Remarks on the non-Riemannian sector in Double Field Theory

Taking O(D,D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {O}(D,D)$$\end{document} covariant field variables as its truly fundamental constituents, Double Field Theory can accommodate not only conventional supergravity but also non-Riemannian gravities that may be classified by two non-negative integers, (n,n¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n,\bar{n})$$\end{document}. Such non-Riemannian backgrounds render a propagating string chiral and anti-chiral over n and n¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{n}$$\end{document} dimensions respectively. Examples include, but are not limited to, Newton–Cartan, Carroll, or Gomis–Ooguri. Here we analyze the variational principle with care for a generic (n,n¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n,\bar{n})$$\end{document} non-Riemannian sector. We recognize a nontrivial subtlety for nn¯≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n\bar{n}\ne 0}$$\end{document} that infinitesimal variations generically include those which change (n,n¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n,\bar{n})$$\end{document}. This seems to suggest that the various non-Riemannian gravities should better be identified as different solution sectors of Double Field Theory rather than viewed as independent theories. Separate verification of our results as string worldsheet beta-functions may enlarge the scope of the string landscape far beyond Riemann.


Introduction
This paper is a sequel to [1] which proposed to classify all the possible geometries of Double Field Theory (DFT) [2][3][4][5][6][7] by two non-negative integers, (n,n). The outcome -which we shall review in Sect. 2 -is that only the case of (0, 0) corresponds to conventional supergravity based on Riemannian geometry. Other generic cases of (n,n) = (0, 0) do not admit any invertible Riemannian metric and hence are non-Riemannian by nature. Strings propagating on these backgrounds become chiral and anti-chiral over n andn dimensions respectively.
In this work we attempt to explore the dynamics of the generic (n,n) sector in Double Field Theory. We analyze with care the relevant variational principle and recognize a nontrivial subtlety: when nn = 0, the resulting Euler-Lagrangian equations of motion depend whether the variations of the action keep the values of (n,n) fixed or not. This rather unexpected subtle discrepancy contrasts DFT with the traditional approaches to the various non-Riemannian gravities.
The organization of the present paper is as follows.
In the remaining of this Introduction, to put the present work into context and set up notation, we describe DFT as the O(D, D) completion of General Relativity along with a relevant doubled string action.
In Sect. 2, we review the (n,n) classification of the non-Riemannian DFT geometries from [1].
In Sect. 3, we revisit the variational principle in DFT and confirm that the known Euler-Lagrangian equations, or 'Einstein Double Field Equations' (1.3) are still valid for non-Riemannian sectors.
In Sect. 4, now keeping (n,n) fixed, we reanalyze the variational principle and show that the full Einstein Double Equations are not necessarily implied when nn = 0. We explain the discrepancy, and further propose a non-Riemannian differential tool kit as a 'bookkeeping device' to expound the equations.
We conclude in Sect. 5, followed by Appendix A and B.

Double Field Theory as the O(D, D) completion of General Relativity
While the initial motivation of Double Field Theory was to reformulate supergravity in an O(D, D) manifest manner [2][3][4][5][6][7] ( [59][60][61] for reviews), through subsequent further developments [62][63][64][65], DFT has evolved and can be now identified as a pure gravitational theory that string theory seems to predict foremost 1 and may differ from General Relativity as it is capable of describing non-Riemannian geometries [1]. Specifically, DFT is the string theory based, O(D, D) completion of General Relativity (GR): taking the O(D, D) symmetry as the first principle, DFT geometrises not merely the Riemannian metric but the whole massless NS-NS sector of closed string as the fundamental gravitational multiplet, hence 'completing' GR. Further, the O(D, D) symmetry principle fixes its coupling to other superstring sectors (R-R [68][69][70][71], R-NS [72], and heterotic Yang-Mills [73][74][75]). Having said that, regardless of supersymmetry, it can also couple to various matter fields which may appear in lower dimensional effective field theories [72,76,77], just as 1 At least formally let alone its phenomenological validity, c.f. [66,67].
GR does so. In particular, supersymmetric extensions have been completed to the full (i.e. quartic) order in fermions for D = 10 cases powered by '1.5 formalism' [78,79], and the pure Standard Model without any extra physical degrees of freedom can easily couple to DFT in an O(D, D) symmetric manner [80]. Schematically, governed by the O(D, D) symmetry principle, DFT may couple to generic matter fields, say collectively ϒ, which should be also in O(D, D) representations: 1 16π G e −2d S (0) + L matter (ϒ, ∇ A ϒ).
( which, with its inverse J AB , is going to be always used to lower and raise the O(D, D) vector indices (Latin capital letters). It splits the doubled coordinates into two parts,  [81] and also the Energy-Momentum tensor, T AB [65], of which the former and the latter are respectively off-shell and on-shell conserved. Equating the two, they comprise the O(D, D) completion of the Einstein field equations, or the Einstein Double Field Equations (EDFEs) [65,82], We summarize the basic geometrical notation of DFT in Table 1, 2 while the DFT-diffeomorphisms are generated by the so-called generalized Lie derivative [3,7]: acting on a tensor density with weight ω T , (1.4) In particular, being a scalar density with weight one (ω T = 1), the exponentiation e −2d is the integral measure of DFT. It is noteworthy and relevant to this work that, all the geometrical notation of the covariant derivative, ∇ A , and the 2 The expression of S AB in Table 1 is newly derived from [63] using  Integral measure e −2d (weight one scalar density) We remind the reader that the O(D, D) indices are raised with J AB . Upon imposing the section condition, the generalized Lie derivative (1.4) is closed by commutators [3,7], The section condition is mathematically equivalent to the following translational invariance [8,83], where the shift parameter, A , is derivative-index-valued, meaning that its superscript index should be identifiable as a derivative index, for example A = s ∂ A t . This insight on the section condition may suggest that the doubled coordinates of DFT are in fact gauged by an equivalence relation, Each gauge orbit, i.e. equivalence class, represents a single physical point. As a matter of fact in DFT, the usual infinitesimal one-form of coordinates, dx A , is not DFTdiffeomorphism covariant, However, if we gauge the one-form by introducing a derivative-index-valued connection, we can have a DFTdiffeomorphism covariant one-form, provided that the gauge potential transforms appropriately, (1.10) It is also a singlet of the coordinate gauge symmetry (1.8): which enjoys symmetries like global O(D, D), target spacetime DFT-diffeomorphisms, worldsheet diffeomorphisms, Weyl symmetry, and the coordinate gauge symmetry. 3 All the background information is encoded in the DFT-metric, H AB .

Review of [1]: classification of the non-Riemannian DFT geometries
The section condition can be generically solved, up to O(D, D) rotations, by enforcing the tilde coordinate independency: , we note that the tilde coordinates are indeed gauged: With respect to this choice of the section, the well-known parametrization of the DFT-metric and the DFT-dilaton in terms of the conventional massless NS-NS field variables [88,89], reduces DFT to supergravity. In this case, the single expression of the EDFEs (1.3) unifies all the equations of motion of the three fields, {g μ , B μν , φ}. Further, after Gaussian integration of the auxiliary gauge potential, A μ , the doubled-yetgauged string action (1.11) reproduces the standard undoubled string action. Yet, this is not the full story. The above parametrization (2.1) is merely one particular solution to the defining relations of the DFT-metric: DFT and the doubled-yet-gauged string action work well, provided these conditions are fulfilled. For example, instead of (2.1), we may let the DFT-metric coincide with the O(D, D) invariant metric, This is a vacuum solution to DFT, or to the 'matter-free' EDFEs, G AB = 0 (1.3), which is very special in several aspects. Firstly, compared with (2.1), there cannot be any associated Riemannian metric g μν and hence it 3 See also [85] for Green-Schwarz doubled superstring, [66] for doubled point particle, and [86,87] for 'exceptional' extensions.
does not allow any conventional or Riemannian interpretation at all. It is maximally non-Riemannian. Secondly, it is fully O(D, D) symmetric, being one of the two most symmetric vacua of DFT, H AB = ±J AB . Thirdly, it is moduli-free since it does not admit any infinitesimal fluctuation, δH AB = 0 [75]. 4 And lastly but not leastly, upon this background, the auxiliary gauge potential, A μ , appears linearly rather than quadratically in the doubled-yet-gauged string action (1.11). Consequently it serves as a Lagrange multiplier to prescribe that all the untilde target spacetime coordinates should be chiral [8] (c.f. [90,91]), An intriguing insight from [11] is then that, the usual supergravity fields in (2.1) would be the Nambu-Goldstone modes of the perfectly O(D, D) symmetric vacuum (2.3). Given the Riemannian and maximally non-Riemannian backgrounds, (2.1) v.s. (2.3), one may wonder about the existence of more generic non-Riemannian geometries (c.f. [8,10] for other examples and also [22] for 'timelike' duality rotations). This question was answered in [1]: the most general solutions to the defining properties of the DFTmetric (2.2) can be classified by two non-negative integers, (n,n), where i, j = 1, 2, . . . , n,ī,j = 1, 2, . . . ,n and 0 ≤ n+n ≤ D.
(i) While the B-field is skew-symmetric as usual, H μν and K μν are symmetric tensors whose kernels are spanned by linearly independent vectors, X i μ ,X¯ı ν and Y μ j ,Ȳ ν j , respectively, (ii) A completeness relation must be satisfied From the linear independency of the zero-eigenvectors, X i μ ,X¯ı ν , orthogonal/algebraic relations follow (2.8) Intriguingly, the B-field (hence 'Courant algebra') is universally present regardless of the values of (n,n), and contributes to the DFT-metric through an O(D, D) adjoint action: (2.9) whereH corresponds to the 'B-field-free' DFT-metric, and B is an O(D, D) element containing the B-field, It is also worth while to note the 'vielbeins' or 'square-roots' of K μν and H μν : where a, b are (D − n −n)-dimensional indices subject to a flat metric, say η ab , whose signature is arbitrary. Essentially, In fact, the analysis of the DFT-vielbeins corresponding to the (n,n) DFT-metric (2.5) carried out in [1] shows that the local Lorentz symmetry group, i.e. spin group is Spin(t + n, s + n) × Spin(s +n, t +n). (2.14) Here (t, s) is the arbitrary signature of η ab or the nontrivial signature of H μν and K μν satisfying t + s + n +n = D. Of course, once the spin group of any given theory is specified, it is fixed once and for all. Thus, each sum, t +n, s+n, s+n, and t +n, should be constant. For example, the Minkowskian D = 10 maximally supersymmetric DFT [85] and the doubledyet-gauged Green-Schwarz superstring action [79], both having the local Lorentz group of Spin(1, 9) × Spin(9, 1), can accommodate (0, 0) Riemannian and (1, 1) non-Riemannian sectors only (see [12] for examples of supersymmetric non-Riemannian backgrounds). Nevertheless, we may readily relax the Majorana-Weyl condition therein [79,85] and impose the Weyl condition only on spinors, such that the local Lorentz group can take any of Spin(t,ŝ) × Spin(ŝ,t) witht +ŝ = 10. The allowed non-Riemannian geometries will be then (n, n) types with n =n running from zero to min(t,ŝ) [1]. On the other hand, bosonic DFT does not care about spin groups and hence should be free from such constraints. It can admit more generic (n,n) non-Riemannian geometries. Crucially, the (n,n) parametrization of the DFT-metric (2.5) possesses two local symmetries, namely GL(n) × GL(n) rotations and Milne-shift transformations. The GL(n) × GL(n) symmetry rotates the i, j, . . . andī,j, . . . indices of the component fields: with infinitesimal local parameters, w i j andwīj , The Milne-shift symmetry generalizes the so-called 'Galilean boost' in the Newtonian gravity literature [40,41]. It acts with infinitesimal local parameters, V μi andV μī , 5 Remarkably, both transformations, (2.15) and (2.16), leave the DFT-metric invariant, as the two local symmetries are actually parts of the underlying local Lorentz symmetries (2.14). Upon the (n,n) background, the doubled-yet-gauged worldsheet string action (1.11) reduces to 1 2πα which should be supplemented by the chiral and anti-chiral constraints over the n andn directions, These constraints are prescribed by the integrated-out auxiliary gauge potential A A (1.10). 5 Through exponentiations, finite Milne-shift transformations can be achieved, which turn out to get truncated at finite orders, for example [1] for the full list.

Comment 1.
Matching with the content of the non-Riemannian component fields, (2.20) and the undoubled string worldsheet action resulting from (1.11), one can identify the original Newton-Cartan [33][34][35] as (1, 0), Stringy Newton-Cartan [36] as (1, 1), Carroll [37,38] as (D−1, 0), and Gomis-Ooguri [39] as (1, 1): see [1,11,57] for the details of the identifications. Further, the isometry of the (1, 1) flat DFT-metric matches with the non-relativistic symmetry algebra such as Bargmann algebra [10], while the notion of T-duality persists to make sense in the non-relativistic string theory [47]. These all seem to suggest that DFT may be the home, i.e. the unifying framework, to describe various known as well as yet-unknown non-Riemannian gravities. 6 Having said that there are also a few novel ingredients from DFT, such as the local GL(n)×GL(n) symmetry (2.15), the notion of 'Milne-shift covariance' as we shall discuss below (2.24), (2.26), and the very existence of the DFT-dilaton of which the exponentiation, e −2d , gives the integral measure in DFT being a scalar density with weight one, It is worth while to generalize the decomposition (2.9) to an arbitrary DFT tensor, Under diffeomorphisms, while the DFT tensor T A 1 ···A n is surely subject to the generalized Lie derivative (1.4), the circled quantity,T A 1 ···A n , is now governed by the undoubled ordinary Lie derivative which can be conveniently obtained as the truncation of the generalized Lie derivative by choosing the section,∂ μ ≡ 0, and setting the parameter, ξ A = (0, ξ μ ) asξ ν ≡ 0: (2.23) 6 Similarly, inequivalent parametrizations of the DFT-vielbeins, or Uduality-covariant generalized metric, correspond to the conventional distinctions between IIA and IIB [79,85], or IIB and M-"theories" [92].
Further, by construction, a DFT tensor is Milne-shift invariant. Yet, the circled one is Milne-shift covariant in the following manner, Explicitly, for a DFT vector, (2.25) That is to say, the circled quantities, For consistency, we also note for the O(D, D) invariant metric, Here G AB and T AB are respectively the stringy or O(D, D) completions of the Einstein curvature [81] and the Energy-Momentum tensor [65], as summarized in Table 1. The above result is easy to obtain once we neglect a boundary contribution arising from a total derivative [63]: and take into account a well-known identity which the infinitesimal variation of the DFT-metric should satisfy [7,62,94], Table 1. Equation (3.3) holds because the DFT-metric is constrained to be a symmetric O(D, D) element (2.2), see also (3.5) below. This is the reason why in the variation of the action (3.1) δH AB is contracted with a projected quantity, i.e. (PGP) AB − 8π G(PTP) AB . Equation (3.1) is then supposed to give the EDFEs, G AB = 8π GT AB (1.3) [65], as the two variations, δH AB and δd, give the projected part and the trace part separately, To answer this, we shall directly identify the truly independent degrees of freedom in the infinitesimal fluctuations of an arbitrary (n,n) non-Riemannian DFT-metric, as (3.12). We shall confirm that the full Einstein Double Field Equations are still valid for non-Riemannian sectors, either trivially due to projection properties or nontrivially from the genuine variational principle.

Variations of the DFT-metric around a generic (n,n) background
Here we shall identify the most general form of the infinitesimal fluctuations around a generic (n,n) DFT-metric (2.5).
The fluctuations must respect the defining properties of the DFT-metric (2.2) and hence satisfy That is to say, the trace of the DFT-metric, H A A = 2(n −n), is invariant under continuous deformations.
Without loss of generality, like (2.9), we put With this ansatz, the former condition in (3.5) is met and the latter gives We need to solve these three constraints. For this, we utilize the completeness relation (2.13), and decompose each of {α, β, γ } into mutually orthogonal pieces, where, since α, β are symmetric, We remind the readers that, using the (D−n−n)-dimensional flat metric, η ab , we freely raise or lower the indices, a, b. Now, with the decomposition (3.9), it is straightforward to see that (3.8) implies Thus, the independent degrees of freedom for the fluctuations consist of In total, as counted sequently as As the number of the equations and the fluctuations are the same, we may well expect that the former should be implied by the variational principle generated by the latter. Below, we confirm this directly through explicit computation, without using the DFT-vielbeins.

Einstein Double Field Equations still hold for non-Riemannian sectors
Now, we proceed to organize the variation of the action induced by that of the (n,n) DFT-metric (3.1) in terms of the independent degrees of freedom for the fluctuations (3.12). We apply the prescription (2.22) and write a pair of circled 'B-field-free' symmetric projectors, C = 0, and useful identities, We also introduce a shorthand notation for the Einstein Double Field Equations, Hereafter, hatted quantities contain generically the H-flux, but, like the circled ones, there is no apparent bare B-field in them.
It is now straightforward to compute the variation in (3.1), Each term is independent and thus, from the variational principle, should vanish individually on-shell, In total, as counted sequently as, there is D 2 − (n −n) 2 number of independent on-shell relations, or EDFEs, in consistent with (3.13). Up to the completeness relations (2.7), (2.13), and the identities (3.17), the first and the seventh in (3.21), the first and the eighth, the third and the fifth, the third and the sixth, the second and the last, the fourth and the last imply respectively, Finally, the first and the last, the second and the fifth, the third and the last, the fourth and the fifth give (P EP) μ ν = 0, (P EP) μν = 0, In this way, all the components of (P EP) AB vanish and the full EDFEs persist to be valid universally for arbitrary (n,n) backgrounds.
Comment. From (3.17), off-shell relations hold among the components of the EDFEs, such that the full EDFEs are satisfied if (3.26)

What if we keep (n,n) fixed once and for all?
As it is a outstandingly hard problem to construct an action principle for non-Riemannian gravity (c.f. [45,46,48] for recent proposals), we may ask if the DFT action restricted to a fixed (n,n) sector might serve as the desired target spacetime gravitational action, c.f. (4.21). In this section, seeking for the answer to this question, we reanalyze the variational principle of DFT, crucially keeping (n,n) fixed. To our surprise, we obtain a subtle discrepancy with the previous section where the most general variations of the DFT-metric were analyzed. We shall see that, when the values of (n,n) are kept fixed and nn = 0, not all the components of the EDFEs (3.26) are implied by the variational principle.

Variational principle with fixed (n,n)
We start with (3.1) which gives the variation of the general DFT action induced by the DFT-metric. With fixed (n,n), the variation of the DFT-metric therein should comprise the variations of the (n,n) component fields: Further, from their defining relations, (2.6), (2.7), the variations of the (n,n) component fields are not entirely independent. They must meet From (2.12), we also note which imply in particular, It is then evident from (4. The variational principle implies either from the second line of (4.6), or alternatively from the third line of (4.6), Although (4.7) and (4.8) appear seemingly different, they are -as should be -equivalent. In fact, they are both equivalent to (4.9) which are, from (3.25), further equivalent to more concise ones, δ H μν , δK ρσ , δX i μ , δY ν j , δX¯ı ρ , δȲ σ j , δB μν , (4.11) contribute actually to the α, β, γ variables defined in the generic variation of the DFT-metric (3.7), With (3.9), one can identify the contributions thoroughly: (4.13) and This is consistent with the general result of (3.11). However, one surprise is that α iī must be trivial when the (n,n) component fields (4.11) are varied while keeping (n,n) fixed.
To identify the significance of the α iī parameter, we focus on the induced transformation of H μν , Geometrically the deformation of 2Y ı α iī is 'orthogonal' to H μν , and thus we expect it should reduce the kernel of H μν . To verify this explicitly, we solve for the eigenvectors of H μν with zero eigenvalue, Without loss of generality, utilizing the completeness relation, K μa H νa + X i μ Y ν i +X¯ı μȲ ν ı = δ μ ν , we decompose the zero-eigenvector, (4.17) substitute this ansatz into (4.16), and acquire the conditions the coefficients should satisfy, This shows that there are in total (n − rank [α iī ]) + (n − rank [α iī ]) = n +n − 2 × rank [α iī ] number of zeroeigenvectors. Moreover, from the invariance, δH A A = 0 (3.6), we note that the deformation by the α iī parameter actually changes the type of the 'non-Riemannianity' as This essentially explains why α iī vanishes in (4.13) where the (n,n) component field variables are varied with fixed values of (n,n), or fixed 'non-Riemannianity'. It is intriguing to note that the deformation makes the DFT-metric always less non-Riemannian. 8

Non-Riemannian differential geometry as bookkeeping device
This subsection is the last one before Conclusion, and is somewhat out of context. At first reading, readers may glimpse (4.21) in comparison with (4.20), and skip to the final Sect. 5.
While the various (n,n) non-Riemannian geometries are universally well described by DFT through O(D, D) covariant tensors -as summarized in Table 1 (4.20) Here in this last section, we propose an undoubled non-Riemannian differential tool kit, such as covariant derivative and curvature, for an arbitrary (n,n) sector. It descends from the DFT geometry, or the so-called "semi-covariant formalism" [63], and generalizes the standard Riemannian geometry underlying (4.20) in a consistent manner. It breaks the manifest O(D, D) symmetry spontaneously, but preserves the ordinary diffeomorphisms, B-field gauge symmetry, and the GL(n)×GL(n) local symmetries as desired. In particular, it enables us to extend the Riemannian expression of (4.20) in a way 'continuously' to the generic (n,n) non-Riemannian case, We commence our explanation. First of all, D μ is our proposed 'upper-indexed' covariant derivative: which preserves both the undoubled diffeomorphisms (2.23) and the GL(n) × GL(n) local symmetries (2.15) as is equipped with proper connections: for undoubled ordinary diffeomorphisms, and for GL(n) × GL(n) rotations, We also denote a diffeomorphism-only preserving covariant derivative by and write for (4.22) and (4.24), Taking care of both spacetime and GL(n) × GL(n) indices, D μ acts on general tensor densities in a standard manner: On the other hand, D μ cares only the spacetime indices and ignores any GL(n) × GL(n) indices, For example, we have explicitly It is instructive to see that the far right resulting expressions in (4.29) are clearly covariant under both diffeomorphisms and GL(n)×GL(n) local rotations, as the ρ, σ indices therein are skew-symmetrized and also contracted with H μρ , (K H) ν σ .
However, without the GL(n) × GL(n) connections, we note (4.30) and this breaks the GL(n) × GL(n) local symmetry. Further, for the DFT-dilaton we should have where we have explicitly (4.32) Because H μν and K ρσ are generically degenerate, the conventional relation (2.1) between the DFT-dilaton, d, and the string dilaton, φ, cannot hold. We stick to use the DFT-dilaton all the way. 9 The connections do the job as they transform properly under the diffeomorphisms (2.23), (2.25) and the GL(n) × GL(n) local rotations (2.15), In particular, X i μ μν λ ,X¯ı μ μν λ , and H ρ[λ μ]ν ρ are covariant tensors which might be viewed as "torsions".
Finally, we define an upper-indexed Ricci curvature, which is diffeomorphism and GL(n)×GL(n) covariant, as it comes from the following commutator relation that is clearly also covariant, A scalar curvature follows naturally, which debuted in (4.21).
Our covariant derivative is "compatible" with the (n,n) component fields in a generalized fashion: Another characteristic is that, if we add one more torsion linear in the H-flux to the -connection, where H λμν is a diffeomorphism covariant, GL(n) × GL(n) invariant, and Milne-shift invariant H-flux, The GL(n) × GL(n) connections (4.26) are inert to the addition of the H-flux-valued-torsion (4.38) as After all, in terms of a hatted covariant derivative, (4.42) we can dismantle the DFT curvatures into a H-flux-free (circled) term and evidently H-flux-valued ones:  (4.43) where, as it should be obvious from our notation, we set S AB := (B −1 ) A C (B −1 ) B D S C D , and the circled quantities are all H-flux free: from Table 1 or [63,65], (4.44) and, with (3.16), (4.45) While we organize the H-flux-valued parts in terms of the hatted covariant derivative, like (4.41), we have The only nontrivial distinction lies in Since e −2d H λμν carries a unit weight, its contraction with the ordinary derivative, ∂ λ (e −2d H λμν ), is also by itself diffeomorphism covariant. In this way, every single term in From (3.26), vanishing of all the five quantities in (4.43) characterizes the (n,n) vacuum geometry of DFT.

Comment 1.
It is worth while to note (4.49) and rewrite the 'kinetic term' of the DFT-dilaton in (4.21), Consequently, the proposed covariant derivative (4.25) and Ricci curvature (4.34) reduce to the standard covariant derivative and Ricci curvature in Riemannian geometry, Appendix B sketches how we have arrived at the above proposal of the non-Riemannian differential tool kit starting from the semi-covariant formalism of DFT. In any case, our proposal is meant to provide a bookkeeping device to expound the EDFEs into smaller modules and to single out the H-fluxes. The actual computation of the variations of the action, even with (n,n) fixed, are still powered by the semi-covariant formalism, specifically (3.2).

Conclusion
The very gravitational theory string theory predicts may be the Double Field Theory with non-Riemannian surprises, rather than General Relativity based on Riemannian geometry. The underlying mathematical structure of DFT unifies supergravity with various non-Riemannian gravities including (stringy) Newton-Cartan geometry, ultra-relativistic Carroll geometry, and non-relativistic Gomis-Ooguri string theory. The non-Riemannian geometries of DFT can be classified by two non-negative integers, (n,n) [1].
We have analyzed with care the variational principle. We have shown that the most general infinitesimal variations of an arbitrary (n,n) DFT-metric have D 2 − (n −n) 2 number of degrees of freedom, which matches with the dimension of the underlying coset [11],

O(D,D)
O(t+n,s+n)×O(s+n,t+n) (3.14). Through action principle, these variations imply the full Einstein Double Field Equations (3.22), (3.24). However, nn number of them change the value of (n,n), i.e. the type of non-Riemannianity (4.19). Consequently, if we keep (n,n) fixed once and for all, the variational principle gets restricted and fails to reproduce the full EDFEs: the specific part, Y μ i (P EP) μνȲ ν ı , does not have to vanish on-shell (4.9). 10 The EDFEs are supposed to arise as the string worldsheet beta-functions [97,98]. For the doubled-yet-gauged string action (1.11) upon an arbitrarily chosen (n,n) background, the (n,n)-changing variations of the DFT-metric would correspond to marginal deformations. We must stress that these deformations could not be realized by merely varying the background component fields with fixed (n,n) (4.13), c.f. [52,54,56]. Nevertheless, it is natural to expect that nn number of Y μ i (P EP) μνȲ ν ı arise as the corresponding betafunctions too. That is to say, at least for nn = 0, the quantum consistency with the worldsheet string theory seems to forbid us to fix (n,n) rigidly. We conclude that the various non-Riemannian gravities should be identified as different solution sectors of Double Field Theory rather than viewed as independent theories. Quantum consistency of the non-Riemannian geometries calls for thorough investigation, which may enlarge the scope of the string theory landscape far beyond Riemann.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: This work is genuinely theoretical.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- 10 As can be seen from (4.44), Y Consequently, with the completeness relation (2.7), the identities from (3.17), and (A.2), we note It follows that With ∂ A = (0, ∂ μ ) and ξ A = (0, ξ μ ) (2.23), using Eq. (2.43) of [65], we get under diffeomorphisms, Alternative combination of (B.1), rather than (B.5), can give different type of covariant derivatives, (B.13) However, these can act only on one-form fields, and appear not so useful.