such that its tangential part α T = ι * α is Killing of constant norm on ∂M .In this case, the exterior differential d ∂M and codifferential δ ∂M commute with ∆ α T as we have seen in Proposition 3.5.Hence, as in [26, Thm.5], we will estimate the first eigenvalue λ α T 1,p (∂M ) ′ of the magnetic Laplacian ∆ α T restricted to exact forms in terms of the p-curvatures.Theorem 6.2.Let (M n , g) be a compact Riemannian manifold with smooth boundary ∂M and let α be a differential 1-form on M such that α T is a Killing form on ∂M of constant norm.Assume that B [p],α ≥ |α| 2 and that the pcurvatures σ p (∂M ) > 0 for some 1 ≤ p ≤ n 2 .Then the first eigenvalue λ α T 1,p (∂M ) ′ satisfies the inequality Proof.Let ω = d ∂M β be a complex exact p-eigenform of ∆ α T associated to the eigenvalue λ α T 1,p (∂M ) ′ .From [3,Lem. 3.1] (see also [28,Lem. 3.4.7]),there exists a complex (p − 1)-form β such that δ M d M β = 0, δ M β = 0 on M and ι * β = β on ∂M .The form β is unique up to a Dirichlet harmonic form, that is an element in H p−1 R (M ).Notice here that β cannot be a Dirichlet harmonic form since this would lead to ω = 0. Let the p-form ω := d M β on M .Clearly, the form ω satisfies the following system: Applying the magnetic Reilly formula in Theorem 5.  Re ⟨ν⌟ω, Therefore by integrating this last inequality and multiplying it by σ n−p (∂M ), Inequality (6.2) reduces to Finally, by using the fact that ω is a closed eigenform for the magnetic Laplacian ∆ α T , we have which is the desired estimate.This finishes the proof of the theorem.

A gap estimate between first eigenvalues
In the next result, we adapt the computations in [14,Thm. 2.3] to find a gap estimate between the eigenvalues of different degrees λ α 1,p (M ) and λ α 1,p−1 (M ).For this, we will assume the manifold (M n , g) is isometrically immersed into Euclidean space R n+m and consider the magnetic Laplacian with Dirichlet boundary conditions, in contrast to [14] where absolute boundary conditions are taken.Recall that for a given normal vector field Z to M , the Weingarten tensor II Z is the endomorphism of T M given by where X, Y are tangent to M and II is the second fundamental form of the immersion.As in Equation (3.5), we will use the extension II

[p]
Z of the Weingarten tensor to p-differential forms.Theorem 6.3.Let (M n , g) be a compact manifold with smooth boundary that is isometrically immersed into the Euclidean space R n+m .Let α be a smooth 1-form on M .Then, for all 1 ≤ p ≤ n, the eigenvalues of the magnetic Dirichlet Laplacian on M satisfy where λ min (A) is the smallest eigenvalue of a symmetric operator A and {f 1 , . . ., f m } is a local orthonormal basis of T M ⊥ .
Proof.The proof follows along the lines of [14].For each j = 1, . . ., n + m, the unit parallel vector field ∂ xj on R n+m splits as where ι is the isometric immersion.For any p-eigenform ω of ∆ α associated to λ α 1,p (M ) with Dirichlet boundary condition, the (p − 1)form (∂ xj ) T ⌟ω clearly satisfies the Dirichlet boundary condition.Hence, by the characterization (3.2) of the first eigenvalue applied to (∂ xj ) T ⌟ω, we have for each j, .3) In the following, we will take the sum over j and compute each term separately.For this, we let {e 1 , . . ., e n } denote a local orthonormal frame of T M .Recall that any complex p-form β on M can be written as β = 1 p n s=1 e * s ∧ (e s ⌟β), and therefore, n s=1 ⟨e s ⌟β, e s ⌟γ⟩ = p⟨β, γ⟩ for any complex p-forms β, γ.Now, the sum over j of the l.h.s. of (6.(6.4) which is then a symmetic endomorphism on T M .Hence, it follows that In the last equality, we use the fact that n i=1 e i ⌟(A(e i )⌟) = 0 for any symmetric endomorphism A of T M .Therefore, we compute Hence, we deduce that (6.5) In the last equality, we apply (6.4) for δ α ω instead of ω.Now using Cartan's formula and the identity L X ⊥ ω for any parallel vector field X ∈ R n+m proven in [14, formula (4.3)],where II

[p]
X ⊥ is defined in (3.5), we write In the last equality, we use the relation X⌟(α ∧ ω) = α(X)ω − α ∧ (X⌟ω) for any vector field X and the definition of the magnetic covariant derivative ∇ α X = ∇ M X + iα(X).Now, we want to take the norm in (6.6) and sum over j.We have Hence, we deduce that ft ) 2 ω, ω⟩dµ g , which ends the proof.
Corollary 6.4.Let (M n , g) be a domain in Euclidean space R n and let α be a 1-form on M .Then, for all p ≥ 1, the eigenvalues of the magnetic Dirichlet Laplacian satisfy In particular, the following estimate holds, where λ 0 (M ) is the first eigenvalue of the scalar Laplacian with Dirichlet boundary condition.
Proof.Since M is a domain in Euclidean space, the second fundamental form and the curvature operator of M vanish.Therefore, Theorem 6. Recall here that −iA [p],α is a symmetric tensor field where A α (X) = X⌟d M α for all X ∈ T M .Now, by the second inequality in (3.7), we have iA [p],α ≤ p||A α || ≤ p||d M α|| ∞ .This finishes the first part.The second part is easily proved by taking successive p's.
Corollary 6.5.Let (M n , g) be a domain in the round unit sphere S n and let α be a 1-form on M .Then, for all p ≥ 1, the eigenvalues of the magnetic Dirichlet Laplacian satisfy In particular, the following estimate holds, where λ 0 (M ) is the first eigenvalue of the scalar Laplacian with Dirichlet boundary condition.
Proof.We use the isometric immersion of S n → R n+1 for which the second fundamental form is the identity.The proof is then a direct consequence of Theorem 6.3 using the fact that, on the round sphere, B A Spectral computations for magnetic Laplacians for functions on Berger spheres A.1 Eigenvalue decomposition of the ordinary Laplacian on the standard 3-sphere The following considerations are based on the arguments given in [17, pp. 27].
For further details see also [25,. Let 1} be the 3-dimensional unit sphere and let g be the standard metric on S 3 of curvature one.We can also think of S 3 as the Lie group of all unit quaternions via the identification (z 1 , z 2 ) → z 1 + jz 2 ∈ H 2 .Let Y 2 , Y 3 , Y 4 be the left-invariant extensions of the tangent vectors i, −k, −j ∈ T 1 S 3 .In this case, the vectors form an orthonormal basis of T (z1,z2) S 3 at every point (z 1 , z 2 ) = (x 1 + y 1 i, x 2 + y 2 i) ∈ S 3 .