Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature

This paper studies well-definedness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian centre of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that if the norm of the derived scheme (resp. iterated derived scheme) is smaller than the corresponding dilation factor then the adapted scheme converges. In this way, we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.

Open access funding provided by Austrian Science Fund (FWF).

The authors acknowledge the support of the Austrian Science Fund (FWF): This research was supported by the doctoral programme Discrete Mathematics (grant no. W1230) and by the SFB-Transregio programme Discretization in geometry and dynamics (grant no. I705).

Authors and Affiliations

1. Institut f. Geometrie, TU Graz, Kopernikusgasse 24, 8010, Graz, Austria

   Svenja Hüning & Johannes Wallner

Corresponding author

Correspondence to Svenja Hüning.

Communicated by: Tom Lyche

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Appendix

We give the proofs of the two Lemmas 5 and 7 which use the concept of Jacobi fields. Therefore, we first briefly recall their definition and properties. Finally, we prove Proposition 12.

1.1 A.1 Jacobi fields

Denote by c(u, s) a one-parameter family of geodesics parametrised by s, with u being the parameter along each geodesic. Let \(c^{\prime }(u,s):=\frac {d}{du}c(u,s)\). Then for fixed s = s₀, \(J(u)=c^{\prime }(u,s_{0})\) is a Jacobi vector field along the geodesic c(⋅, s₀). It is known that Jacobi fields are solutions of the linear second-order differential equation \(J^{\prime \prime }+R(c^{\prime },J)c^{\prime }=0\), with R denoting the Riemann curvature tensor. For any given geodesic, there is a linear space of Jacobi vector fields whose dimension is \(2\dim M\). Jacobi fields are an important tool in global Riemannian geometry because on the one hand, geodesics cannot be shortest curves if they admit Jacobi fields with two zeros, and on the other hand, the behaviour of Jacobi fields is guided by R. For this reason, Jacobi fields are a well-investigated topic. We refer to the textbook of do Carmo [1] for a general introduction.

1.2 A.2 Proofs

Proof of Lemma 5

Recall the definition of f_α by (9). Let γ : [0, 1] → M be a geodesic and

For any s, the geodesic c_t(⋅, s) connects x_σ(t) with γ(s). Those geodesics exist and are unique since M is Cartan- Hadamard. Additionally, let ct′(u, s) := dduc_t(u, s) and ċ_t(u, s) := ddsc_t(u, s). By construction, dist (x_σ(t), γ(s)) = ∥ct′(u, s)∥. For each t, s the vector field J(u) = ċ_t(u, s) along the geodesic u↦c_t(u, s) is a Jacobi field. Since

we obtain

Here, ∇ ∂s denotes the covariant derivative along the curve γ(s). In the following, we use the facts that ∥ct′(u, s)∥ does not depend on s, ∇ ∂sct′(u, s) = ∇ ∂uċ_t(u, s) and finally that ∇ ∂uct′(u, s) = 0 since c is a geodesic. This leads to

Since ċ_t(0, s) = 0, we finally obtain

Observe that ct′(1, s) = − expγ(s)− 1x_σ(t) (by definition of the exponential map) and ċ_t(1, s) = γ̇(s) (by construction) are independent of t. Therefore,

By the definition of the gradient, we conclude that

Using (8), we see that

To obtain the inequality above, we used the following relations between the Jacobi field and its derivative

where J^tan (resp. J^norm) denotes the tangential (resp. normal) part of the Jacobi field; see Appendix A in [15] for more details. Here, we used the fact that the sectional curvature of M is bounded above by zero. □

Remark 1

We note that a direct further differentiation of (1) yields

Thus,

This equality is used in the next proof.

To prove the next result, we make use of the representation of f_α (resp. f_β) as in (9) in terms of the function σ (resp. τ). Before we give the proof of Lemma 7, we illustrate the idea by means of our main example:

Example 1

From Example 4, we know that

Similarly, we obtain

with β₋ = 2 32, β₊ = 34 32 and

In order to get the desired result in Lemma 7, we estimate the distance between the gradients of the functions f_α and f_β at the point x^∗ = av(α, x) (as explained in more detail in the proof of the Lemma 7). To be able to do so, we make use of Lemma 5 and convert the resulting four integrals into two. In this case, we get

with

Note that the construction of the function ν is similar to the one of σ in (9).

We are now ready to give the proof of Lemma 7 which follows the structure in [23] and the ideas of [15].

Proof of Lemma 7

To obtain a lower bound for the absolute value of the gradient of 12f_α(x), we make use of Theorem 1.5. in [15] . Let γ be the geodesic starting from x^∗ and ending in x and let c_t(u, s) = expx_σ(t) (u ⋅ expx_σ(t)− 1γ(s)) be the family of geodesics from x_σ(t) to γ(s). We apply the Cauchy-Schwarz inequality and the fact that gradf_α(x^∗) = 0 by definition of x^∗ to obtain

By Remark 2, we conclude

with ċ_t(u, s) = ddsc_t(u, s). As in the proof of Lemma 5, let J(u) = ċ_t(u, s) denote the Jacobi field along the curve u↦c_t(u, s). The dependence on s and t is not indicated in the notation. We have J(1) = γ̇(s) and J^′(1) = ∇ ∂uċ_t(1, s). Using (8), we obtain

The last inequality follows in the same way as in the proof of Lemma 5. By the definition of the geodesic γ, we have ∥γ̇(s)∥ = dist(x, x^∗) and conclude that

By definition of x^∗, we have gradf_α(x^∗) = 0. Together with Lemma 5, we obtain

We define the sequence δ = (δ_j)_{j=−m,…, m+ 1} by δ_j = α_j − β_j. Let ν be the function constructed as σ in (9) with respect to the coefficients δ, i.e. the value of ν is constant in intervals of length |δ_j| and given by the corresponding index. Denote by δ₋ (resp. δ₊) the sum of the absolute values of the negative (resp. nonnegative) coefficients of δ. Equation (3) implies that δ₋ = δ₊. As in (9), we rewrite the sum above as an integral

With the help of (3), we conclude that

To obtain the third inequality above, we used the fact that in Cartan-Hadamard manifolds, the exponential map does not decrease distances, see for example [16].

It remains to show that

To do that, we split the sequence of coefficients δ in two sequences η¹, η² defined by

Similarly to the construction in the proof of Lemma 3 of [23], we consider the function 𝜖₁ given by

Analogously, we define 𝜖₂ for the sequence η². We finally obtain

This concludes the proof of Lemma 7. □

Finally, we give the proof of Proposition 12.

Proof of Proposition 12

Assume that t₁, t₂ ∈ ℝ with |t₁ − t₂| < 1. Then, there exists an integer k ∈ ℤ such that 2^−k− 1 ≤|t₂ − t₁|≤ 2^−k. As in the proof of Theorem 8, let c_k be the union of geodesic segments c_k|[ i2^k, i+ 1 2^k ] connecting the points T^kx_i and T^kx_i+ 1. Together with (11), we obtain

Using (14), we have

for all t ∈ ℝ. Summarising the previous two observations leads to

Since |t₂ − t₁|≤ 2^−k, taking the logarithm shows that \(\mu ^{k+1} \leqslant \mu ^{-\log _{2}(|t_{2}-t_{1}|)}\). We conclude that

with ι = − log μ log 2 = 1 − log ∥S^∗∥ log 2. Here, the last equality holds because μ = 1 2∥S^∗∥. □

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