Fractional Sobolev metrics on spaces of immersions

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler–Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.

The variational formulations allow one to study analytical properties of the PDEs in relation to geometric properties of the underlying infinite-dimensional Riemannian manifold [52,49,4,5,13,34].Most importantly, local well-posedness of the PDE, including smooth dependence on initial conditions, is closely related to smoothness of the geodesic spray on Sobolev completions of the configuration space [23].This has been used to show local well-posedness of PDEs in many specific examples, cf. the recent overview article [38].An extension of this successful methodology to wider classes of PDEs requires an in-depth study of smoothness properties of partial and pseudo differential operators with non-smooth coefficients such as those appearing in the geodesic spray or, more generally, in the Euler-Lagrange equations.This is the topic of the present paper.
Contribution.This article establishes local well-posedness of the geodesic equation for fractional order Sobolev metrics on spaces of diffeomorphisms and, more generally, immersions.A simplified version of our main result reads as follows: Theorem.On the space of immersions of a closed manifold M into a Riemannian manifold (N, ḡ), the geodesic equation of the fractional-order Sobolev metric is locally well-posed in the sense of Hadamard for any p ∈ [1, ∞).This follows from Theorems 4.4 and 4.6.The result unifies and extends several previously known results: • For integer-order metrics, local well-posedness on the space of immersions from M to N has been shown in [11].However, the proof contained a gap, which was closed in [50] for N = R n , and which is closed in the present article for general N .The strategy of proof, which goes back to Ebin and Marsden [23], is to show that the geodesic spray extends smoothly to certain Sobolev completions of the space.Our generalization to fractional-order metrics builds on recent results about the smoothness of the functional calculus of sectorial operators [6].• For N = R n , the set of N -valued immersions becomes a vector space, which simplifies the formulation of the geodesic equation; see Corollary 5.3.The treatment of general manifolds N requires a theory of Sobolev mappings between manifolds, which is developed in Section 2.2.Moreover, in the absence of global coordinate systems for these mapping spaces, we recast the geodesic equation using an auxiliary covariant derivative following [11]; see Lemma 2.6 and Theorem 4.3.• For M = N our result specializes to the diffeomorphism group Diff(M ), which is an open subset of Imm(M, M ).On Diff(M ) we obtain local wellposedness of the geodesic equation for Sobolev metrics of order p ∈ [1/2, ∞); see Corollary 5.1.Analogous results have been obtained by different methods (smoothness of right-trivializations) for inertia operators that are defined as abstract pseudo-differential operators [24,10,3].• For M = S 1 , our result specializes to the space of immersed loops in N .For loops in N = R d , local well-posedness has been shown by different methods (reparameterization to arc length) in [8].Our analysis extends this result to manifold-valued loops and also to higher-dimensional and more general base manifolds M .
2. Sobolev mappings 2.1.Setting.We use the notation of [11] and write N for the natural numbers including zero.Smooth will mean C ∞ and real analytic C ω .Sobolev regularity is denoted by H r , and Sobolev spaces H s H r of mixed order r in the foot point and s in the fiber are introduced in Theorem 2. 4.
Throughout this paper, without any further mention, we fix a real analytic connected closed manifold M of dimension dim(M ) and a real analytic manifold N of dimension dim(N ) ≥ dim(M ).
2.2.Sobolev sections of vector bundles.[6, Section 2.3] We write H s (R m , R n ) for the Sobolev space of order s ∈ R of R n -valued functions on R m .We will now generalize these spaces to sections of vector bundles.Let E be a vector bundle of rank n ∈ N >0 over M .We choose a finite vector bundle atlas and a subordinate partition of unity in the following way.Let (u i : ) i∈I be a finite atlas for M , let (ϕ i ) i∈I be a smooth partition of unity subordinated to (U i ) i∈I , and let ψ i : E|U i → U i × R n be vector bundle charts.Note that we can choose open sets [15,Appendix H3]).Then we define for each s ∈ R and f ∈ Γ(E) Then • Γ H s (E) is a norm, which comes from a scalar product, and we write Γ H s (E) for the Hilbert completion of Γ(E) under the norm.It turns out that Γ H s (E) is independent of the choice of atlas and partition of unity, up to equivalence of norms.We refer to [58,Section 7] and [28, Section 6.2] for further details.
The following theorem describes module properties of Sobolev sections of vector bundles, which will be used repeatedly throughout the paper.
2.3 Theorem.Module properties.[6,Theorem 2.4] Let E 1 , E 2 be vector bundles over M and let s 1 , s 2 , s ∈ R satisfy Then the tensor product of smooth sections extends to a bounded bilinear mapping The following theorem describes the manifold structure of Sobolev mappings between finite-dimensional manifolds.It is an elaboration of [46, 5.2 and 5.4] and an extension to the Sobolev case of parts of [41,Section 42].2.4 Theorem.Sobolev mappings between manifolds.The following statements hold for any r ∈ (dim(M )/2, ∞) and s, s 1 , s 2 ∈ [−r, r]: (a) The space H r (M, N ) is a C ∞ and a real analytic manifold.Its tangent space satisfies in a natural (i.e., functorial) way with foot point projection given by π )' is a real analytic manifold and a real analytic vector bundle over H r (M, N ).Similarly, spaces such as L(H s1 H r (M, T N ), H s2 H r (M, T N )) are real analytic vector bundles over H r (M, N ).(c) The space Met H r (M ) of all Riemannian metrics of Sobolev regularity H r is an open subset of the Hilbert space Γ H r (S 2 T * M ), and thus a real analytic manifold. Proof.
(a) Let us recall the chart construction: we use an auxiliary real analytic Riemannian metric ĝ on N and its exponential mapping exp ĝ; some of its properties are summarized in the following diagram: Without loss we may assume that V N ×N is symmetric: [41, 42.7] and since H r (M, N ) is continuously embedded in C 0 (M, N ), a suitable C 0 -norm neighborhood of g ∈ H r (M, N ) contains a real analytic f ∈ C ω (M, N ), thus f ∈ U g , and by symmetry of V N ×N we have g ∈ U f .The chart changes, The tangent bundle T H r (M, N ) is canonically glued from the following vector bundle chart changes, which are real analytic by Lemma A.5 again: These identify T H r (M, N ) canonically with H r (M, T N ) since where we used the vertical lift vl : T N × N T N → T T N which is given by vl(u x , v x ) = ∂ t | t=0 (u x + t.v x ); see [45, 8.12 or 8.13].The corresponding foot-point projection is then T N is fiber respecting real analytic by the module properties 2.3.Note that d fiber u f2,f1 • s is then an H r -section of the bundle L(f * 1 T N, f * 1 T N ) → M , which may be applied to the H ssection h by the module properties 2.3.These extended chart changes then glue the vector bundle Thus, the space of metrics is open.
2.5.Connections, connectors, and sprays.This sections reviews some relations between connections, connectors, and sprays.It holds for general convenient manifolds N , including infinite-dimensional manifolds of mappings, and will be used in this generality in the sequel (see e.g. the proofs of Theorems 4.3 and 4.4).(a) Connectors.[45, 22.8-9] Any connection ∇ on T N is given in terms of a connector K : T T N → T N as follows: For any manifold M and function h : M → T N , one has ∇h = K • T h : T M → T N .In the subsequent points we fix such a connection and connector on N .(b) Pull-backs.[45, (22.9.6)]For any manifold Q, smooth mapping g : Q → M and Z y ∈ T y Q, one has ∇ T g.Zy s = ∇ Zy (s • g).Thus, for g-related vector fields Z ∈ X(Q) and X ∈ X(M ), one has ∇ Z (s • g) = (∇ X s) • g, as summarized in the following diagram: (c) Torsion.[45, (22.10.4)]For any smooth mapping f : M → N and vector fields X, Y ∈ X(M ) we have Tor (d) Sprays.[45, 22.7] Any connection ∇ induces a one-to-one correspondence between fiber-wise quadratic C α mappings Φ : T N → T N and C α sprays S : T N → T T N .Here ∇ ∂t c t = Φ(c t ) corresponds to c tt = S(c t ) for curves c in N .Equivalently, in terms of the connector K, the relation between Φ and S is as follows: The diagram on the left introduces the projections T (π N ) and π T N , which define the two vector bundle structures on T T N .The diagram on the right shows that Φ and S are related by Φ = K • S.
The following lemma describes how any connection on T N induces via a productpreserving functor from finite to infinite-dimensional manifolds [37,40] a connection on the mapping space H s H r (M, T N ).The induced connection will be used as an auxiliary tool for expressing the geodesic equation; see Theorem 4.3.2.6 Lemma.Induced connection on mapping spaces.Let r ∈ (dim(M )/2, ∞), s ∈ [−r, r], and α ∈ {∞, ω}.Then any C α connection on T N induces in a natural (i.e., functorial) way a C α connection on H s H r (M, T N ).
Proof.Note that T N → H s H r (M, T N ) is a product-preserving functor from finitedimensional manifolds to infinite-dimensional manifolds as described in [40] and [41,Section 31].Furthermore, note that T H s H r (M, T N ) = H r,s,r,s (M, T T N ), where (r, s, r, s) denotes the Sobolev regularity of the individual components in any local Applying the functor H s H r (M, •) to the connector K : T T N → T N gives the induced connector The induced connector is C α by Lemma A.5.

Sobolev immersions
This section collects some results about the differential geometry of immersions with Sobolev regularity.More specifically, it describes the Sobolev regularity of the induced metric, volume form, normal and tangential projections, and fractional Laplacian, as well as variations of these objects with respect to the immersion.Here the term fractional Laplacian is understood as a p-th power of the operator 1 + ∆, where ∆ is the Bochner Laplacian and p ∈ R; see [6, Section 3].
3.1 Lemma.Geometry of Sobolev immersions.The following statements hold for any r ∈ (dim(M )/2 + 1, ∞) and any smooth Riemannian metric ḡ on N : an open subset of the real analytic manifold H r (M, N ).(b) The pull-back metric is well defined and real analytic as a mapping The Riemannian volume form is well defined and real analytic as a mapping (d) The tangential projection ⊤ : T Imm(M, N ) → X(M ) and the normal projection They extend real analytically for any real number s ∈ where is a real analytic section of the bundle Proof.
This defines a real analytic map We write T N as a sub-bundle of a trivial bundle N ×V and denote the corresponding inclusion and projection mappings by This allows one to define a projection from N × V onto T N and further onto the normal bundle of f , which is real analytic as a map This construction can be repeated for any open set Ũ such that T Ũ is parallelizable, and the resulting projections p f coincide on U ∩ Ũ .Thus, one obtains a real analytic map By the module properties 2.3, this induces a real analytic map These maps fit into the commutative diagrams The maps i * and π * are real analytic, as shown in part (b') of the proof of Lemma A.5.Therefore, the map (e) There is a bundle E over N such that T N ⊕ E is a trivial bundle, i.e., T N ⊕ E ∼ = N × V for some vector space V .We endow the bundle E with a smooth connection and the bundle N × V ∼ = T N ⊕ E with the product connection.By construction, the inclusion i : T N → N ×V and projection π : N ×V → T N respect the connection.At the level of Sobolev sections of these bundles, this means that the natural inclusion and projection mappings fit into the following commutative diagram with p = 1: As the functional calculus preserves commutation relations, this extends to all p.Thus, we have reduced the situation to the bottom row of the diagram, where the fractional Laplacian acts on H s (M, V ).In this case real analytic dependence of the fractional Laplacian on the metric has been shown in [6,Theorem 5.4].Now the claim follows from the chain rule and (b).
The following lemma describes the first variation of the metric and fractional Laplacian.The key point is that the variation in normal directions is more regular than the variation in tangential directions.This will be of importance in Theorem 4.6.The lemma is formulated using an auxiliary connection ∇ on N , e.g., the Levi-Civita connection of a Riemannian metric ḡ on N .
3.2 Lemma.First variation formulas.Let ḡ be a smooth Riemannian metric on N , and let ∇ be a C α connection on N for α ∈ {∞, ω}.(a) For any r ∈ (dim(M )/2 + 1, ∞) and s ∈ [2 − r, r], the variation of the pull-back metric extends to a real analytic map , the variation of the pull-back metric in normal directions extends to a real analytic map Laplacian in normal directions extends to a C α map where ∇ is the induced connection on GL(H r Imm r (M, T N ), H 1−r Imm r (M, T N )) described in Lemma 2.6, and L(H r Imm r (M, T N ), H 1−r Imm r (M, N )) is the vector bundle over Imm r (M, N ) described in Theorem 2.4.
Proof.We will repeatedly use the module properties 2.3.
(a) follows from the following formula for the first variation of the pull-back metric [11,Lemma 5.5]: Splitting the above formula into tangential and normal parts of m yields Now the claim follows from the real analyticity of the projection ⊥ in Lemma 3.1.
(c') We claim for any bundle E over M with fixed fiber metric and fixed connection (i.e., not depending on g) that the following map is real analytic: To prove the claim we proceed similarly to [6,Lemma 3.8].As the connection on E does not depend on the metric g, Here ∇ g is the covariant derivative on T * M ⊗ E. The proof of [6, Lemma 3.8] and some multi-linear algebra show that D g,m ∇ g is tensorial and real analytic as a map Moreover, the following maps are real analytic by [6, Lemmas 3.2 and 3.5]: Together with the module properties 2.3 this establishes (c').
(c") Using (c') we will now study the smooth dependence of fractional Laplacians.In particular we claim for any bundle E over M with fixed fiber metric and fixed connection and any p ∈ (1, r − 1] that the following map is real analytic: The claim is a generalization of [6, Lemma 5.5] to perturbations m with even lower Sobolev regularity and uses the fact that the connection on E does not depend on the metric g.Let X, Y, Z be the spaces of operators given by Note that the conditions r > 2 and p > 1 ensure that X, Y , and Z are intersections of operator spaces on distinct Sobolev scales, as required in [6, Theorem 4.5] Moreover, let U ⊆ X be an open neighborhood of 1 + ∆ g with g ∈ Met H r−1 (M ) such that the holomorphic functional calculus is well-defined and holomorphic on U in the sense of [6,Theorem 4.5].Then the desired map is the composition of the following two maps: The first map is real analytic by Lemma 3.1.(e)and (c').The second map has to be interpreted via the following identity, which is shown in the proof of [6, Lemma 5.5] using the resolvent representation of the functional calculus: The right-hand side above is the composition of the following maps, which are again real analytic by [6, Theorem 4.5]: This proves (c").Note that (c") extends to p = 1 thanks to (c').
(c) As in the proof of Lemma 3.1.(e),we write i and π for the inclusion and projection mappings of T N , seen as a sub-bundle of a trivial bundle T N ⊕E ∼ = N ×V with C α product connection.If we consider i * and π * as real analytic sections of operator bundles, Imm r (M, T N ), H r−2p Imm r (M, N × V )), then the covariant derivative of the fractional Laplacian can be expressed as follows: The maps i * and π * are real analytic, and consequently their covariant derivatives are C α .According to Lemma 2.6, the canonical connection D on the vector space V induces a real analytic connection on the bundle of bounded linear operators . By general principles, this connection differs from ∇ by a C α tensor field, often called the Christoffel symbol.Thus, it suffices to show that the following map is C α : As D is the canonical connection, this is equivalent to the following map being C α : . By (b) with s = 2p − r, the variation of the pull-back metric in normal directions is real analytic as a map ).Thus, (c) follows from (c") and the chain rule.

Weak Riemannian metrics on spaces of immersions
The main result of this section is that the geodesic equation of Sobolev-type metrics is locally well posed under certain conditions on the operator governing the metric.The setting is general and encompasses several examples, including in particular fractional Laplace operators.
4.1.Sobolev-type metrics.Within the setup of Section 2.1, we consider Sobolevtype Riemannian metrics on the space of immersions f : M → N of the form where ḡ is a C α Riemannian metric on N for α ∈ {∞, ω}, and where P is an operator field which satisfies the following conditions for some p ∈ [0, ∞), some r 0 ∈ (dim(M )/2 + 1, ∞), and all r ∈ [r 0 , ∞): (a) Assume that P is a C α section of the bundle , where GL denotes bounded linear operators with bounded inverse.(b) Assume that P is Diff(M )-equivariant in the sense that one has for all ϕ ∈ Diff(M ), f ∈ Imm r (M, N ), and h ∈ T f Imm r (M, N ) that (c) Assume for each f ∈ Imm r (M, N ) that the operator P f is nonnegative and symmetric with respect to the H 0 (g) inner product on T f Imm r (M, N ), i.e., for all h, k ∈ T f Imm r (M, N ): (d) Assume that the normal part of the adjoint Adj(∇P ) ⊥ , defined by for all f ∈ Imm(M, N ) and m, h, k ∈ T f Imm, exists and is a C α section of the bundle of bilinear maps Here ∇ denotes the induced connection (see Lemma 2.6) of the Levi-Civita connection of ḡ.
4.2 Remark.In [11, Section 6.6] we had more complicated conditions, and we implicitly claimed that they imply the conditions in Section 4.1 above.There was, however, a significant gap in the argumentation of the main result.Namely, we did not show the smoothness of the extended mappings on Sobolev completions.This article closes this gap and extends the analysis to the larger class of fractional order metrics.
We now derive the geodesic equation of Sobolev-type metrics.Recall that the usual form of the geodesic equation is f tt = Γ f (f t , f t ), where the time derivatives f t and f tt as well as the Christoffel symbols Γ are expressed in a chart.This raises the problem that the space Imm(M, N ) lacks canonical charts, unless N admits a global chart.However, Imm(M, N ) carries a canonical connection, namely, the one induced by the metric ḡ on N , which has been described in Lemma 2.6.This auxiliary connection, which will be denoted by ∇, allows one to write the geodesic equation as ∇ ∂t f t = Γ f (f t , f t ), where Γ is a difference between two connections and therefore tensorial.In the special case where N is an open subset of Euclidean space, this coincides with the usual derivative ∇ ∂t f t = f tt ; cf.Corollary 5.3.
if and only if it satisfies the geodesic equation This also holds for smooth curves in Imm r (M, N ) for any r ≥ r 0 .
Here we used the following notation: g = f * ḡ is the pull-back metric and ♯ = g −1 its associated musical isomorphism, the operator P is seen as a map P : Imm → GL(T Imm, T Imm), its composition with f is denoted by P f : R → GL(T Imm, T Imm), its covariant derivative with respect to the connection on L(T Imm, T Imm) induced by ∇ is denoted by ∇P : T Imm → L(T Imm, T Imm), the canonical vector field on R is denoted by ∂ t : R → T R, the time derivative f t = ∂ t f is viewed as a map f t : R × M → T N in the expression ∇f t : R × M → T * M ⊗ f * T N and as a map f t : R → T Imm elsewhere, the spatial derivative T f is viewed as a map T f : R × M → T * M ⊗ f * T N , and the map ∇T f : R × M → T * M ⊗ T * M ⊗ f * T N is the second fundamental form.
Proof.We will consider variations of the curve energy functional along one-parameter families f : (−ε, ε) × [0, 1] × M → N of curves of immersions with fixed endpoints.The variational parameter will be denoted by s ∈ (−ε, ε), the time-parameter by t ∈ [0, 1].Then the first variation of the energy E(f ) can be calculated as follows: As the connection respects ḡ and is a derivation of tensor products, and as the operator P f is symmetric, we have We will treat each of the three summands above separately, making extensive use of properties 2.5 of the (induced) connection ∇: (a) For the first summand we have by the definition of the adjoint that 1 2 To calculate the tangential part of the adjoint, thereby establishing its existence, we need the following formula for the tangential variation of P , which holds for any vector field X on M : where F l X t denotes the flow of the vector field X at time t and where we used the equivariance of P in the step from the second to the third line.Using this and the symmetry of P we get 1 2 Thus we obtain the following formula for the first summand of the variation of E: (b) As P f is symmetric and the covariant derivative on Imm(M, N ) is torsionfree (see Section 2.5), i.e., we get for the second summand Integration by parts for ∂ t yields To further expand the last term we use the following formula for the variation of the volume form [11, Lemma 5.7]: , where ∇T f is the second fundamental form and where ∇ * denotes the adjoint of the covariant derivative.Using the first of the above formulas we obtain for the second summand: (c) Using the second version of the variational formula for the volume in the third summand in the variation of the energy yields 1 2 Taken together, the calculations of (a)-(c) yield Setting ∂ s E(f ) = 0 for arbitrary perturbations f s yields the geodesic equation on the space Imm(M, N ) of smooth immersions.This statement extends to the space Imm r (M, N ) of Sobolev immersions because the right-hand side of the geodesic equation is continuous in f ∈ C ∞ ([0, 1], Imm r (M, N )), as shown in part (a) of the proof of Theorem 4.4.
We next show well-posedness of the geodesic equation using the Ebin-Marsden approach [23] of extending the geodesic spray to a smooth vector field on T Imm r for sufficiently high r and showing that the solutions exist on a time interval which is independent of r.The solutions depend C α on t and on the initial condition f t (0) ∈ T Imm r (M, N ).(b) The Riemannian exponential map exp P exists and is C α on a neighborhood of the zero section in T Imm H r , and (π, exp P ) is a diffeomorphism from a (smaller) neighborhood of the zero section to a neighborhood of the diagonal in Imm r (M, N ) × Imm r (M, N ).Proof.
(a) This can be shown as in [11,Theorem 6.6].Let Φ(f t ) denote the right-hand side of the geodesic equation, i.e., A term-by-term investigation using the conditions 4.1 and the module properties 2.3 shows that Φ is a fiber-wise quadratic C α map Here the condition p ≥ 1 is needed to ensure that the term P −1 ḡ(P h, h) Tr g (∇T f ) is again of regularity H r .The map Φ corresponds uniquely to a C α spray S via the induced connection described in Lemma 2.6.In more detail: The right-hand side diagram in the proof of Section 2.5.(d)holds for any manifold N with connector K. Thus, replacing (N, K) by (Imm r (M, N ), K * ), one obtains the diagram The spray S is C α because the connection K * and the map Φ are C α .Therefore, by the theorem of Picard-Lindelöf, S admits a C α flow Proof.In the proof of Theorem 4.3, the geodesic equation is derived by expressing the first variation ∂ s E of the energy functional using the Levi-Civita connection of ḡ.If the auxiliary connection ∇ is used instead, then the following additional terms appear in the formula for ∂ s E: Accordingly, letting Ψ denote the right-hand side of the original geodesic equation with ∇P replaced by ∇P , i.e., the geodesic equation becomes One verifies as in the proof of Theorem 4.4 that the right-hand side, seen as a function of f t , is a fiber-wise quadratic real analytic map T Imm r (M, N ) → T Imm r (M, N ).As the auxiliary connection ∇ is real analytic, this implies that the corresponding spray is real analytic, as well; see Section 2.5.Since the spray is independent of the auxiliary connection ∇, one may proceed as in the proof of Theorem 4.4.
The following theorem shows that (scale-invariant) fractional-order Sobolev metrics satisfy the conditions in Section 4.1.This implies local well-posedness of their geodesic equations by Theorem 4.4.Further metrics considered in the literature include curvature weighted metrics and the so-called general elastic metric [33], which can also be formulated in the present framework [12].The proof takes advantage of the fact that the adjoint in the geodesic equation 4.3 has been split into normal and tangential parts.The normal part has the correct Sobolev regularity thanks to Lemma 3.2.The tangential part incurs a loss of derivatives, but the bad terms cancel out with some other terms in the geodesic equation as shown in part (a) of the proof of Theorem 4.3.4.6 Theorem.The following operators satisfy the conditions in Section 4.1 with α = ω for any p ∈ [1, ∞) and r 0 ∈ (dim(M )/2 + 2, ∞) ∩ [p + 1, ∞): and Thus, the geodesic equations of these metrics are well posed in the sense of Theorem 4.4.
Proof.We will prove this result only for the first field of operators because the proof for the second one is analogous.We shall check conditions (a)-(d) of Section 4.1.
(b) Diff(M )-equivariance of (1 + ∆ f * ḡ ) is well-known for smooth f and follows in the general case by approximation, noting that the pull-back along a smooth diffeomorphism is a bounded linear map between Sobolev spaces of the same order of regularity [31,Theorem B.2].As the functional calculus preserves commutation relations, this implies the Diff(M )-equivariance of (1 + ∆ g ) p .
(c) is well-known for smooth f, h, k and follows in the general case by approximation using the continuity of f → •, • H 0 (f * ḡ) established in [6,Lemma 3.3] and the continuity of f → P f .
(d) Recall from Lemma 3.2 that the derivative of P f in normal directions extends to a real analytic map . Equivalently, the following map is real analytic: . Dualization using the H 0 (g) duality shows that the adjoint is real analytic Imm r (M, T N )).In particular, the adjoint is real analytic Imm r (M, T N )).4.7 Remark.For Sobolev metrics of integer order p ∈ N >0 , condition (d) of Section 4.1 can be verified directly by a term-by-term investigation of the following explicit formula for the normal part of the adjoint [11, Section 8.2], assuming that ∇ = ∇ is the Levi-Civita connection of ḡ: Here g = f * ḡ, ∆ = ∆ g , ∇ = ∇ g , and R ḡ denotes the curvature on (N, ḡ).This direct calculation is consistent with the more general argument of Theorem 4.6.

Special cases
This section describes several applications of the general well-posedness result, Theorem 4.4.First, we consider the geodesic equation of right-invariant Sobolev metrics on the diffeomorphism group Diff(M ).In Eulerian coordinates, this equation is called Euler-Arnold [2] or EPDiff [29] equation and reads as In Lagrangian coordinates, the equation takes the form shown in the following corollary.The conditions for local well-posedness in this corollary agree with the ones in [3], where metrics governed by a general class of pseudo-differential operators are investigated.The proof is an application of Theorem 4.4 to Diff(M ), seen as an open subset of Imm(M, M ).Moreover, the proof extends Theorem 4.4 to lower Sobolev regularity using some cancellations which are due to the vanishing normal bundle.The notation is as in Theorem 4.4.
5.1 Corollary.Diffeomorphisms.A smooth curve ϕ : [0, 1] → Diff(M ) is a critical point of the energy functional It remains to verify these conditions for the specific operator P = (1 + ∆) p .Condition (a) for p ≥ 1 follows from Lemma 3.1, and condition (a) for p ∈ [1/2, 1) is verified as follows.We split the operator P ϕ in two components, As 1 + p ≥ 1, Lemma 3.1 shows that the operator (1 + ∆ ϕ * ḡ ) 1+p is a real analytic section of the bundle for any r such that r − 2p − 2 ≥ 1 − r, i.e., r ≥ p + 3/2.Similarly, under even weaker conditions, the operator (1 + ∆ ϕ * ḡ ) −1 is a real analytic section of the bundle . By the chain rule, the operator P ϕ is real analytic as required in condition (a).Conditions (b) and (c) can be verified as in the proof of Theorem 4.6.
Next we consider reparametrization-invariant Sobolev metrics on spaces of immersed curves, i.e., we consider the special case M = S 1 .Our interest in these spaces stems from their fundamental role in the field of mathematical shape analysis; see e.g.[9,62,36,56,7] for R n -valued curves and [42,54,18,55] for manifoldvalued curves.For curves in R n local well-posedness of the geodesic equation for integer-order metrics has been shown in [47].This has recently been extended to fractional-order metrics in [8].The following corollary of our main result further generalizes this to fractional-order metrics on spaces of manifold-valued curves: if and only if it satisfies the geodesic equation Proof.To see this we first choose a second vector bundle F → M such that E ⊕ M F is a trivial bundle, i.e., isomorphic to M × R n for some n ∈ N. Then Γ H r (E) is a direct summand in H r (M, R n ), so that we may assume without loss that E is a trivial bundle, and then, that it is 1-dimensional.So we have to identify C ∞ (R, H r (M, R)).But in this situation we can just apply Theorem A.1 for the set V ⊂ H s (M, R) ′ consisting just of all point evaluations ev x : H r (M, R) → R.
A.3 Lemma.Function spaces of mixed smoothness.Let U be an open subset of a finite-dimensional vector space, let r ∈ (dim(M )/2, ∞), let α ∈ {∞, ω}, and let C α (U ) = lim ← −p E p be the representation of the complete locally convex space C α (U ) as a projective limit of Banach spaces E p .Then where ⊗ is the injective, projective, or bornological tensor product, or any tensor product in-between, and where H r (M, C α (U )) is defined as the projective limit lim ← −p H r (M, E p ).
The lemma justifies the following notation, which shall be used in Lemma A.5 below.If E 1 and E 2 are vector bundles over M , and U ⊆ E 1 is an open neighborhood of the image of an H r section, then we write Γ H r (C α (U, E 2 )) for the set of all fiber-preserving functions F : U → E 2 which have regularity H r C α in every C α vector bundle chart of E 1 .Loosely speaking, these are sections of regularity H r in the foot point and regularity C α in the fibers.
Proof.The space C ∞ (U ) is nuclear by [57, Corollary to Theorem 51.4], and the space C ω (U ) is nuclear as a countable inductive limit of nuclear spaces of holomorphic functions [41,Theorem 30.11].Let ⊗ ε , ⊗ π , and ⊗ β be the injective, projective, and bornological completed tensor products, respectively.Then where the first equality holds because C α (U ) is nuclear, and the second equality holds by [41,Proposition 5.8] using that H r (M ) is a normed space, and C ω (V ) is an (LF)-space and therefore bornological.Thus, all tensor spaces C α (U ) ⊗H r (M ) are equal.Moreover, by [57,Theorem 44.1], and by [32,Corollary 16.7.5],where H denotes holomorphic functions and Ũ are open neighborhoods of U in the complexification of the underlying vector space.Let ∆ 2 be the natural norm on L 2 functions [22, 7.1].Then The following map is real analytic: For any f ∈ C ω (U ) and h ∈ H r (M, U ), the composition f • h coincides with the Riesz functional calculus f (h), which is defined as follows [21,Theorem 4.7].As the spectrum σ(h) equals the range of h, which is a compact subset of U , there is a set of positively oriented curves Γ = {γ 1 , . . ., γ n } in U \ σ(h) such that σ(h) is inside of Γ, and C \ U is outside of Γ [21,Proposition 4.4].Then one defines f (h) as the following Bochner integral over the resolvent of h: For any fixed Γ, this integral is well-defined and real analytic as claimed.
(b) The following map is real analytic thanks to (b') and the boundedness of multiplication where (a ⊗ f ) • (Id M , h) denotes the map x → a(x)f (h(x)).Equivalently, by the real analytic exponential law [41, 11.18], the following map is real analytic: This map is bilinear and real analytic, and therefore bounded.By the universal property of the bornological tensor product ⊗ β [41, 5.7], it descends to a bounded linear map The domain of this map equals H r C ω (M × U ) by Lemma A.3.
A.5 Lemma.Push-forward of sections.Let E 1 , E 2 be vector bundles over M , let U ⊂ E 1 be an open neighborhood of the image of a smooth section, let F : U → E 2 be a fiber preserving function, and let r ∈ (dim(M )/2, ∞).Then Γ H r (U ) is open in Γ H r (E 1 ), and the following statements hold: (a) If F is smooth or belongs to Γ H r (C ∞ (U, E 2 )), then the push-forward F * is smooth: (b) If F is real analytic or belongs to Γ H r (C ω (U, E 2 )), then the pushforward F * is real analytic.The notation Γ H r (C ∞ (U, E 2 )) and Γ H r (C ω (U, E 2 )) is explained in Section A.3.

Proof.
(a) Let c : R ∋ t → c(t, •) ∈ Γ H r (U ) be a smooth curve.As r > dim(M )/2, it holds for each x ∈ M that the mapping R ∋ t → F x (c(t, x)) ∈ (E 2 ) x is smooth.By the Faà di Bruno formula (see [25] for the 1-dimensional version, preceded in [1]  For each x ∈ M and α x ∈ (E 2 ) * x the mapping s → s(x), α x is a continuous linear functional on the Hilbert space Γ H r (E 2 ).The set V 2 of all of these functionals separates points and therefore satisfies the condition of Theorem A.1.We also have for each p ∈ N >0 , t ∈ R, and x ∈ M that ∂ p t F x (c(t, x)), α x = ∂ p t F x (c(t, x)), α x .Using the explicit expressions for ∂ p t F x (c(t, x)) from above we may apply Lemma A.2 to conclude that t → F (c(t, )) is a smooth curve R → Γ H r (E 2 ).Thus, F * is a smooth mapping, and we have shown (a).
(b') We claim that (b) holds when F is fiber-wise linear.Then F can be identified with a map in F ∈ Γ H r (L(E 1 , E 2 )).For any h ∈ Γ H r (E 1 ), the composition F • h equals the trace F .h, which is real analytic in h by the module properties 2.3.
(b) To prove the general case, we write E 1 and E 2 as sub-bundles of a trivial bundle M × V .The corresponding inclusion and projection mappings are real analytic mappings of vector bundles and are denoted by Then the set Ũ := π −1 1 (U ) ⊆ M × V and the map F := i 2 • F • π 1 fit into the following commutative diagrams: y y y y

Appendix B. A real analytic no-loss no-gain result
The following lemma is a variant of the no-loss-no-gain theorem of Ebin and Marsden [23], adapted to the real analytic sprays on spaces of immersions as in the setting of Theorem 4.4.The proof is a minor adaptation of the proof in [23]; see also [16].
B.1 Lemma.Real analytic no-loss no-gain.Let r 0 > dim(M )/2 + 1 and let α ∈ {∞, ω}.For each r ≥ r 0 , let S r be a Diff(M )-invariant C α vector field on T Imm r (M, N ) such that T i r,s • S r = S s • i r,s where i r,s : T Imm r (M, N ) → T Imm s (M, N ) is the C α -embedding for r 0 ≤ s < r.By the theorem of Picard-Lindelöf each S r has a maximal C α -flow Fl S r : U r → T Imm r (M, N ) for an open neighborhood U r of {0} × T Imm r (M, N ) in R × T Imm r (M, N ).
Then U r = U s ∩ (R × T Imm r (M, N )) for all r 0 + 1 ≤ r and r 0 ≤ s ≤ r.Thus, there is no loss or gain in regularity during the evolution along any S r for r ≥ r 0 +1.

Proof.
(a) We shall use the following result [23, Lemma 12.2]: Any h ∈ H r (M, T N ) such that T h • X ∈ H r (M, T T N ) for all X ∈ X(M ) satisfies h ∈ H r+1 (M, T N ).
(b) For h ∈ T Imm r (M, N ) let J r h be the open interval such that U r ∩(R×{h}) = J r h × {h}, i.e., J r h is the maximal domain of the integral curve of S r through h in T Imm r (M, N ); see [41, 32.14].Since i r,s • Fl S r t = Fl S s t • (see [41, 32.16]), for h ∈ T Imm r (M, N ) we have J r h ⊆ J s h for r 0 ≤ s < r.(c) Claim.For h ∈ T Imm r+1 (M, N ) we have J r h = J r+1 h .Since S r is invariant under the pullback action of Diff(M ), we have for h ∈ T Imm r+1 (M, N ) and any X ∈ X(M ) that Fl S r t (h • Fl X u ) = Fl S r t (h) • Fl X u .Differentiating both side we get )) = T (Fl S r t )(T h • X) Since T h • X ∈ H r (M, T T N ) we see that T (Fl S r t (h)) • X ∈ H r (M, T T N ).By result (a) we get Fl S r t (h) ∈ T Imm r+1 (M, N ), and thus J r h ⊇ J r+1 h .The converse inclusion is (b).
(d) Let r 0 + 1 ≤ s < r < s + 1 and let h ∈ T Imm r (M, N ).Then J r h ⊆ J s h ⊆ J r−1 h = J r h , where the inclusions follow from (b), (b), and (c), respectively.Thus we have J r h = J s h = J r−1 h .

4. 4
Theorem.Local well-posedness of the geodesic equation.Assume the conditions of Section 4.1 with p ≥ 1.Then the following statements hold for all r ∈ [r 0 , ∞): (a) The initial value problem for geodesics has unique local solutions in Imm r (M, N ).

( c )
The neighborhoods in (a)-(b) are uniform in r and can be chosen open in the H r0 topology.Thus, (a)-(b) continue to hold for r = ∞, i.e., on the Fréchet manifold Imm(M, N ) of smooth immersions.

Fl
S : U → T Imm r (M, N ) for a maximal open neighborhood U of {0} × T Imm r (M, N ) in R × T Imm r (M, N ).The neighborhood U is Diff(M )-invariant thanks to the Diff(M )-equivariance of S. (b) follows from (a) as in [11, Theorem 6.6], and (c) follows from Lemma B.1 by writing Imm(M, N ) as the intersection of all Imm r0+k (M, N ) with k ∈ N ≥0 .4.5 Corollary.Theorem 4.4 with α = ω remains valid if the assumptions in Section 4.1 are modified as follows: the metric ḡ is only C ∞ , and the connection ∇ in condition (d) is replaced by an auxiliary connection ∇, which is induced by a torsion-free C ω connection on N , as described in Lemma 2.6.

5. 2
Corollary.Curves.A smooth curve c : [0, 1] → Imm(S 1 , N ) is a critical point of the energy functional thanks to the Sobolev embedding theorem.(a) follows from the more general statement Lemma A.5.(a).(b') As an intermediate step, we claim that the following map is real analytic:

Γ
H r ( Ũ ) F * G G Γ H r (M × V ) All maps in the diagram on the left are real analytic by definition.The map ( F ) * is real analytic by Lemma A.4.(b) applied component-wise to the trivial bundle M × V , and the maps (i 1 ) * and (π 2 ) * are real analytic by (b').Therefore, F * = (π 2 ) * • ( F ) * • (i 1 ) * is real analytic, which proves (b).
ϕ t ) vol g dt if and only if it satisfies the geodesic equation∇ ∂t ϕ t =P −1 ϕ − T ϕ ḡ(P ϕ ϕ t , ∇ϕ t ) ♯ − (∇ ϕt P )ϕ t − Tr g ḡ(∇ϕ t , T ϕ) P ϕ ϕ t .The geodesic equation is well-posed in the sense of Theorem 4.4 if P satisfies conditions (a)-(c) of Section 4.1 for some p ∈ [1/2, ∞) and all r ∈ [r 0 , ∞) with r 0 ∈ (dim(M )/2 + 1, ∞).In particular, this is the case if P = (1 + ∆) p with Proof.The formula for the geodesic equation follows from Theorem 4.3 because the terms Adj(∇P ) ⊥ and ∇T f = (∇T f ) ⊥ vanish.To show well-posedness of the geodesic equation, note that condition (d) of Section 4.1 is trivially satisfied because Adj(∇P ) ⊥ vanishes.Moreover, note that the condition p ∈ [1, ∞) in Theorem 4.4 can be replaced by the weaker condition p ∈ [1/2, ∞) because the term ∇T f , which is of second order in f , vanishes.This can be seen by a term-by-term investigation of the right-hand side of the geodesic equation as in the proof of Theorem 4.4.Therefore, the geodesic equation is well-posed for any operator field P satisfying conditions (a)-(c) of Section 4.1 for some p ∈ [1/2, ∞) and all r ∈ [r 0 , ∞) with r 0 ∈ (dim(M )/2 + 1, ∞), as claimed.
A.2 Lemma.Smooth curves in Sobolev spaces of sections.Let E be a vector bundle over M , and let ∇ be a connection on E. Then it holds for each r ∈ (dim(M )/2, ∞) that the space C ∞ (R, Γ H r (E)) of smooth curves in Γ H r (E) consists of all continuous mappings c : R × M → E with p • c = pr 2 : R × M → M For each p ∈ N ≥0 , the curve ∂ p t c has values in Γ H r (E) so that ∂ p t c : R → Γ H r (E), and t → ∂ t c(t, ) H r is bounded, locally in t.
t , where the first equality holds because ε ≤ ∆ 2 ≤ π [22, 7.1], the second one by the definition of tensor products of locally convex spaces[22, 35.2], and the third one because the fractional Laplacian (1 + ∆ g ) : H r (M ) → L 2 (M ) with respect to any auxiliary Riemannian metric g ∈ Met(M ) is an isometry and becauseL 2 (M, E p ) = L 2 (M ) ⊗ ∆2 E pby the definition of ∆ 2 [22, 7.2].A.4 Lemma.Push-forward of functions.Let U be an open subset of R, and let r ∈ (dim(M )/2, ∞).Then H r (M, U ) is open in H r (M, R), and the following statements hold.(a) The following map is smooth: by 55 years), we have for each p ∈ N >0 , t ∈ R, and x ∈ M that∂ p t F x (c(t, x)) =