Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces

We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.


Introduction
We are interested in solving linear inverse problems F u = y, (1.1) where F : U 1 → U 2 is a bounded operator between Hilbert spaces of functions defined on surfaces. Note that the functions in spaces U 1 and U 2 might be defined on different surfaces M 1 and M 2 respectively. Especially we pay attention to applications (see also Section 3 and 4) where U 1 denotes a space of functions from a surface M 1 into the vector bundle or into the tangent vector bundle. We assume that solving (1.1) is ill-posed and requires some regularization to approximate the solution in a stable way. The regularization method of choice for solving (1.1) is Tikhonov regularization, which consists in approximating a solution of (1.1) by a minimizer of the functional T α,y δ (u) := F u − y δ 2 U2 + αR(u) (1.2) over U 1 . Here y δ denotes some approximation of the exact data y, from which we assume that y − y δ U2 ≤ δ.
The analysis of Tikhonov regularization has advanced from a theory for solving ill-posed operator equations in a Hilbert space setting, both in a finite and an infinite dimensional setting (see for instance [2,3,4,14,23]), to more complex situations, when F is nonlinear [8,9,11,15] or with sophisticated regularization [17,19,20,22,24,27,26], and in statistical setting [18]. The analysis of finite dimensional approximations of regularizers in infinite dimensional spaces (see for instance [5,24,30]) is highly relevant for this work. The results of [24] are already quite general, but do not cover approximations of solutions of (1.1) by functions defined on approximating surfaces. Note that for functions defined on subsets of an Euclidean space, the finite dimensional functions approximating infinite dimensional functions are also defined on the Euclidean space, and this is not the case anymore if the underlying manifold is approximated. In this paper the differential geometrical concept of a pullback is used to compare functions defined on different surfaces.
The second issue of this paper is, that the solution of (1.1) may be a function with range in a vector bundle. Consequently, when the surface is approximated also the vector bundle, or in other words, the range of the function is varying by approximation. To resolve this issue we consider vector fields represented with ambient bases. It is then presented in the paper that the analysis of regularization for vector fields on surfaces can be incorporated into a general framework developed for vector valued functions on surfaces.
We consider several applications and perform numerical tests. One example concerns reconstructing the magnetization from measurements of the magnetic potential [10,25,31]. Another numerical test example concerns vector field denoising. Our analysis on regularizing tangent vector fields in ambient spaces provides a different point of view to existing work on tangent vector field regularization (see for instance [21,28,29]).
The paper is organized as follows: In Section 2 we first review standard regularization results in an infinite dimensional setting, and then perform a convergence analysis of Tikhonov regularization taking into account approximated surfaces, which cannot be handled with the existing theory. In Section 3, we introduce proper spaces for functions with range in vector bundles, and discuss applications of the regularization theory for recovering vector fields in ambient spaces. We make an additional discussion on recovering tangent vector fields there. Finally, in Section 4, we present some examples and verify the theoretical results numerically. The underlying geometric concepts, including all the basic notations, are summarized in the Appendix A.
2 Regularization for functions defined on surfaces Below we first review results from the regularization literature (see [3,11,22]), which provide well-posedness, convergence and stability of Tikhonov regularization. For this purpose we summarize the basic assumptions needed throughout this paper.

Standard regularization theory
We review some regularization results from the literature: Theorem 2.2 (Chapter 3, [22]). Let F , R, D, U 1 and U 2 satisfy Assumption 2.1, and y and y δ satisfy (1.3).
• Assume that α > 0. Then there exists a minimizer of T α,y δ . If (y k ) k∈N is a sequence converging to y in U 2 , then every sequence has a weakly convergent subsequence in U 1 , and the limit is a minimizer of T α,y .
• If there exists u 0 ∈ D such that Then there exists an R minimizing solution u † . That is Let (y k ) k∈N satisfy y − y k U2 ≤ δ k and set α k = α(δ k ). Then every sequence (u k = argmin {T α k ,y k (u) : u ∈ D}) has a weakly convergent subsequence, and the limit is an R-minimizing solution.
• Moreover, if in addition u † satisfies a source condition, which assumes that there exists an element W ∈ U 2 , such that Here ∂R(u † ) denotes the subgradient of R at u † , and F * : Then, the R-minimizing solution satisfies where B R (·, ·) denotes the Bregman distance with respect to the convex functional R, see for instance [17].

Convergence analysis taking into account surface perturbations
In the following we study Tikhonov regularization, consisting in minimizing the functional T α,y δ in (1.2), where F : U 1 → U 2 is an operator between function spaces where we delete the superscript 0 in case σ = 0. That is · 1 = · 1,0 . We need a few assumptions on the approximating surfaces.
be a sequence of surfaces approximating the surfaces M i for i = 1, 2, then we assume the following properties and estimates hold: • For every i = 1, 2 and every σ ≥ 0, every surface M i,σ can be parametrized by a patch m i,σ : with the same parameter domain Ω i , and m i,σ is a bijection.
• The operators and the inverse are uniformly bounded. Here by bounded we mean that there exists a real constant C such that • Let the family of operators F σ : U 1,σ → U 2,σ , σ > 0 and the operator F : with σ > 0 be the family of regularization functionals, then there exists C M ∈ R such that holds uniformly for all u ∈ A M ⊂ U 1 , with A M := {x 1 ∈ U 1 : R(x 1 ) ≤ M } .

Remark 2.4 (On Assumption 2.3).
For surfaces which can not be parametrized by a single domain (such as a sphere), we assume that the domain can be covered by patches, and the above assumption has to be satisfied on every patch.
For given data y δ σ we consider the regularization strategy consisting in minimization of the Tikhonov functional for u σ ∈ U 1,σ , Theorem 2.6. Let the Assumptions 2.1, 2.3 hold. Moreover, assume that α := α(σ, δ) → 0 for σ → 0, and δ → 0 and that Let (u k := u α k ,σ k ,δ k ) be a sequence of minimizers of T k := T α k ,σ k ,y k , defined in (2.12), with y k := y δ k σ k satisfying (2.10). In addition we denote by the pullback of u k onto M 1 , and T i,k := T i,σ k for i = 1, 2 and k ∈ N.
• Moreover, assuming the source condition (2.3) holds, then, the R-minimizing solution has the convergence rate by Bregman-distance Proof.
• To prove the first item let u k denote the minimizer of T k . Moreover, we use the abbreviations F k = F σ k , R k = R σ k , and u † k = T 1,k u † . Then, according to the definition of a minimizer, we have (2.15) Since the vector field u † solves F u † = y, and (2.5) in Assumption 2.3 and Equations (2.8), (2.11) hold, it follows that Applying (2.9), we find Sinceǔ k := T −1 1,k u k , it follows from (2.6), (2.7), (2.18) and (2.19) that (2.20) From the assumptions on the parameters (2.13) it follows after division of the inequality by α k and taking the limit k → ∞ afterwards that Similarly as in [22,Theorem 3.26] it can be seen that for some fixed α 0 , there is N 0 ∈ N + the set {T α0,y (ǔ k ) : k ≥ N 0 } is uniformly bounded. Then by the weak sequential compactness of the level sets of T α0,y (Assumption 2.1), we have {ǔ k } is bounded in U 1 . Thus there is a weakly convergent subsequence of {ǔ k }, for which we denote the weak limit byū ∈ U 1 .

Using (2.21) it follows that
Fū − y This in particular shows thatū ∈ U 1 solves (1.1). The weakly lower semicontinuity of the functional R(·) implies that which because u † is an R-minimizing solution tells us that R(ū) = R(u † ), and thusū is also an R-minimizing solution of (1.1).
• To prove the convergence rate result we reconsider (2.20) and the family of Moreover, from (2.11) it follows that We can apply the results of [24, Theorem 2.6] with the triangle inequality to taking care of some additional error terms on the right side of (2.20), then we get Remark 2.7. We note that if the parameters are chosen in the following way, , then we derive the standard convergence rates Especially, if we choose R(·) = · 2 1 , then we have Theorem 2.6 is a generalization of Theorem 2.2. Actually Theorem 2.2 is a trivial case of Theorem 2.6 when σ ≡ 0.
In the following, we shortly discuss a case example in which U i (i = 1, 2) are the Sobolev spaces on surfaces (see [16] for instance).
We have T i,σ and T −1 i,σ are bounded. Moreover, let and assume that γ i : R + → R is uniformly bounded, monotonically increasing, and satisfies the convergence γ i (σ) → 0 as σ → 0 . For a general operator equation (1.1) of which the data satisfies (1.3), we consider its Tikhonov regularization approximation (1.2) with R(·) = · Remark 2.9. In particular if k i = 2 and m i,σ is a uniform cubic spline approximation with grid size h < 1 of m i which is C 2 -smooth, then the condition (2.22) holds and the estimate (2.23) can have an explicitly form, see for instance [1], that is Remark 2.10. Note the condition (2.8) has to be further checked for individual operators F and F σ . The source condition (2.3) may require more smoothness on the non-disturbed surfaces M 1 and M 2 . For example: if F : where ∆ M denotes the Laplace-Beltrami operator. It asks that m 1 ∈ W 2,∞ (Ω 1 ), which is one order higher regularity than W 1,∞ (Ω 1 ) given in (2.22).

Application to recover vector fields in ambient spaces
In this section, we are studying an ill-posed operator equation, of which the solution is a functions with range in the vector bundle over a surface. A typical case is like a tangent vector field u : M → T M (see Appendix A). As a consequence, in case of surface perturbations, also the range of the function u is perturbed and we get the approximation To take this into account we consider vector fields represented by the basis in ambient space R d+1 of the surface M, which consist of vector valued functions u : M → R d+1 . We start by introducing some appropriate function spaces. The relevant geometric notations are summarized in Table 2 from the Appendix A.

Spaces of functions with range in the vector bundle
Before introducing the function spaces we outline a basic assumption first: for some appropriate constant C c . Here and |·| denotes the Frobenius norm of a matrix.
In fact, (3.1) is a uniform bound on the extrinsic curvature of the surface M. The surface gradient operator ∇ M · should not be confused with the covariant derivative ∇· (see Definition A.3 in Appendix). Note that the latter is only defined for functions with range in the tangent bundle and does not involve the metric of the surface. ∇ M n is sometimes referred as shape operator in the literature.
respectively. Here s(x) denotes the d-dimensional surface measure. Note that if U 1 , U 2 are scalar valued functions, then · denotes multiplication of numbers and for vectors and matrices · denotes component wise multiplication. |·| (without any subscript) denotes the Euclidean norm of a vector or the Frobenius norm of a matrix. We define the sets where T x M and N x M are the tangent and normal spaces, respectively (see Appendix A). The associated inner products and norms for respectively, are defined by where where the definitions of P τ and P n can be found in Appendix A (A.3).
In the following we prove that these spaces are in fact Hilbert spaces. Moreover, we prove some equivalent relations to the standard Sobolev space H 1 (M) which is another way to denote the space On the other hand Let u = P τ u and n = P n u, then u = u + n, and thus .
Thus, also the orthogonality is proven.
In the following we prove auxiliary results, which show embeddings of the space H 1 (M).
Proof. From the definition of (3.3) it follows that Then, by using Lemma A.5, and triangle inequality it follows that it follows from (3.1) and Cauchy-Schwarz inequality on R d+1 that From this it directly follows that Moreover, from the definition of H 1 (M), (3.3), it follows that Then, by using Lemma A.5, triangle inequality, and the fact that the Frobenius norm is sub-multiplicative it follows that (3.8) By using that for a matrix nu T the spectral and the Frobenius norm are identical and satisfy nu T = |n| |u| = |u| [12], we get from (3.8) Combining (3.7) and (3.9), it then follows that In summary, we have Moreover, we have the equivalence between H 1 (M) and H 1 (M). Proof. We first show that This follows from Assumption 3.1, Definition 3.2 and Lemma 3.4 together with Since P τ u+P n u = u, and by orthogonality of P τ u and P n u in L 2 (M) it follows that u . Then, from Lemma A.5, it follows that and therefore from the definition of H 1 (M) (3.3) it follows that Then by triangle inequality, and using the estimates used in (3.8) in Lemma 3.4, it follows that  With the discussion above, we can conclude that the spaces introduced in Definition 3.2 are actually Hilbert spaces.
where the Tr is the trace of a matrix.

Regularization theory for vector fields
We proceed to discuss the regularization theory with respect to the specific spaces X 1 (M 1 ) = H 1 (M 1 ) and X 2 (M 2 ) = L 2 (M 2 ), which are useful for many applications. In practice, it is common to consider the following type of regularization functional where R(·) is a real, non-negative, local Lipschitz and convex function, and thus is proper, convex and weakly lower semi-continuous. Since the spaces are fixed, we can have a precise smoothness characterization on surfaces M 1,σ and M 2,σ . Remark 3.10. We point out that in order to have compatibility between the smoothness of surfaces and the regularity of the spaces, we ask the parametrization map m 1,σ ∈ W 2,∞ (Ω 1 ) on the surfaces for spaces H 1 (M 1,σ ), but only ask m 1,σ ∈ W 1,∞ (Ω 1 ) for spaces H 1 (M 1,σ ) (c.f. Example 2.8).
In company with Assumption 2.1 and Corollary 3.9, we can apply the results from Theorem 2.2 and Theorem 2.6 to the regularization of problem (1.1) associated with vector fields represented in ambient coordinates.

Recovering tangent vector fields
We restrict ourselves to solving problem (1.1), where the operator F is applied to functions with range in the tangent bundle only. We rewrite (1.1) as (3.11) in this particular case F u = y, for u ∈ U 1 . (3.11) The ambient approach requires to extend the operator F and the associated Hilbert space U 1 to the space U 1 which is for vector fields represented in ambient coordinates.
For the sake of simplicity, we select the concrete representation For all u ∈ H 1 T (M 1 ) we have P n u = 0, hence it follows that Note that |·|   We consider solving the system of equations We call (3.12) the ambient operator equation for tangent vector fields. The second equation of the system ensures that u is tangential.
Tikhonov regularization for solving the ambient operator equation (3.12) consists in minimization of the energy functional

2.ũ
T (M1) seminorm minimizing solution of (1.1) if and only if it is a |·| 2 H 1 (M1) seminorm minimizing solution of (3.12). To prove convergence of the regularization method we still need the following compactness results. Proof. For every u ∈ H 1 (M 1 ), by Definition 3.2 and Lemma 3.3, we have where u ∈ H 1 T (M 1 ) and n ∈ H 1 N (M 1 ). By Definition 3.11, the operator F fulfils . Now, we use the Peter-Paul inequality with > 0 and get and consequently it follows that it follows from (3.14) that and subsequently, we have This estimate shows that every level set of T α,y is uniformly bounded in H 1 (M 1 ) and thus has a weakly convergent subsequence in H 1 (M 1 ). Because of the weaklower semi-continuity of the norms and seminorms and the boundedness of F it follows that the limit is also an element of the level set, which gives the assertion.
The conditions in Assumption 2.3 and 3.1 are satisfied by Corollary 3.9, except the estimate (2.8) on operators.
Lemma 3.12 and 3.13 guarantee that Assumption 2.1 is satisfied for the ambient operator equation (3.12). Assumptions 2.3 and 3.1 are verified as well because of Corollary 3.9, Lemma 3.14. Then we can extend Theorem 2.2 and Theorem 2.6 for (3.12), the ambient operator equation for tangent vector fields, with exact and disturbed surfaces respectively.

Examples
In the following we present examples of applying Tikhonov regularization for an ill-posed problem and an image problem of which the solutions are vector fields defined on surfaces.

Magnetization reconstruction
We consider a modelling for an inverse problem of reconstructing the Earth's magnetizations from measurement of the magnetic potential (see [31] for a recent reference), which consists in solving the operator equation Here M 1 = S 3 1 denotes the surface of the Earth, u : S 3 1 → R 3 denotes the vectorial magnetization of the Earth and y : S 3 2 ⊂ R 3 → R denotes the magnetic potential data on the (satellite) orbit M 2 = S 3 2 . Moreover, ·, · denotes the Euclidean inner product in R 3 and ∇ y denotes the gradient in Euclidean space with respect to y. We assume that the interior of the satellite orbit strictly contains S 3 1 . For the sake of simplicity of presentation we assume a 2D-setting, that u, y are constants in one Euler angle of S 3 1 and S 3 2 respectively. Then Equation (4.1) simplifies to where S 1 and S 2 denote the rectifiable, planar curves, which are the restrictions of S 3 1 and S 3 2 to the 2-dimensional Euclidean plane and ·, · denotes the Euclidean inner product in R 2 . For simplicity, we ignore a constant ( 1 2π ) multiplication with the integral. In the left image of Figure 1, the geometry of the experiment is sketched. We plot simulated data y according to some test data u † , and some noisy data, y δ , which is obtained by adding Gaussian noise to y. As a case example, we assume that the lodestones are distributed only on a part Figure 1: The left image illustrates the problem setting. The right image shows some noisy magnetic potential data (with NSR=0.5) corresponding to the mag- of the upper half of the sphere S + 1 of the earth. We denote functions in H 1 (S 1 ), which have support on the upper hemi-sphere, by H 1 (S + 1 ). This is numerically convenient, since this assumption allows to parametrize all functions with just one patch. We consider now F as the operator which is finite, because S 2 and S 1 have a positive distance. Then, because of We point out that, in general, the 2-dimensional function u in (4.2) cannot be uniquely reconstructed from a 1-dimensional equation. If we restrict attention to tangential fields, that is u † is tangential to S 1 , then the dimensions match, and one obtain a unique solution. Here and in the later we will restrict to this case. Motivated by this, we consider regularization by the Tikhonov functional On a one dimensional curve, using Definition 3.2 the explicit form of the regularization functional becomes where ∂ s denotes the derivative along the curve S 1 with respect to arclength, and τ is then a unit tangent vector field of S 1 .

Numerical tests
In our numerical test example we assume that S + 1 is the upper half part of a circle of radius 1 and S 2 is an ellipse with short radius 2 and long radius 3. Let S + 1 be parametrized as the function graph [x 1 , where t ∈ [−0.9, 0.9]. We use the uniform cubic B-splines to approximate S + 1 and note it as S + 1 [h 1 ]. S 2 is approximated piecewise linearly by polygons S 2 [h 2 ]. We first test an example of direct reconstruction without regularization, and the results (in Figure 2) shows that the problem is highly ill-posed and the solution by a least square inversion completely losses the expected information.
Taking into account the discretization of the surfaces, we have the functional (4.3) in an approximated form The corresponding optimality condition is where τ h and n h denotes the unit tangent and the unit normal vector field on S + 1 [h 1 ] respectively, and F h is the dual operator of F h in L 2 sense, that is In the implementation, we use linear finite element methods for solving the problem (4.4). In the first example, we reconstruct the magnetization from the noisy data produced from a tangent vector field 1 the coordinates in ambient space. The results are presented in Figure 3, where we show a selection of plots by varying the noise level δ and the parameter α, as well as the discretization scale h. The parameters are chosen to satisfy δ k = C 1 α k = C 2 h k with C 1 and C 2 are constants, such that the assumption (2.13) is fulfilled. The numerical results in Figure 3 are in accordance with the first statement of Theorem 2.6.
Another set of tests is made to distinguish the behaviour of the two seminorms which are defined in (3.2) for regularization. In this example, we set the ideal solution to be u † (x) = [10x 2 + 5x 1 , 5x 2 − 10x 1 ] T with [x 1 , x 2 ] T ∈ S + 1 , that is a vector field composed with constant amplitudes on both tangent and normal fields of S + 1 , and we let the noise signal ratio be N SR = 0.5 in the data. Note that in this example, we do not enforce the tangential constraint, that is we are approximating the minimizer of The numerical results are shown in Figure 4, where the results regularized by using the square of H 1 (S 1 ) semi-norm (4.5) are also provided for comparison R(u) = |u| 2 H 1 (S1) . (4.5) We visualize for the regularization parameter which gives the best approximation of the solution. We find that in this particular example, it is not possible to find a good reconstruction by minimizing with the functional (4.5).

Convergence rates for vector field denoising
We consider denoising of a 2d-vector-field y δ = u δ defined on a 1-dimensional curve S. In this example, we have the coincidence of the two surfaces, that is M 1 = M 2 = S, and F : H 1 (S) → L 2 (S) is the embedding operator.
Assuming a sequence of approximating curves (S h ) h≥0 of S and data (y δ h ) h≥0 defined on S h , respectively, Tikhonov regularization consists in minimizing the functional min In our tests S is the graph of a sine function [x 1 , x 2 ] T = [t, sin(t)] T with t ∈ [0, 2π) (cf. Figure 5). S h is approximated by uniform cubic B-splines. In this way, S h is a C 2 approximation which satisfies Corollary 3.9.
We test for the synthetic solution In this particular example the source condition (2.3) reads as follows: τ ∂ 2 s (8x 2 )) + n∂ 2 s (4 cos(x 1 )) ∈ L 2 (S), which is easy to verify. For the implementation, we approximate u by piecewise linear functions with a uniform step size h u . We denote the discretization size of S h by h s . Then S h contributes a surface disturbance to S with an error bound of the order of γ(h s ) according to our assumption. The results are shown in Table 1, where we selectively show the residuals computed in the experiments. We halve all the parameters for the selected steps, including the regularization parameter α k , the noise level δ k and the discretization sizes h s,k and h u,k . The parameters then satisfy δ 2 k α k = O(δ k ), and the order of O(γ k ) is very close to the order of O(α k ) in the tests, that is . We find that the numerical rates coincide with the results obtained in the second statement of Theorem 2.6

Conclusion
In this paper we have studied Tikhonov regularization for solving ill-posed operator equations where the solutions are functions defined on surfaces, and especially we emphasize on the functions with range in vector bundles. Such problems appear in a variety of applications such as recovering magnetization from magnetic potential, vector fields denoising on surfaces and so on. We extended the existing theory on approximation of infinite dimensional Tikhonov regularized solutions for ill-posed operator equations to the surface setting. The theory has been generalized to the case of vector fields, where they are represented by vector valued functions associated to the coordinates in ambient spaces of the surfaces. The additional features of this theory are that it allows to take into account perturbations of the surface and the vector bundle, and it provides an analysis on the convergence and convergence rates which may give an optimal regularization parameter choice.
respectively. We denote by the Jacobian of the parametrization at x = m(ζ) ∈ M. The parameters in Ω are denoted by ζ. The derivatives with respect to these parameters are always denoted by ∂. The metric tensor g is related to the parametrization m by g(ζ) = (∂m(ζ)) T ∂m(ζ) ∈ R d×d (A.1) on M at x = m(ζ). We call a vector field a tangent vector field ifṽ : M → R d+1 with range in the tangent bundle T M, that is,ṽ(x) ∈ T x M for all x ∈ M. Every tangent vectorṽ(x) ∈ T x M can be represented in terms of the tangential basis ( For two tangent vector fieldṽ(x) andũ(x), using (A.2), we have the relation We denote by n(x) = (n 1 (x), n 2 (x), · · · , n d+1 (x)) T ∈ R d+1 the unit normal vector of M at x ∈ M with fixed orientation, and by P τ and P n the projections onto T M, N M at a point x ∈ M, respectively. They are represented by matrices P n (x) = n(x)n(x) T ∈ R (d+1)×(d+1) and P τ (x) = I − P n (x) ∈ R (d+1)×(d+1) , (A.3) where I is the (d+1)×(d+1) identity matrix. Because P τ and P n are orthogonal projections on the surface they are satisfying the pointwise estimate max {|P τ u| , |P n u|} ≤ |u| .
In Figure 6 we give a schematic representation of the surface M, its parametrization m over the coordinate domain Ω, as well as the vector spaces T x M and N x M. where |A| denotes the Frobenius norm.
Definition A.2. For a scalar field v : M → R, we define its surface gradient at a point x = m(ζ) by The surface gradient fulfills the relation Given a vector valued function v : M → R d+1 , the definition of the gradient with respect to the surface ∇ M consists in taking the gradient of each scalar component v i , i.e., for x = m(ζ) we have Every vector field v : M → R d+1 can be decomposed into the two vector fields P τ v and P n v which have ranges in T x M and N x M respectively for all x ∈ M.
To conclude the appendix, we present a few auxiliary results, which are used to characterize spaces in Section 3.
Lemma A.4. The gradient operator ∇ M is independent of the parameterization, and fulfils standard rules of differentiation, such as the product rule 1. Let v, w : M → R be differentiable, then ∇ M (vw) = v∇ M w + w∇ M v.
2. Moreover, let v, w, z : M → R d+1 , then Lemma A.5. Let n(x) = (n 1 (x), · · · , n d+1 (x)) T the unit normal vector field on M, and let v : M → R d+1 be a differentiable vector field, then 1. The following formulas hold n T ∇ M n = 0, P τ ∇ M n = ∇ M n and P τ n = 0 , 2. Moreover, the seminorm can be represented by Proof. For the first item, since n is a unit normal vector field, n T (x)n(x) = 1, ∀x ∈ M.
Moreover, P τ (x) = I − n(x)n T (x), and thus from the previous it follows that P τ (x)∇ M n(x) = I∇ M n(x) − n(x)n T (x)∇ M n(x) = ∇ M n(x).
We apply the product rules in Lemma A.4 and properties of n derived above, which show that and P n ∇ M (P n v) = P n ∇ M ((n T v)n) = P n (nn T ∇ M v + nv T ∇ M n + (n T v)∇ M n) = P n ∇ M v + nv T ∇ M n.
Adding the two identities, and using the orthogonality of P τ and P n on L 2 (M), we get the second item from Definition 3.2: This concludes the proof.