Symmetries of supergeometries related to nonholonomic superdistributions

We extend Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions.

Assuming dim g is finite, the above bound is sharp, meaning that there exists a standard model with symmetry superalgebra s equal to g: the homogeneous supermanifold G/P gives a geometric structure of type (D, q) with a maximal space of automorphisms, meaning that G is the automorphism supergroup (or differs from it by a discrete quotient) and dim G is the maximal possible dimension of such a supergroup. Theorem 1.2. Let (M, D, q) be a bracket-generating, strongly regular filtered G 0 -structure with a finite-dimensional Tanaka-Weisfeiler prolongation g = pr(m, g 0 ). If M o has finitely many connected components, then Aut(M, D, q) is a Lie supergroup. If M o is connected, then dim Aut(M, D, q) dim g in the strong sense as above.
As noted above the dimension bound is sharp. In fact we have dim Aut(M, D, q) dim s and the Lie superalgebra Lie(Aut(M, D, q)) is the subalgebra of s consisting of the complete supervector fields. (We recall that any supervector field possesses a local flow in a suitable sense and it is called complete if its maximal flow domain is R 1|1 × M, cf. [30,15]. Moreover, it is complete if and only if the associated vector field on the reduced manifold M o is so.) Thus in many cases the inequality is strict.
The structure of the paper is as follows. After introducing the main tools for working with geometric structures on supermanifolds in §2, we will show that the main ideas behind the classical results can be carried over to the super-setting. However, special care should be taken with the reduction of the structure group and usage of superpoints in frame bundles. We manage this through a geometric-algebraic correspondence, elaborated for principal bundles. In §3, we recall the algebraic prolongation following the ideas of Tanaka-Weisfeiler and construct the prolonged frame bundle with an absolute parallelism. Introduction of normalization conditions via the generalized Spencer complex is inspired by a previous work by Zelenko [43]. One of our main technical features is the geometric realization as supermanifolds of the sheaves of frames introduced in [2] (in the context of G-structures). This is crucial to carry out the inductive geometric prolongation argument.
In §4, we give the proof of the main theorems, using the constructed frame bundles, and discuss supersymmetry dimension bounds. Furthermore we exploit a relation of the prolongation to the Lie equation and note that the symmetry algebra s of a filtered geometric structure can be obtained by a filtered subdeformation of g, i.e., by passing to a graded Lie subsuperalgebra and changing its filtered structure while preserving its associated-graded. We also discuss the maximal supersymmetry models there. Some applications, in particular new symmetry bounds, are given in §5. This covers holonomic supermanifolds, equipped with affine, metric, symplectic, periplectic and projective structures, as well as nonholonomic ones such as exceptional G(3)-contact structures, equations of super Hilbert-Cartan type, super-Poincaré structures and some scalar odd ODEs.
The automorphism supergroup in the case of G-structures was studied by Ostermayr [32] though this reference does not contain the supersymmetry dimension bounds. Our class of geometries is considerably larger, and in addition we consider the infinitesimal symmetry superalgebra that gives a finer dimension bound. We can also vary smoothness in the real case, to which we restrict for simplicity, and our results hold true in the complex analytic or algebraic cases too, as well as in the mixed case (cs manifolds, allowing for real bodies and complex odd directions) considered in [32]. Indeed, our arguments do not rely on any Batchelor realization of M = (M o , A M ) [5], which is well-known to fail for most classes of supermanifolds.
For algebraic computations of prolongations, related to certain geometric structures, we refer to the works of Leites et al [27,33] (see also references therein).
Finally, we remark that while Theorems 1.1 and 1.2 are formulated for strongly regular distributions (with possible reductions), we expect them to hold in the general case, allowing singularities. This would superize the result of [22]. One should only require the existence of a dense set of localizations where the derived sheaves give rise to distributions.

B -
For details on the background material on supermanifolds we refer to [7,11,27,35,40]. Here we elaborate the geometric-algebraic correspondence for the description of fiber, vector and principal bundles. To illustrate it, here are three definitions of tangent bundles: Approaches (ii) and (iii) appear e.g. in [2,16]; we will elaborate upon (i) below. It will be shown that the geometric approach (i) and the algebraic approach (iii) are equivalent, while the reduced tangent bundle T M| M o encodes less information than T M or TM. We will also establish a similar geometric-algebraic correspondence for principal bundles, crucial for our developments.

Lie supergroups and their actions.
A Lie supergroup is a supermanifold G = (G o , A G ) endowed with a multiplication morphism m : G × G → G, an inverse morphism i : G → G and a unit morphism e : R 0|0 → G with usual compatibilities, which make G a group object in the category of supermanifolds. The reduced manifold G o is a classical Lie group.
The associated functor of points G[−] : SMan op → Group is particularly useful in the case of linear Lie supergroups. For example, consider the general linear Lie supergroup G = GL(V) associated to a supervector space V = V0 ⊕ V1 of dim V = (p|q). The set of S-points G[S] = Hom(S, G)0 of G is the group with a i j , d α β ∈ A¯0(S), b i β , c α j ∈ A¯1(S) for all 1 i, j p, 1 α, β q (2.1) of the even invertible (p|q) × (p|q) matrices with entries in A(S). This group acts on the set of S-points of V where V = V0 ⊕ V1 is thought as the linear supermanifold V = (V0, C ∞ . By Yoneda, we then have an action morphism of supermanifolds α : One may similarly define a linear supergroup G ⊂ GL(V) with a morphism α : G × V → V that satisfies the usual properties of a linear action. See [7] for more details. The chart approach developed below is better adapted, but we will mainly be interested in the structures of finite type, where the automorphisms form a genuine Lie supergroup.

Fiber bundles and sections.
Recall that a morphism π : [26]. Locally, this is a product of supermanifolds, and this is the basis of the following.
and a diffeomorphism (local trivialization) ϕ : π −1 (U) → U×F such that pr U • ϕ = π. A morphism of fiber bundles π 1 : E 1 → M 1 and π 2 : E 2 → M 2 is defined via the commutative diagram isomorphisms of trivial fiber bundles over the identity is called a cocycle if ϕ ii = ½ U ii ×F and ϕ ij = ϕ ik • ϕ kj where all three are defined. Since the ϕ ij cover the identity in the first component, they can be equivalently written as ϕ ij : U ij × F → F, by abusing the notation, or even as morphisms ϕ ij : U ij → Aut(F). We refer to the latter as "transition morphisms".  [4,Prop. 4.9]. The idea is to glue E from the local data Pull-back fits in a commutative diagram, where the fiber bundle morphism ψ ♯ is determined by ψ: Recall that a subsupermanifold is called closed if its reduction is a closed submanifold.
is the closed subsupermanifold given as the pullback It can be specified more concretely via [4,Prop. 3.4]: the algebra of global superfunctions of the fiber is A(E The fiber E x is (non-canonically) diffeomorphic to the typical fiber F.
The reduced fiber bundle is defined by and it is a fiber bundle over M o with typical fiber F, according to Definition 2.2. Its defining cocycle ( ϕ ij ) o : (U ij ) o → Aut(F)¯0 is the reduced morphism of the cocycle of E. Definition 2.7. An even section of a fiber bundle π : E → M is a morphism of supermanifolds The same notion applies to superdomains U ⊂ M and we denote the set of all even sections by Γ E (U)0 = σ : U → π −1 (U) | π • σ = ½ U . (A superspace of sections can also be introduced using the functor of points [34, §4.3], but we won't need that for our arguments.) 2.4. Vector bundles on supermanifolds. Let V be a finite-dimensional supervector space.

Definition 2.8 (Geometric approach).
A geometric vector bundle is a fiber bundle π : E → M with typical fiber V such that the transition morphisms are fiberwise linear, i.e., they take values in a linear supergroup: ϕ ij : U ij → G ⊆ GL(V). If G is a proper subsupergroup, then π is called a G-vector bundle.
More concretely, any cocycle ϕ ij : In other words, Note also that fiberwise linearity for the reduced bundle is a weaker condition than that for the geometric vector bundle because the off-diagonal blocks of the above matrix are odd, hence vanish upon evaluation [4,Ex. 4.15].
A morphism of geometric vector bundles E 1 , E 2 is a fiber bundle morphism ψ : E 1 → E 2 that is fiberwise linear. More concretely, let π 1 : E 1 → M 1 and π 2 : E 2 → M 2 be geometric vector bundles with typical fibers V and W, respectively. Denote by α : Hom(V, W) × V → W the natural composition morphism of supermanifolds. Then, similar to [4,Def. 4.12], we define a morphism of vector bundles in a local component ϕ : Proposition 2.3 and Corollary 2.4 specialize straightforwardly to geometric vector bundles. Next, formula (2.3) shows that the assignment U o → Γ E (U)0 describing local even sections, gives a sheaf of right (A M )0-modules on M o . (This can be converted to a left module with the usual rule of signs.) This sheaf however is not locally free (e.g., for E = T M, M = R 0|2 (θ 1 , θ 2 ), the module of even supervector fields θ α ∂ θ β is not free over (A M )0 = 1, θ 1 θ 2 ), but it can be enlarged to a locally free sheaf U o → Γ E (U) = Γ E (U)0 ⊕ Γ E (U)1 of (right) A M -modules of rank (p|q) as follows.
First we define the parity change vector bundle π : ΠE → M as the geometric vector bundle with typical fiber ΠV determined by the transition morphisms Π ϕ ij : This leads to an alternative definition of a vector bundle. For simplicity of exposition, we assume that M o is connected for the remaining part of §2.

Definition 2.10 (Algebraic approach).
(1) A locally free sheaf E on M o of (right) A M -modules of finite rank is called an algebraic vector bundle over M.  Proof. This is [4,Prop. 4.22], to which we refer the reader.
Recall that a coherent sheaf The algebraic vector bundles are therefore the locally free coherent sheaves.
Let ϕ : M → N be a morphism and F a sheaf on M o of A M -modules. Then the direct image sheaf ϕ * F is the sheaf of A N -modules over N o given by the law ϕ * F : and the homomorphism ϕ * : A N → (ϕ o ) * A M . The kernel, cokernel and direct image of a morphism of coherent sheaves are coherent sheaves. The inverse image sheaf ϕ −1 o G of a sheaf G on N o exhibits some relatively subtle features and it is easier to define directly in terms of stalks: given In this situation, the inverse image sheaf is defined as the sheaf of A M -modules by the formula If G is locally free, then the sheaf ϕ * G is locally free as well.
There is a natural adjunction correspondence between morphisms of sheaves of modules: is a morphism of supermanifolds and ψ : Proof. Only the part of the proof on morphisms has to be modified and this is based on the following observations.
(i) The pullback of geometric vector bundles correspond to the inverse image of locally free coherent sheaves.
(ii) A morphism of geometric vector bundles from E 1 to E 2 always factorizes through the pull-back bundle as Therefore it can be viewed as a morphism of supermanifolds ψ ♭ : M 1 → M 2 paired with a morphism of geometric vector bundles ψ : E 1 → ψ * ♭ E 2 covering the identity ½ M 1 . The claim then follows from the claim on morphisms of Proposition 2.11.
Definitions 2.8 and 2.10 can therefore be used interchangeably, but care must be given to distinguish between morphisms of vector bundles and sheaves. For instance the kernel and the direct image of a morphism of vector bundles can only be interpreted as coherent sheaves in general. Henceforth we will use the nomenclature vector bundle without specification.

Principal bundles on supermanifolds.
Let G = (G o , A G ) be a Lie supergroup. Definition 2.14 (Geometric approach). A geometric principal bundle with structure group G is a fiber bundle π : P → M with typical fiber G such that the transition morphisms take values in the Lie supergroup acting on itself by left multiplication: ϕ ij : U ij → G ⊂ Aut(G).
The right action of G on a local trivialization U i ×G is given by ½ U i ×m : and it extends to a well-defined action morphism of supermanifolds α : P × G → P satisfying π • α = π • pr P . Moreover the ϕ ij are G-equivariant, i.e., we have the commutative diagram and by the Yoneda lemma this is equivalent to the identity ϕ ij = m • ϕ ij × ½ G . Definition 2.15. The fundamental vector field ζ X ∈ X(P) associated to X ∈ g is the supervector field defined by (½ P ⊗ X) • α * = α * • ζ X . Equivalently, given any local trivialization π −1 (U) ∼ = U × G, it is the left-invariant supervector field on G corresponding to X.
Let π 1 : P 1 → M 1 , π 2 : P 2 → M 2 be geometric principal bundles with structure groups G 1 and G 2 , respectively, and let γ : G 1 → G 2 be a homomorphism of Lie supergroups. We define a γ-morphism of principal bundles to be a fiber bundle morphism ψ : P 1 → P 2 that is γ-equivariant. More concretely, this is expressed as the commutative diagram Equivalently a local component ϕ : U 1 × G 1 → G 2 of a γ-morphism has the following form ϕ = m 2 • g × γ : U 1 × G 1 → G 2 , for some morphism g : U 1 → G 2 .
If G = G 1 = G 2 and γ = ½ G , we simply say that ψ is a morphism of G-principal bundles.
If P is locally simply transitive, then ϕ * P is locally simply transitive as well.

Definition 2.19.
A γ-morphism of algebraic principal bundle P 1 → P 2 is a pair (ψ, ψ ♭ ) where ψ ♭ : M 1 → M 2 is a morphism of supermanifolds and ψ : P 1 → ψ * ♭ P 2 a γ-morphism. To link geometric principal bundles with algebraic principal bundles, it is sufficient to consider even sections as defined in Definition 2.7. For any morphism g : U → G and any even section σ ∈ Γ P (U)0 we define another section by σ · g = α • σ, g , so the assignment U o → Γ P (U)0 is sheaf of right G M -sets. By the Yoneda lemma, one easily sees the following: Lemma 2.20. For any σ, τ ∈ Γ P (U)0 there exists a unique morphism g : U → G such that τ = σ · g.
In other words G M (U o ) = G[U] acts simply transitively on Γ P (U)0 if it is nonempty. Since this condition is always satisfied for small superdomains U ⊂ M we conclude the following. Conversely, given an algebraic principal bundle P, there is an open cover U i i∈I of M such that G M (U i ) o acts simply transitively on P (U i ) o for all i ∈ I. In other words, we have an identification t i : , whence the morphisms ϕ ij = g ij : U ij → G that satisfy the cocycle conditions. Therefore we obtain a geometric principal bundle π : P → M. Proof. We already proved the correspondence between objects. The correspondence between morphisms is similar to that of the proof of Theorem 2.13: (i) The pullback of a geometric principal bundle correspond to the inverse image sheaf.
(ii) A γ-morphism of geometric principal bundles from P 1 to P 2 always factorizes through the pull-back bundle as Therefore it can be viewed as a morphism of supermanifolds ψ ♭ : M 1 → M 2 paired with a γ-morphism of geometric principal bundles ψ : P 1 → ψ * ♭ P 2 covering the identity ½ M 1 : M 1 → M 1 . One then concludes as in Theorem 2.13.
2.6. Subbundles. Let F ′ ⊂ F be a subsupermanifold. (1) the transition morphisms ϕ ij : U ij → Aut(F) of E factor through Aut(F, F ′ ), (2) π : E ′ → M is itself a fiber bundle with typical fiber F ′ , whose transition morphisms are obtained by postcomposing with the restriction map Aut(F, This has a clear specification for vector and principal bundles: • For vector subbundles the typical fibers V ′ ⊂ V are supervector spaces and we consider GL(V) ⊃ GL(V, V ′ ) → GL(V ′ ) instead of general automorphisms; • For principal subbundles we have instead a Lie subsupergroup G ′ ⊂ G. These are geometric subbundles. There are also algebraic subbundles, of which we specify only the vector version, leaving algebraic principal subbundles to the reader.
is locally constant, thus constant. Consequently E ′ is a a locally free coherent sheaf, proving our claim.
Note that a vector subbundle is not the same as an injective morphism of vector bundles. Indeed, the latter may not be locally a direct factor. Similarly to the previous subsections one can prove the equivalence of algebraic and geometric definitions of subbundles.
We can also define associated bundles. Let π : P → M be a geometric principal bundle determined by transition morphisms g ij : U ij → G. For any representation ρ :

P S
We begin by revising prolongations of G-structures on supermanifolds using our setup and then will pass to the filtered version.
The sheaf of groups GL M : acts naturally on the right on the set of frame fields via is parametrized as in §2.1. This action is locally simply transitive, hence Fr M is an algebraic principal bundle over M with structure group GL(V). (The minus sign in (3.3) follows from the sign rule, so that (3.3) is indeed an action.) Higher order frame bundles Fr k M → M (k 1, with Fr 1 M = Fr M ) can be introduced via the description of jet superbundles of [18], which does not rely neither on (topological) points or the functor of points. Let J k (R m|n , M) be the vector bundle of jets from R m|n to M, which is defined as a supermanifold of homomorphisms of appropriate algebras in [18, §6]. This is a geometric vector bundle over the product R m|n × M. The open subsupermanifold The corresponding sheaf is obtained via the geometric-algebraic correspondence. Below we adapt a different approach (which is though similar in spirit) to introduce higher frame bundles in the non-holonomic situation.
3.2. G-structures on supermanifolds. In geometric language, a G-structure for G ⊂ GL(V) is a reduction of the frame bundle as in Example 2.16; following §2.6, this corresponds to a subbundle F G ⊂ Fr M . In algebraic language this is a subsheaf F G ⊂ Fr M on which the subsheaf G M ⊂ GL M acts locally simply transitively from the right.
The soldering form ϑ ∈ Ω 1 (F G , V) is given by where ξ ∈ X(F G ), π = (π o , π * ) : F G → M is the natural projection and F a local field of frames. More precisely, the R.H.S. is a short-hand for the operation detailed in the following.

Lemma 3.1. The soldering form is a well-defined even
. Now, the sheaf of π-superderivations is isomorphic to the inverse image sheaf , so we may express Ξ as follows: is an element of (A F G ) p ⊗ V, as expected, and it is easy to see that it does not depend on the fixed expression (3.4).
The other claims are obvious -e.g., G-equivariancy can be checked as in the classical case thanks to Yoneda lemma.
The following exact sequence defines the first prolongation of g ⊂ gl(V): the Spencer skew-symmetrization operator (in the supersense), that is δ(w⊗α⊗β) = w⊗(α⊗β−(−1) |α| |β| β⊗α), and g (1) = Ker(δ) = g⊗V * ∩V ⊗S 2 V * . If g F G = F G × g → F G is the trivial vector bundle over F G with fiber the Lie superalgebra g and, by abuse of notation, we denote the corresponding locally free sheaf on (F G ) o with the same symbol, then we have the exact sequence of sheaves Let π * : TF G → π * TM be the differential of π : Since Ker(π * ) and Im(δ) are locally free sheaves, all normalizations and horizontal distributions are as well, see §2.6. Any horizontal distribution gives an isomorphism H ∼ = π * TM.
is the right-inverse to the projection π * . The torsion of the horizontal distribution H is then defined by can be computed by the Cartan formula. In other words, we have a morphism of sheaves from Λ 2 π −1 o TM to V F G , which clearly extends to Λ 2 π * TM. Since H ∼ = π * TM and the soldering form ϑ ∈ Ω 1 (F G , V) induces an isomorphism of sheaves ϑ| H : H → V F G , the torsion can be in turn identified with a global even section of the trivial vector bundle over to be the sheaf on (Fr 0 ) o given by for any open subset V o of (Fr 0 ) o . The sheaf of Abelian groups G (1) (1) [V] on (Fr 0 ) o acts simply-transitively on Fr 1 from the right, so this gives an affine bundle Fr 1 → Fr 0 by the geometric-algebraic correspondence.
Further prolongations follow the same scheme (literally as in [38]) and yield the tower of prolongations A G-structure F G is called of finite type if this tower stabilizes.

Remark 3.3.
Alternatively, a geometric structure can be defined via its Lie equations [23]: instead of frames one considers the supermanifold J 1 (V, M) of 1-jets of maps V → M, and the defining equation is a subsupermanifold Prolongations are defined as differential ideals where the f i are defining equations of E 1 , and D σ are iterated total derivatives.
Symmetries of G-structures may be introduced via automorphism supergroups as in §2.2, but a more concrete description is in terms of super Harish-Chandra pairs. To this, we recall that the differential ϕ * : * Fr M and we note that the Lie superalgebra g ⊗ A(U o ) acts from the right on T m|n M (U).
The symmetries of G-structures are majorized by the tower of principal bundles (3.9) by the classical construction of Sternberg [38] (see also [17]), extended to the supercase in [32]: it is proven there that automorphisms of a finite type G-structure F G on a supermanifold M form a Lie supergroup Aut(M, F G ). We will generalize this in what follows.

Superdistributions and algebraic prolongations. A distribution on a supermanifold
M is a graded A M -subsheaf D = D¯0⊕D¯1 of the tangent sheaf TM that is locally a direct factor. As explained in §2.6, any such sheaf is locally free, so we may consider the associated vector bundle D over M. The latter induces a reduced subbundle D| M o ⊂ T M| M o , but as usual with evaluations, D| M o does not determine D. We focus here on the algebraic perspective.
The weak derived flag of D is defined as follows: where each term is a graded A M -subsheaf of TM. We assume the bracket-generating property D µ = TM for some µ > 0, and also that D is regular, i.e., all subsheaves D i are locally direct factors in TM.

Example 3.5.
For many examples of (strongly) regular superdistributions, see [24]. We give here a superdistribution that is not regular. It is a superextension of the Hilbert-Cartan equation depending on two odd variables. (In [24], we discussed a more general extension with G(3)-symmetry.) Consider the supermanifold R 5|2 with coordinates (x, u, p, q, z | θ, ν), endowed with the following superdistribution of rank (2|1): We directly compute The latter is clearly not a superdistribution, due to the presence of the supervector field θ∂ u .
In the case of a regular D we get an increasing filtration D i of TM by superdistributions, which is compatible with brackets of supervector fields: for each superdomain U ⊂ M we have In particular, the bracket on gr(TM) is A M -linear and thus descends to a Lie superalgebra bracket on the supervector space m( Concretely, a regular superdistribution is strongly regular if it has a local basis of supervector fields adapted to the weak derived flag and whose brackets, after the appropriate quotients, are given by the structure constants of m (which are real constants).
From now on, we assume all arising superdistributions to be strongly regular. Note that by construction m is fundamental, i.e., generated by m −1 . We will also assume that m is nondegenerate, i.e., g −1 contains no central elements of m if µ > 1. (Typically, one has z(m) = g −µ .) The Tanaka-Weisfeiler prolongation of m is the maximal Z-graded Lie superalgebra g = i∈Z g i such that It is denoted g = pr(m). The proof of the existence and uniqueness of pr(m) from [39,41] extends verbatim to the Lie superalgebra case. Concretely g 0 = der gr (m) and g i for i > 0 are defined recursively by the condition (applies also for i = 0) It is easy to verify that pr(m) = i −µ g i is a Lie superalgebra.
There are several variations on this construction. The most popular one is related to a reduction to a subalgebra g 0 ⊂ der gr (m). Then (i) in the definition of the prolongation is changed to g 0 = m ⊕ g 0 and (ii) remains with the same formula but ∀i > 0. The resulting prolongation superalgebra is denoted by g = pr(m, g 0 ). A more sophisticated reduction is as follows. Assume we have already computed the prolongation to the level ℓ > 0 and let g ℓ as g 0 -module be reducible: Then we can reduce g −µ ⊕ · · · ⊕ g ℓ−1 ⊕ g ℓ to g −µ ⊕ · · · ⊕ g ℓ−1 ⊕ g ′ ℓ and prolong for i > ℓ by adapting the range of the map u in (3.11). The result will be denoted by pr(m, . . . , g ′ ℓ , . . . ), where we list all reductions, or simply g = ⊕ ∞ i=−µ g i if no confusion arises. An example of this higher order reduction is projective geometry, cf. the classical case in [23,Example 3], which we will also discuss in the super-setting in §5.1.5.
The generalized Spencer complex of a reduced prolongation algebra g = ⊕ ∞ i=−µ g i is the Lie superalgebra cohomology complex Λ • m * ⊗ g with the Chevalley-Eilenberg differential where d is the Z-degree of a cochain, and it also admits a parity decomposition into even and odd parts as a supervector space. It follows from definitions that H i,1 (m, g) = 0 if and only if g i is the full prolongation of m⊕g 0 ⊕· · ·⊕g i−1 , therefore H 0,1 (m, g) = ⊕ i 0 H i,1 (m, g) encodes all possible reductions.
3.4. Filtered geometric structures. Now we superize the notion of filtered geometric structure as developed in [39,31,23]. Let D be a strongly regular, fundamental, non-degenerate distribution on a supermanifold M. The corresponding zero-order frame bundle is a principal bundle π : Here we denoted by m M = M×m → M the trivial vector bundle over M with the fiber m and, by abuse of notation, the associated locally free sheaf with the same symbol. The structure group of the bundle is the Lie supergroup G 0 = Aut gr (m) which, by the Harish-Chandra construction, can be identified with the pair Aut gr (m)0, der gr (m) formed by the Lie group of degree zero automorphisms of m and the Lie superalgebra of degree zero superderivations of m. Since m is fundamental, the structure group G 0 embeds into the Lie supergroup GL(g −1 ) and Fr 0 can be realized as a sheaf of special A M -linear isomorphisms from (g −1 ) M to D.
More generally a first-order reduction is given by a G 0 -reduction F 0 ⊂ Pr 0 (M, D) with structure group a Lie subsupergroup G 0 ⊂ Aut gr (m), which again can be thought of as an inclusion of super Harish-Chandra pairs (G 0 )0, g 0 ⊂ Aut gr (m)0, der gr (m) . (Often such reductions are given by order 1 invariants, e.g., tensors or their spans. As first example, an OSp(m|2n)-reduction in the case D = TM corresponds to an even supermetric q ∈ S 2 T * M.) In the next section we will construct higher order frame bundles Fr i = Pr i (M, D), which fit into a tower of principal bundles with projections (3.13) where the principal bundle Fr i → Fr i−1 has Abelian structure group g i , for all i > 0. The bottom projections have the structure of fiber bundles over M: for every superdomain U ⊂ Fr i−1 . This corresponds to a vector bundle, whose fiber can be further reduced but it is not relevant here. For any first-order reduction F 0 ⊂ Pr 0 (M, D) we denote the prolongation bundles by for all i > 0. They also fit into a tower of principal bundles analogous to (3.13). For higherorder reductions, we restrict to a subbundle F i ⊂ F (i) 0 for some i > 0, and the geometric object q responsible for this reduction will have higher order. (E.g., a projective superstructure is given by an equivalence class of superconnections. The associated Lie equations for symmetry, cf. Remark 3.3, are of second order.) Further reductions can be imposed in a similar way on the prolongations F (i) j . In the rest of the paper, in order not to overload notations, we will mostly concentrate on pure prolongations or first order reductions. However, the results apply in the general situation.

Definition 3.7.
A filtered geometric structure (M, D, F) on a supermanifold M consists of a strongly regular, fundamental, non-degenerate distribution D on M and possibly some reductions F of the tower (3.13). If F are encoded by a tensorial or higher-order structure q, we will also use the notation (M, D, q).
3.5. Geometric Prolongation. Now we shall construct the higher (super) frame bundles partially following the revision by Zelenko [43] of the constructions by Sternberg [38] and Tanaka [39] (beware: our notations differ from theirs). Our approach is novel in the following: we construct the tower of bundles F ℓ , ℓ 0, and the frames ϕ H ℓ on them, using the entire Spencer differential (instead of a reduced one) and recognize the choices of complements as the space of 0-and 1-cochains therein (with freedom being co-boundaries).
3.5.1. First prolongation. Thanks to §3.4, we assume that the bundle π 0 : F 0 → M is already constructed. Via pullback by dπ 0 , the filtration on TM induces a filtration on TF 0 : where we also set T k F 0 = TF 0 for all k < −µ, T k F 0 = 0 F 0 for all k > 0 and, for simplicity, omit the inverse image symbol π * 0 in front of each of the sheaves on M in the bottom row of (3.15). Via dπ 0 , we then have as sheaves of negatively Z-graded Lie superalgebras on F 0 , with π * 0 gr(TM) referring to the inverse image sheaf. There is the canonical (A F 0 -linear, even) vertical 0-trivialization given on fundamental vector fields by γ 0 (ζ X ) = X for all X ∈ g 0 . Let U ⊂ M be a superdomain and consider a section of F 0 on U, which is identified with an . We also call it a horizontal 0-frame. Working with stalks of inverse image sheaves as in the proof of Lemma 3.1, one easily checks that the following is well-defined.
Concretely, one may compute ϑ 0 by restricting to a superdomain U × G 0 ∼ = π −1 0 (U) ⊂ F 0 trivialized by a fixed frame ϕ 0 and recalling that all other frames are obtained by the action of G 0 : ϕ 0 · g := α • ϕ 0 , g for any morphism g : U → G 0 , with α : F 0 × G 0 → F 0 the right action. The soldering form is G 0 -equivariant. We also note that for i < 0, is invertible, and that ϑ 0 = {ϑ i 0 } i<0 is an isomorphism of sheaves of Z-graded Lie superalgebras over A F 0 . In particular, using ϑ 0 and γ 0 , we obtain a full frame of gr(TF 0 ). (We caution the reader that this does not identify gr(TF 0 ) with A F 0 ⊗(m⊕g 0 ) as Lie superalgebras, as the bracket on gr(TF 0 ) is not A F 0 -linear if at least one entry is a vertical supervector field.) For each k ∈ Z, we set T k 0 := T k F 0 . Let i < 0 and consider the following exact sequence of sheaves on F 0 : The sequence (3.18) makes sense for i = 0 as well, in which case it simply says that H 0 0 := T 0 0 . By the splitting lemma, H i 0 is the kernel of a left-inverse h i 0 to ι i 0 or, equivalently, the image of the right-inverse k i In terms of the maps h 0 = {h i 0 } i 0 determined by H 0 , we define the 1st structure function on the entries v k ∈ g k , k < 0, by and then extend by Since the filtration on TF 0 is respected by the Lie bracket, then the input of ϑ 0 above lies in gr i+j+1 (TF 0 ) = T i+j+1 0 /T i+j+2 0 , which is mapped by ϑ 0 to g i+j+1 . In particular, c H 0 is well-defined, it has even parity and Z-degree 1, i.e., it maps g i ⊗g j to g i+j+1 . We let Λ 2 m * ⊗g = k∈Z (Λ 2 m * ⊗g) k be the natural decomposition of Λ 2 m * ⊗ g into Z-graded components, so that c H 0 ∈ A F 0 ⊗ (Λ 2 m * ⊗ g) 1 . The space m * ⊗ g has an analogous decomposition and clearly (m * ⊗ g 1 ⊂ m * ⊗ (m ⊕ g 0 ).
Let us take another complement H 0 = { H i 0 } i 0 and the 1-frame ϕ H 0 . By construction, for for some morphism ψ : Suppressing upper indices for simplicity and denoting by Ψ : m F 0 → gr(TF 0 ) the morphism obtained composing ψ with the identifications (3.16)-(3.17), we get for all v l ∈ g l , l −1: where the last equality follows from the definition of structure function and the fact that the soldering form ϑ 0 is a G 0 -equivariant morphism of Lie superalgebras.
This gives the following method to restrict the H 0 's. Take a complement N 1 ⊂ (Λ 2 m * ⊗ g) 1 to δ(m * ⊗ g 1 and denote the corresponding sheaf over F 0 by N 1 = A F 0 ⊗ N 1 . Then we define the sheaf Pr 1 (M, D, F 0 ) over F 0 by equivalently the collection of the associated 1-frames (3.20). By (3.22)-(3.23) and Lemma 3.9, this is a principal bundle π 1 : Pr 1 (M, D, F 0 ) → F 0 over F 0 with Abelian structure group G 1 = exp(g 1 ) consisting of all ψ ∈ m * ⊗ g 1 in the kernel of the Spencer operator δ, i.e., of all elements of the first prolongation g 1 = g (1) 0 . The affine bundle Pr 1 (M, D, F 0 ) may have a further reduction resulting in the first frame bundle F 1 ⊆ Pr 1 (M, D, F 0 ). By (3.20), a section ϕ 1 of F 1 over V ⊂ F 0 can be equivalently thought as an element ϕ 1 : (g 0 ) F 0 → gr [1] (TF 0 ) such that H 0 = Im(ϕ 1 ).

3.5.2.
Higher frame bundles. The higher frame bundles are constructed similarly. We will not specify structure reductions anymore, denoting (reduced or non-reduced) frame bundles by the same symbol F i .
Since the framework of supermanifolds does not allow to work at a fixed point, what we will really need is the pull-back bundle π * ℓ F ℓ → F ℓ with its canonical section. In other words ϕ ℓ : and π * ℓ T i ℓ−1 for, respectively, negative and non-negative indices. In this subsection, we construct the new horizontal subspaces H ℓ = {H i ℓ } i ℓ , which includes the construction for the non-negative indices 0 i ℓ.
Via pullback by dπ ℓ , the filtration (3.25) on TF ℓ−1 induces a filtration on TF ℓ : where, as usual, we omit inverse image sheaf symbol π * ℓ for the sheaves in the bottom row. It is important to note for later use that the filtration on TF ℓ is respected by the Lie bracket only for non-positive filtration indices (because of the Leibniz rule). For instance, the vertical subbundle T ℓ ℓ is integrable and it also preserves all T i ℓ for −µ i ℓ − 1, since the latter bundle is induced via pull-back. Similarly one has for all 0 m ℓ and −µ n m. Note the isomorphism gr <ℓ (TF ℓ ) ∼ = → π * ℓ gr(TF ℓ−1 ) as sheaves of A F ℓ -modules. The soldering form ϑ ℓ = {ϑ i ℓ } i<ℓ ∈ Hom(gr <ℓ (TF ℓ ), (g <ℓ ) F ℓ ) on F ℓ is defined by composing this isomorphism with the soldering form and the vertical trivialization on F ℓ−1 , i.e., it is the pull-back via π ℓ of the forms (3.26). We also have a canonical vertical ℓ-trivialization γ ℓ : T ℓ ℓ → A F ℓ ⊗ g ℓ . Consider the following two exact sequences of sheaves over F ℓ (the sequence over F ℓ−1 lifts via the inverse image operation, the notation of which we suppress again), with i < 0: For all i < 0, the differential dπ ℓ induces the map a i ℓ : , which is an isomorphism. We then define the middle vertical map by b i ℓ := a i ℓ •  i ℓ . As in §3.5.1, we will later see in Lemma 3.11 that these sequences split, with dashed lines indicating leftinverses h i j to ι i j and right-inverses k i For all 0 i ℓ − 1, we let b i ℓ : T i ℓ → π * ℓ T i ℓ−1 be the projection induced by the differential, whose kernel is T ℓ ℓ . We then have the following exact sequences and set H ℓ ℓ = T ℓ ℓ . Dashed lines indicate the respective inverses h i j and k i j to ι i j and j i j .

Lemma 3.10. Given
) for all 0 s ℓ + 1 and i + s ℓ, We omit the proof by induction of (i) for the sake of brevity. Claim (ii) follows from (i) considering i + s instead of i and taking the quotient by T i+ℓ+2 ℓ /T i+s+ℓ+2 ℓ . We note that The following result is then a straightforward consequence of (ii).

Proposition 3.11. Given
is the natural projection. In particular are the complements to T i+ℓ+1 ℓ /T i+ℓ+2 ℓ and T ℓ ℓ , respectively.
Let us take another complement H ℓ = { H i ℓ } i ℓ constructed as before and the associated (ℓ + 1)-frame ϕ ℓ+1 . By construction, for any for some morphism ψ : (g ℓ−1 ) F ℓ → (g ℓ ) F ℓ of sheaves of A F ℓ -modules. It is clear that the components are elements of even parity, with the first component having Z-degree (ℓ + 1). In other words ψ − is an even element of A F ℓ ⊗ m * ⊗ g ℓ+1 and ψ + of A F ℓ ⊗ (g + ℓ−1 ) * ⊗ g ℓ . Conversely, given any such 3.5.3. Normalization conditions. In this section, we detail the normalization conditions to be enforced on the (ℓ + 1)-frames. Since the Lie bracket is compatible with the filtration on TF ℓ only for non-positive filtration indices, we first need to collect some finer properties satisfied by the frames.

Lemma 3.12.
Let ζ ∈ T k ℓ with 0 k ℓ and ϕ ℓ+1 : (g ℓ ) F ℓ → gr [ℓ+1] (TF ℓ ) be an (ℓ + 1)-frame. Then, for v i ∈ g i , i < 0: (3.39) Given a choice of complements H ℓ = {H i ℓ } i ℓ , we define the (ℓ + 1)-th horizontal structure function c − H ℓ ∈ A F ℓ ⊗ (Λ 2 m * ⊗ g <ℓ ) on the entries v k ∈ g k , k < 0, by and extending by A F ℓ -linearity to the entries from (g k ) F ℓ = A F ℓ ⊗g k . Evidently [T i ℓ , T j ℓ ] ⊂ T i+j ℓ as i, j < 0. However, the Lie bracket is compatible with the filtration on TF ℓ only for the nonpositive filtration indices, so the fact that (3.40) is well-defined deserves an additional explanation: we show that the input of ϑ ℓ •h ℓ above is a well-defined element in T i+j Lemma 3.13. The horizontal structure function c − H ℓ is well-defined. Proof. Recall i, j < 0. If both i + ℓ + 2 and j + ℓ + 2 are non-positive, the claim follows immediately from the general properties of the Lie bracket. Otherwise we may assume, say, by the general property of the Lie bracket, and the same result follows from Lemma 3.12 if j + ℓ + 2 > 0.
Note that c − H ℓ has Z-degree (ℓ + 1), i.e., it is an element of C ℓ+1,2 (m, g) F ℓ . As in Lemma 3.9: Lemma 3.14. Under a change of complement, the structure function transforms as c − We know from Proposition 3.11 that for 0 k ℓ−1, a complement of T ℓ ℓ in T k ℓ is ⊕ ℓ−1 s=k H s ℓ , so there is a projection pr k s from T k ℓ to H s ℓ ∼ = A F ℓ ⊗ g s for any k s ℓ − 1, where the last isomorphism is given by the soldering form. The analogous projection from T k ℓ to T k ℓ /T k+ℓ+2 ℓ and then to H s ℓ for any k s ℓ + k is defined for all k < 0. We note that Ker(pr k s ) ⊃ T s+1 ℓ . Again we need some finer properties of the frames. Lemma 3.15. Let ϕ ℓ+1 : (g ℓ ) F ℓ → gr [ℓ+1] (TF ℓ ) be an (ℓ + 1)-frame and i 0. We then have [ζ, ϕ ℓ+1 (v i )] ∈ T k ℓ for all ζ ∈ T k ℓ with i < k ℓ. Note that the claim of Lemma 3.15 is automatically satisfied also for k i, due to (3.30). Let g + ℓ−1 = g 0 ⊕ · · · ⊕ g ℓ−1 as before. The (ℓ + 1)-th vertical structure function where v i ∈ g i with i < 0, and v j ∈ g j with 0 j ℓ−1. By Lemma 3.12 one sees that the input of ϑ i+ℓ ℓ • pr i+j i+ℓ is in T i+j ℓ with some ambiguity, which in this case lies in

Canonical parallelism and comparison with Zelenko's approach.
Theorem 3.17. Let (M, D, q) be a filtered structure of finite type, i.e., the Tanaka prolongation stabilizes: g = g −µ ⊕ · · · ⊕ g 0 ⊕ · · · ⊕ g d (with g d = 0, but g d+1 = 0). Then there exists a fiber bundle π : P → M of dimP = dim g and an absolute parallelism Φ ∈ Ω 1 0 (P, g), which is natural in the sense that any equivalence transformation f : M → M ′ lifts to a unique map F : P → P ′ preserving the parallelisms Φ and Φ ′ .
Proof. Let P = F d and consider the structure Φ d consisting of ϕ d+1 (v i ), which are supervector fields for i −1 and truncated supervector fields for i < 0. Let us prolong further, but first note that the map F k → F k−1 is a principal bundle with trivial fiber g k = 0 for any k d+1, hence a diffeomorphism. Take j = d+µ−1. Then ϕ j+1 | g i : Thus we get the required non-truncated supervector fields. The frames ϕ j+1 : A F j ⊗ g → T j give an absolute parallelism Φ on P := F j ∼ = F d . Indeed a basis on g, respecting the parity and Z-grading, gives a basis of supervector fields on P.
Remark 3.18. In general, the fiber bundle π : P → M is not principal, so the parallelism Φ lacks equivariancy and it is not a Cartan superconnection, but it suffices for dimensional bounds. If the normalizations can be chosen invariantly w.r.t. the Lie supergroup G 0 ⋉ exp(g 1 ⊕ · · · ⊕ g d ), then we expect that π : P → M is a principal bundle and a Cartan superconnection exists. (We have verified that this is true in the d = 0 case.) In this case, the step of our construction involving vertical structure functions and normalizations would not be required: one may simply take the fundamental vector fields of the principal action.
Lemma 3.19. The map p is injective when restricted to the kernel of δ on Λ 2 m * ⊗ g. In particular, it is injective when restricted to δ(m * ⊗ g) and Ker(∂) = Ker(δ).
Since p is also the projection to A along the Z-graded complement in Λ 2 m * ⊗ g, the following result follows straightforwardly.
Consequently, any complement Z is obtained as Z = p(N) for some complement N. However, it is not true that any N is of the indicated type N = Z ⊕ B, yet for our purposes this distinction plays no role. In concrete cases, when the prolongation g 0 acts completely reducibly this may turn important in finding an invariant complement.  (i) An automorphism ϕ ∈ Aut(M, D, F)0 of a filtered structure is an element ϕ ∈ Aut(M)¯0 such that: -it preserves the distribution: ϕ * (D) ⊂ (ϕ o ) −1 * D, so it induces an isomorphism of Pr 0 (M, D) (which, by abuse of notation, we simply denote by ϕ * ); -in the case of a first order reduction F = F 0 ⊂ Pr 0 (M, D), we also require that 0 and, if there are higher order reductions, we also require that it preserves them: such that L X (D) ⊂ D, and it successively preserves the structure reductions, namely: Assume the filtered structure is of finite type. By the naturality of the constructions, any automorphism of the filtered structure on M lifts to a symmetry of the bundle π : P → M constructed in Theorem 3.17 and it preserves the absolute parallelism Φ on it. Likewise, any infinitesimal symmetry of the filtered structure on M lifts to an infinitesimal symmetry of the bundle π : P → M preserving the absolute parallelism Φ. In particular, the dimension of the symmetry superalgebra s is bounded by the dimension of the infinitesimal symmetries of Φ. To prove dim s dim P (in the strong sense) and complete the proof of Theorems 1.1 and 1.2, it is sufficient to establish a bound on the dimension of the symmetries of the absolute parallelism.

4.1.2.
Dimension of the symmetry superalgebra. By fixing a basis of g, the absolute parallelism Φ corresponds to a coframe field {ω β } on P, where the index β run over both the even and odd indices. Let {e α } be the dual frame, i.e., e α , ω β = (−1) |α||β| ω β (e α ) = δ β α . Here α runs through both even and odd indices as well. The following result was originally established in [32,Lemma 13]. We give here a simplified proof that does not use the concept of flow for supermanifolds.

Lemma 4.2.
Let {e α } be a frame on a supermanifold P = (P o , A P ) with connected reduced manifold. Fix a point x ∈ P 0 . Then any infinitesimal automorphism υ ∈ X(P) of the frame is determined by its value at x.
Proof. The statement is equivalent to the claim that ev x (υ) = 0 implies υ = 0.
Consider the ideal J = (A P ) 2 1 ⊕ (A P )1 of A P generated by nilpotents. Then for any k > 0 the map is injective. In other words, if f ∈ J k+1 , then there exists an odd α such that e α (f) ∈ J k . Now if υ is in J k ⊗ TP but not in J k+1 ⊗ TP for some k > 0, then the Lie equation L υ e α = 0 cannot hold for all odd α because there exists one for which it is wrong already modulo J k ⊗ TP. This tells us that the evaluation map ev : X(P) → Γ (T P| P o ) is injective on the symmetries.
Set υ = ev υ ∈ Γ (T P| P o ). Taking the Lie equation L υ e α = 0 modulo J ⊗ TP for an even α we get a pair of reduced Lie equations ev L υ0 e α = 0 , ev L e α υ1) = 0 , where f α ∈ C ∞ P o for any even α. Hence the value of a parallel section at one point determines the section everywhere. In summary, the map υ → ev x υ is injective.
Let us now observe how the dimension of the solution space is constrained. The structure equations [e α , e β ] = c γ αβ e γ ⇔ dω γ = − 1 2 (−1) |α||β| ω α ∧ ω β c γ αβ involve structure superfunctions c γ αβ ∈ A P of parity |α| + |β| + |γ|. An infinitesimal symmetry is a supervector field υ = a δ e δ ∈ X(P) such that L υ e α = 0 for all α. Equivalently it must preserve the coframe, so we get for all γ, which we rewrite as This is a complete PDE on the superfunctions a γ 's and the dimension bound dim P is achieved if and only if the compatibility conditions d 2 a γ = 0 holds. We can see this explicitly in local coordinates x α on P.
where we denoted σ γ ǫα = (−1) |υ||ǫ|+|α||β|+|α||ǫ| κ β ǫ c γ αβ . The compatibility conditions given by the vanishing of the supercommutator of ∂ ǫ ′ and ∂ ǫ ′′ on a γ are If the parenthetical expression vanishes for all indices, then any initial value for the a α 's produces a unique solution υ, and the dimension of the solution space is dim P. If not, then we have to differentiate the L.H.S. of (4.3), substitute (4.2) and study the 0-th order linear equations on the a α 's. When the system stabilizes, the corank of the resulting matrix (i.e., the matrix size minus the size of the largest invertible minor), gives the dimension of the solution space.

Dimension of the automorphism supergroup.
By the results in [32, §4.2], the group of automorphisms Aut(Φ)¯0 of an absolute parallelism Φ on a supermanifold P = (P o , A P ) with connected reduced manifold P o is a (finite-dimensional) Lie group. At first, if we denote by Φ0 the induced absolute parallelism on P o (in the notation of §4.1.2, this is Φ0 = {ev e α } α even ), then the classical argument of [20] proves that Aut(Φ¯0) is a Lie group: Aut(Φ¯0) is mapped to P o as the orbit through x ∈ P o and the stabilizer of a classical absolute parallelism at any point is trivial. Then, the forgetful map Aut(Φ)¯0 → Aut(Φ¯0) is injective with closed image, cf. [32, Lemmas 10 and 11]. It follows from this and Lemma 4.2 that the automorphism supergroup (Aut(Φ)0, aut(Φ)) is a finite-dimensional super Harish-Chandra pair, in other words a Lie supergroup. Here aut(Φ) is the Lie superalgebra of complete infinitesimal automorphisms. We remark that for a pair of complete supervector fields neither their linear combination nor their commutator are complete. (This holds also in the classical case.) However the set of complete supervector fields that are infinitesimal automorphisms of an absolute parallelism form a supervector space, and moreover a Lie superalgebra. This is because the sum and Lie bracket of two infinitesimal automorphisms of Φ0 is still complete by the classical result of [20] and a supervector field υ ∈ X(P) is complete if and only if the associated vector field ev(υ¯0) ∈ X(P o ) on P o is so [15,30]. This shows that the "representability issue" of [32,Thm 15] can be amended: completeness of the infinitesimal automorphisms (i.e., the requirement that inf(Φ) = aut(Φ)) is not an obstruction for the representability of the automorphism supergroup.
By the construction of the absolute parallelism Φ on P, it is not difficult to see that the automorphism supergroup Aut(M, D, q) = (Aut(M, D, q)0, aut(M, D, q)) of a non-holonomic geometric structure (D, q) on M or, more generally, of a filtered structure, is a closed subsupergroup of Aut(Φ) = (Aut(Φ)0, aut(Φ)). Therefore it is a Lie supergroup, whose dimension is bounded by dim P = dim g, and this finishes the proof of Theorem 1.2.
A more careful analysis shows that Aut(M, D, q) is a discrete quotient of Aut(Φ). Indeed, by Theorem 3.17, automorphisms in the base lift to the frame bundle. On the other hand, for any k, automorphisms of the frame bundle F k preserve the fundamental fields from g k and therefore they project to automorphisms of a cover of the frame bundle F k−1 , namely to the quotient of F k by the connected component of unity in the structure group G k . We apply this for k descending from d to 0 and conclude the claim. If the structure groups G k are connected (this is usually the case for k > 0, if no higher order reductions are imposed, because the fibers are affine), then we have the equality Aut(M, D, q) = Aut(Φ).

Remark 4.3.
In the case M o has finitely many connected components, say n ∈ N, one easily modifies the above arguments to get the following dimension bound: Indeed, enumerate the components 1, . . . , n of M o and let σ ∈ S n encodes a (possibly trivial) permutation of components. Then all automorphisms are parametrized as follows: no more than dim g parameters for maps of the 1st component to that number σ(1), no more than dim g parameters for maps of the 2nd component to that number σ(2), etc.

4.2.
Structure of the symmetry superalgebra and maximally symmetric spaces. We now discuss the following statement, which is not primary for the purposes of this paper. We therefore will only sketch the proof, referring the reader for details to the original papers.
Let g = g −µ ⊕ · · · ⊕ g 0 ⊕ · · · ⊕ g d be the Tanaka algebra associated to the filtered structure (M, D, q). Its natural (decreasing) filtration is given by subspaces g i = g i ⊕ · · · ⊕ g d , i −µ, which for µ = 1 is the so-called filtration by stabilizers and for µ > 1 is the weighted (or Weisfeiler) filtration.

Theorem 4.4.
The symmetry algebra s of (M, D, q) embeds into g as a filtered subspace ι : s → g in such a way that the corresponding graded map gr(ι) : gr(s) → g is an injection of Lie algebras.
Proof. Fix a point x ∈ M o and consider the weighted filtration of the stalk TM x that refines the filtration by the maximal ideal in (A M ) x using the weighted filtration induced by the distribution D. This generalizes to the super-setting the second filtration on the symmetry Lie algebra sheaf from [22] and gives the required embedding ι : s → g.
Alternatively, consider the Lie equation governing infinitesimal symmetries of (M, D, q) as a subsupermanifold embedded into the space of weighted super-jets. This provides the solution space s of the equation with the desired filtration, see [23].
This theorem serves as a base to obtain submaximally symmetric models via filtered deformations of large graded subalgebras of g, see [25] for applications in the classical case and [24] for examples in the super case. The so-called flat model is the homogeneous space G/H, with G a Lie supergroup with Lie(G) = g and H its closed subsupergroup with Lie(H) = g 0 . (One can impose simplyconnectedness of G/H though this is not necessary.) The filtration g i on g induces a leftinvariant filtration F i on G and therefore the distribution D = F −1 /F 0 on G/H with the desired derived flag. In addition, all the reductions are invariant w.r.t. G, hence the induced filtered structure is invariant. If q encodes the filtered structure, then inf(G/H, D, q) = g and Aut(G/H, D, q) coincides with the supergroup G or its discrete factor.
The so-called standard model is obtained through a left-invariant structure (D, q), or more generally a filtered structure, on the nilpotent Lie supergroup M = exp(m). This usually does not have the maximal automorphism group, but it is locally isomorphic to the flat model and hence inf(M, D, q) = g. Complete description of other models with maximal symmetry dimension can be obtained via the technique of filtered deformations of s = g.

E
Here we demonstrate how our dimensional bounds work. We emphasise that all our main results are applicable to both real smooth and complex analytic cases, so some examples will be stated over R and some over C. Thus for the symmetry algebra of ∇ we have: The Lie algebra of the even part Aut(M, ∇)0 of the Lie supergroup of affine transformations Aut(M, ∇) consists of supervector fields that are complete. Therefore, its dimension might be smaller than dim s in general.

Super-Riemannian structures.
A super-Riemannian structure on M is given by a nondegenerate even supersymmetric A M -bilinear form q on TM. (In the real case, the even part of q can have any signature.) It is a G 0 -structure with G 0 = OSp(m|n), n ∈ 2Z. For g 0 = Lie(G 0 ) it is known that g 1 = g (1) 0 = 0. The argument straightforwardly generalizes the classical one [38], see [32], which corresponds to the analog of the Levi-Civita connection [16]. Thus (M, q) determines an affine structure.
The Lie superalgebra of Killing supervector fields satisfies The above remark about completeness for affine structures applies to super-Riemannian structures and the isometry supergroup as well.

Almost super-symplectic structures.
An almost super-symplectic structure on M is given by a nondegenerate even super-skewsymmetric bilinear form q on T M. It is a G 0 -structure with G 0 = SpO(m|n), m ∈ 2Z. In this case g 0 = Lie(G 0 ) = spo(m|n) ∼ = osp(n|m) but as representations on V = R m|n ∼ = ΠR n|m these Lie superalgebras are quite different, cf. Remark 2.1. In particular g ⊂ gl(V) is of infinite type unless V is purely odd. Lemma 5.1. We have: g i = g (i) 0 = S i+2 V * (in the super-sense), which is nonzero ∀i 0 if m > 0. The proof of this claim mimics the proof of the classical computation for almost symplectic structure [38] and will be omitted. We note that Above st and ⊤ are the supertranspose and the usual transpose, respectively. We also define some related Lie superalgebras: • special periplectic spe(n) := pe(n) ∩ sl(n|n). This is simple for n 3.
Consider now the case when P is skew-supersymmetric, i.e. P(x, y) = −(−1) |x||y| P(y, x) for all pure parity x, y. The same formula as in (5.1) defines the skew-periplectic Lie superalgebra pe sk (n) and taking P = We may define analogous related Lie superalgebras as above. Clearly the parity change functor V → Π(V) on V = R n|n induces an isomorphism pe(n) ∼ = pe sk (n) as Lie superalgebras, but the differing representation on V is crucial as the following result shows: Proposition 5.3. Let g −1 = V = R n|n . If g 0 ⊃ spe sk (n), then g = pr(g −1 , g 0 ) has infinite odd part.
Proof. Focusing on the "B-part" of (5.3), we see that g 0 contains a rank 1 odd element x ⊗ ω, where x ∈ V1 and ω ∈ V * 0 . Considering the odd elements φ k = x ⊗ ω k+1 for all k > 0, we inductively observe that φ k ∈ g k for all k > 0. 5.1.5. Projective superstructures. Classical projective structures are defined as equivalence classes of affine connections for which geodesics differ by a reparametrization. It is wellknown that every class contains a torsion-free connection. We here omit the discussion of what a supergeodesic is since this is not uniform in the literature ( [16,28]) and simply follow [28] in adapting the classical interpretation of projective equivalence for the torsion-free connections: two torsion-free affine superconnections ∇ and ∇ ′ are equivalent if and only if ∇ − ∇ ′ = Id •ω ∈ Γ (S 2 T * M ⊗ TM) for an even 1-form ω ∈ Ω 1 (M). (The symmetric power is meant in the super-sense.) This is a higher order reduction of the frame bundle. Namely, using the gl(V)-equivariant splitting 1 (trace and trace-free parts), the principal bundle F 1 → F 0 = Fr M is reduced to the (Abelian) structure group g ′ 1 . After this the geometric structure is prolonged. The obtained projective structure has symmetry the entire diffeomorphism group in the case of (even) line. Proof. In dimension (0|1) we have g 1 = 0, so it is clear. Otherwise we claim that g i = (g ′ 1 ) (i−1) = 0 for all i > 1. Indeed the Spencer complex in Z-degree 2 is given by and its first cohomology group vanishes, i.e., Considering the vanishing of the remaining three terms gives A = 0. The first cohomology groups then must vanish in higher Z-gradings too and H 2,1 (V, V ⊕ gl(V) ⊕ g ′ 1 ) = 0 is equivalent to the prolongation claim, cf. [39,24].
Consequently, the symmetry dimension of a projective structure on a supermanifold M with dim M = (1|0) is bounded by dim s dim V + dim gl(V) + dim g ′ 1 = 2m + n 2 + m 2 | 2n + 2mn . Now we will consider examples of filtered structures in the nonholonomic case D TM.

G(3)-contact supergeometry.
Consider a contact distribution C of rank (4|4) on a supermanifold M of dimension (5|4). The induced conformally super-symplectic structure on C reduces the structure group to CSpO(4|4) and it is still of infinite type. A cone structure on C is given by a field of supervarieties in projectivized contact spaces. Namely for x ∈ M o the projective superspace PC| x contains a distinguished subvariety V| x of dimension (1|2) that is isomorphic to the unique irreducible flag manifold of the simple Lie supergroup OSp(3|2), namely V| x ∼ = OSp(3|2)/P II 1 , where p II 1 is the parabolic subalgebra × . We call this subvariety the (1|2)-twisted cubic, because its underlying classical manifold is a rational normal curve of degree 3, which is "deformed" in 2 odd dimensions.
This cone field reduces the structure group to COSp(3|2) ⊂ CSpO(4|4), and now this is of finite type: if g 0 = cosp(3|2) and g = g(3) is the Lie algebra of G(3) then H d,1 (m, g) = 0 if d > 0 by [24,Theorem 3.9], so the maximal prolongation is g = pr(m, g 0 ) (Corollary 3.10 loc.cit.). Such a geometric structure arises on the generalized flag-supervariety G(3)/P IV 1 with marked Dynkin diagram × and in [24,Theorem 4.9] we established that the maximal symmetry dimension (in the strong sense) of supergeometries (M, C, V) as above is (17|14), under the assumption that the geometry is locally homogeneous. Now as a direct corollary of Theorem 1.1 we derive that the assumption of local homogeneity can be removed (we fulfill thus what is written in footnote 5 at page 54 of loc.cit.):  (2|4, 1|2, 2|0). The symbol of a (fundamental, nondegenerate) superdistribution with such growth vector can be one of four types [24,Theorem 5.1], and just of two types if its even part is the standard symbol as for the Hilbert-Cartan equation. Moreover one of them is generic, hence rigid, and it is called SHC type symbol. More explicitly, for a basis of m (listed in the format (even|odd)) g −1 = e 1 , e 2 | θ ′ 1 , θ ′′ 1 , θ ′ 2 , θ ′′ 2 , g −2 = h | ρ 1 , ρ 2 , g −3 = f 1 , f 2 | · , the nontrivial commutator relations of the SHC type symbol are the following:

Super Hilbert-Cartan geometries. Another G(3) supergeometry lives on supermanifolds of dimensions (5|6) and it is given by a superdistribution with growth vector
Such a superdistribution arises on the generalized flag-supermanifold G(3)/P IV 2 with the marked Dynkin diagram × . For the grading corresponding to the parabolic P IV 2 the Lie superalgebra g = Lie(G(3)) contains m as the negative part. In [24,Theorem 3.16] we established H d,1 (m, g) = 0 for all d 0. Hence (Corollary 3.17 loc.cit.) g is the Tanaka-Weisfeiler prolongation of m, i.e., g = pr(m).
The methods of [24] allow to conclude that (17|14) is the maximal symmetry dimension for locally homogeneous distributions with the SHC symbol. Using Theorem 1.1 of the present paper we derive the result in full generality without the local homogeneity assumption.

Super-Poincaré structures.
Let V be a complex vector space with a non-degenerate symmetric bilinear form (·, ·) and S an irreducible module over the associated Clifford algebra. A supertranslation algebra is a Z-graded Lie superalgebra m = m −2 ⊕ m −1 , where m −2 = m0 = V and m −1 = m1 = S ⊕ · · · ⊕ S is the direct sum of an arbitrary number N 1 of copies of S, whose bracket Γ : m −1 ⊗ m −1 → m −2 is of the form (Γ (s, t), v) = B(v · s, t) for v ∈ V, s, t ∈ m −1 . Here B is a non-degenerate bilinear form on m −1 , which is admissible in the sense of [1]. We note that Γ is so(V)-equivariant, so the semidirect sum p = m ⋊ so(V) is a Lie superalgebra, usually referred to as Poincaré superalgebra (complex, N-extended, in dimension dim V).
Real supermanifolds M endowed with a strongly-regular odd distribution D ⊂ TM whose complexified symbol is m appear naturally in "super-space" approaches to supergravity and rigid supersymmetric field theories (see [36,37,12,13,9,10] and references therein). The superdistribution D has been called a super-Poincaré structure in [3] and the main result of that paper is the explicit description of the maximal transitive prolongation of m. Here we recall it for the reader's convenience:  Table 1. sl (2) T . Exceptional prolongations of super-Poincaré algebras. Here the simple roots of degree 1 coincide with the odd simple roots, i.e., those associated to black and gray nodes on the Dynkin diagram.
Since the complexification of the prolongation of a real symbol is the prolongation of the complexified symbol, one may combine Theorems 1.1 and 5.7 to get the bound on the dimension of the symmetry superalgebra of a super-Poincaré structure in dimension dim V 3. For the exceptional cases with g 1 = 0, it is provided by the second column in Table 1, in all other cases it is given by dim s where d = dim V and square brackets refer to the integer part. Furthermore, the subalgebra h 0 of the internal symmetries can be easily described on a case-by-case basis. It splits into the sum of its symmetric part h s 0 and skew-symmetric part h a 0 with respect to B and the condition that elements of h 0 act as derivations of m yields: It is well-known that S is so(V)-irreducible if d is odd and the direct sum of two inequivalent so(V)-irreducible submodules if d is even. By so(V)-equivariancy, a uniform (but not sharp) bound on dim h 0 is thus given by N 2 if d is odd and 2N 2 if d is even.
We conclude this subsection with the following direct consequence of §3.6. Consider, for instance, the 4-and 11-dimensional vector spaces V in Lorentzian signature. The real spinor module S is an irreducible module for the Lorentz algebra so(V) and it is of Clifford real type (i.e., S ⊗ C = S). It follows from Theorem 5.7 that, if we reduce the structure algebra to so(V), the prolongation of the real N = 1 Poincaré superalgebra is just p. Theorem 3.17 and Remark 3.18 then imply that any super-Poincaré structure D with reduced structure group P 0 = Spin(V) has associated a Cartan superconnection on a P 0 -principal bundle π : P → M. This bridges from the "super-space approach" to the socalled "rheonomic approach" of supergravity and supersymmetric field theories (see, e.g., the nice reviews [8,6]). We stress that in the rheonomic approach the axioms of a Cartan superconnection follows from a Lagrangian principle on the absolute parallelism Φ, whereas our general construction affords the existence of the Cartan superconnection, from purely geometric arguments.
It would be interesting to study the normalization conditions on the Cartan superconnection in the cohomological spirit of §3.5.3 and compare them with those traditionally obtained in the rheonomic approach via Lagrangian principles.

Odd Ordinary Differential Equations.
5.4.1. Review of some classical ODE. Classically, ODE are geometrically viewed as submanifolds of a jet space with the inherited structure (via pullback along the inclusion map). This leads to formulating these as manifolds M with a rank 2 distribution C ⊂ T M (having specific symbol m) and a splitting into line fields C = E ⊕ V: • 2nd order ODE y ′′ = f(x, y, y ′ ) (up to point transformations): Introduce local coordinates (x, y, p) on M with C = E ⊕ V = ∂ x + p∂ y + f∂ p ⊕ ∂ p . Then C has symbol m = g −1 ⊕ g −2 = X, e 1 ⊕ e 2 with non-trivial bracket [X, e 1 ] = e 2 . (C is a contact distribution.) • 3rd order ODE y ′′′ = g(x, y, y ′ , y ′′ ) (up to contact transformations): Introduce local coordinates (x, y, p, q) on M 4 with C = E ⊕ V = ∂ x + p∂ y + q∂ p + g∂ p ⊕ ∂ q . Then C has symbol m = g −1 ⊕ g −2 ⊕ g −3 = X, e 1 ⊕ e 2 ⊕ e 3 with non-trivial brackets [X, e 1 ] = e 2 , [X, e 2 ] = e 3 . (C is an Engel distribution.) The splitting indicates a reduction g 0 ֒→ der gr (m). In both cases, dim(g 0 ) = 2 with g 0 ֒→ gl(g −1 ) corresponding to scalings along the two distinguished directions in g −1 .
There are well-known Z-gradings of A 2 ∼ = sl 3 and B 2 ∼ = so 2,3 for which the negative parts g − are the indicated symbol algebras above and the non-negative parts p = g 0 are the respective Borel subalgebras. This implies inclusions of sl 3 and so 2,3 into the respective Tanaka prolongations pr(g − , g 0 ). One can show that these are, in fact, equalities by verifying that H +,1 (g − , g) = 0, as was done by Yamaguchi [42] using Kostant's theorem. (Previously this was done by Tresse, Cartan and Chern using geometric methods.) 5.4.2. 2nd order odd ODE. Consider a 2nd order odd ODE ξ ′′ = F(x, ξ, ξ ′ ), where ξ is an odd function of the even variable x, and F is an odd function. As in the classical case, the space for even X and odd θ 1 , θ 2 satisfying [X, θ 1 ] = θ 2 .
Letting E ij be the matrix with a 1 in the (i, j)-position and 0 elsewhere, we use (Z 1 , Z 2 ) to induce a bigrading on g, and let Z = Z 1 + Z 2 be the induced grading on g. In particular: Here E 21 is even, while E 31 , E 32 are odd. The only non-trivial bracket on g − is [E 32 , E 21 ] = E 31 . We conclude that sl(2|1) includes into pr(g − , g 0 ). From Table 2, we use the differentials δ k : C k (g − , g) → C k+1 (g − , g) to conclude that H +,1 (g − , g) = 0, whence that pr(g − , g 0 ) ∼ = sl(2|1).
This vanishes for solutions of the even 2nd order ODE system and this trivializes the odd ODE (5.6).
Let C ∈ g 3 be even. Write CX = 0, Cθ 1 = cB. We find C = 0 from: Now let C ∈ g 3 be odd. Write CX = cB, Cθ 1 = 0. We find C = 0 from: When G = 0, i.e. ξ ′′′ = 0, we have the symmetries X = S f (satisfying L X E ⊂ E and L X V ⊂ V), expressed in terms of a generating superfunction f (see Appendix A).
In contrast to the 2nd order odd ODE case, 3rd order odd ODE are not in general contacttrivializable, i.e. equivalent to ξ ′′′ = 0. In fact, 3rd order odd ODE have the form ξ ′′′ = a(x)ξ + b(x)ξ ′ + c(x)ξ ′′ + d(x)ξξ ′ ξ ′′ and one can verify that the term d(x) is a relative invariant. (The even part of the contact supergroup is (x, ξ) → (α(x), β(x)ξ) and the verification is straightforward; the odd part does not contribute.) Consequently, general 3rd order odd ODE are not linearizable.
Below we exhibit two examples of 3rd order odd ODEs that are not contact-trivializable. Both have solvable symmetry superalgebras. Symmetries are given in terms of their generating superfunctions (see Appendix A for the Lagrange bracket).

A
The research leading to these results has received funding from the Norwegian Financial Mechanism 2014-2021 (project registration number 2019/34/H/ST1/00636) and the Tromsø Research Foundation (project "Pure Mathematics in Norway").

C I
The authors declare that they have no conflict of interest. A

A. G
In this section, we briefly discuss the jet spaces J r := J r (R p|0 , R 0|1 ) associated with p even variables (x i ) and one odd (dependent) variable ξ. Our interest is only local, so we introduce these spaces (with Cartan distribution C r ⊂ T J r ) from a local perspective only: • r = 0: Coordinates (x i , ξ). No local structure.
A vector field S on J r (R p|0 , R 0|1 ) is contact if L S C r ⊂ C r . By the Lie-Bäcklund theorem, any such vector field is projectable over J 1 , and S is canonically determined from this projection via prolongation. On J 1 , fixing σ = dξ − (dx i )ξ i generating C 1 , any contact vector field S is uniquely determined by its generating superfunction f = ι S σ (which has opposite parity to S since σ is odd), and conversely any local superfunction f = f(x i , ξ, ξ i ) determines a contact vector field: Proposition A.1. Given a superfunction f = f(x i , ξ, ξ i ) with pure parity |f|, its associated contact vector field has parity |f| + 1 and is given by the formula S f = (−1) |f| (∂ ξ i f)D (1) x i + f∂ ξ + (D (1) We have [S f , S g ] = S [f,g] with [f, g] = f(∂ ξ g) + (−1) |f| (∂ ξ f)g + (D (1) x j f)(∂ ξ j g) + (−1) |f| (∂ ξ j f)(D (1) x j g). This result is analogous to [24,Prop.4.3]. Details are left as an exercise for the reader since it is proved similarly.