Scalar curvature and the multiconformal class of a direct product Riemannian manifold

For a closed, connected direct product Riemannian manifold $(M, g)=(M_1\times\cdots\times M_l, g_1\oplus\cdots\oplus g_l)$, we define its multiconformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus \cdots\oplus f_l^2g_l\}$ of all Riemannian metrics obtained from multiplying the metric $g_i$ of each factor $M_i$ by a function $f_i^2>0$ on the total space $M$. A multiconformal class $ [\![ g ]\!]$ contains not only all warped product type deformations of $g$ but also the whole conformal class $[\tilde{g}]$ of every $\tilde{g}\in [\![ g ]\!]$. In this article, we prove that $ [\![ g ]\!]$ carries a metric of positive scalar curvature if and only if the conformal class of some factor $(M_i, g_i)$ does, under the technical assumption $\dim M_i\ge 2$. We also show that, even in the case where every factor $(M_i, g_i)$ has positive scalar curvature, $ [\![ g ]\!]$ carries a metric of scalar curvature constantly equal to $-1$ and with arbitrarily large volume, provided $l\ge 2$ and $\dim M\ge 3$. In this case, such negative scalar curvature metrics within $ [\![ g ]\!]$ for $l=2$ cannot be of any warped product type.


Introduction
The notion of warped products dates back to the work of Bishop-O'Neill [6], in which they showed that a variety of product manifolds carry metrics of negative sectional curvature. Here, for two Riemannian manifolds (B,ǧ) and (F,ĝ), their warped product with respect to a positive function f on the base space B is defined as the Riemannian manifold (B ×F,ǧ⊕f 2ĝ ). Another ubiquitous way of deforming a Riemannian metric by functions is conformal change. Here, for a Riemannian manifold (M, g), we take a positive function f on M and define a new Riemannian manifold (M, f 2 g), in which the angle of two tangent vectors is the same as that of (M, g). Unifying the notions of warped product and conformal change, Koike [26] defined the twisted product of two Riemannian manifolds (M 1 , g 1 ), (M 2 , g 2 ) with respect to positive functions f 1 , f 2 on M 1 × M 2 as the Riemannian manifold (M 1 × M 2 , f 2 1 g 1 ⊕f 2 2 g 2 ). See also Koike [27] for the case of more than two factors; for more details about the terminology, we refer to Remark 8.2. His main interest seemed to be the extrinsic geometry of the leaves M 1 and M 2 , such as total umbilicity. It is the aim of the present article to study twisted product metrics in the sense of Koike from a more intrinsic point of view, with particular focus on scalar curvature.
Here and henceforth, we adopt the following notation. For a smooth manifold M, we define C ∞ (M) = {ϕ : M → R | ϕ is C ∞ smooth}, C ∞ + (M) = {u ∈ C ∞ (M) | u > 0}. For a Riemannian metric g on M, R g and dµ g denote the scalar curvature and the volume element of g, respectively. For ϕ ∈ C ∞ (M), grad g ϕ and ∆ g ϕ are the gradient vector field and the Laplacian of ϕ, respectively, so that M |grad g ϕ| 2 dµ g = − M ϕ(∆ g ϕ)dµ g if M is closed (i.e. compact without manifold boundary).
Firstly, we recall some fundamental results about the scalar curvature of Riemannian metrics within a conformal class. Let (M m , g) be a closed connected Riemannian manifold. The scalar curvature of the conformally related metric g = f 2 g, f ∈ C ∞ + (M), satisfies m−2 |grad g ϕ| 2 + R g ϕ 2 dµ g M ϕ 2 dµ g > −∞ be the smallest eigenvalue of the operator − 4(m−1) m−2 ∆ g +R g and v the corresponding eigenfunction so normalized that min M v = 1. On the one hand, setting u = v in (1.1), we see that the metric v 4/(m−2) g has scalar curvature λ 0 v −4/(m−2) , which has the same sign as λ 0 . On the other hand, ∆ g u attains zero somewhere on M because M (∆ g u)dµ g = 0, and thus R g and Rg cannot have different signs by (1.1). It follows that, provided m ≥ 3, every conformal class [g] on a closed connected manifold contains a metricg whose scalar curvature satisfies either Rg > 0, Rg ≡ 0, or Rg < 0, and these three cases are mutually exclusive. The same statement holds also for m = 1 and 2, where we note that the scalar curvature of a 1-dimensional Riemannian manifold is always constantly equal to zero and that M 2 R g dµ g = 4πχ(M 2 ) in dimension 2 by the Gauss-Bonnet theorem. It is now well known that there exists a metric of constant scalar curvature in every conformal class of a closed connected manifold M m ; this follows from the uniformization theorem for Riemann surfaces if m = 2 and from the resolution of the Yamabe problem if m ≥ 3 (cf. Yamabe [53], Trudinger [46], Aubin [3], Schoen [41]).
Secondly, we recall the following related observations on the scalar curvature of warped product metrics, which can be found in Dobarro-Lami Dozo [12, Theorems 3.1-3.3]. Let (M m , g) = (M m 1 1 , g 1 ) × (M m 2 2 , g 2 ) be a direct product of closed connected Riemannian manifolds, f 2 ∈ C ∞ + (M 1 ), andg = g 1 ⊕ f 2 2 g 2 a warped product metric on M. Then where we set u = f (m 2 +1)/2 2 . Differentiation of (1.2) with respect to vector fields tangent to M 2 shows that R g 2 is constant if Rg is constant. We define λ 0 ∈ R and v ∈ C ∞ + (M 1 ) so that Integration by parts then yields provided both Rg and R g 2 are constant; in particular, Rg − λ 0 and R g 2 must have the same sign. We note that λ 0 > 0, λ 0 = 0, and λ 0 < 0 hold, respectively, if R g 1 ≥ 0 and R g 1 ≡ 0, if R g 1 ≡ 0, and if M 1 R g 1 dµ g 1 ≤ 0 and R g 1 ≡ 0. We summarize the sign restrictions thus obtained for warped product metrics of constant scalar curvature in Table 1. We take the direct product of constant scalar  Table 1 is nonempty and that, rescaling the metric g 2 by constants, there is no sign restriction on the scalar curvature in the two remaining cases in Table 1 that are left blank. The existence question for warped product metrics of constant scalar curvature which are not direct product is also considered in the same article [12]. The scalar curvature of warped product type metrics are also studied in [45,18,16,54,28,17,13,14,15]. We introduce the following perspective that unifies the previous sign restrictions on scalar curvature. For a direct product Riemannian manifold (M, g) = (M 1 × · · · × M l , g 1 ⊕ · · · ⊕ g l ), we define its multiconformal class [[g]] by We emphasize that the multiconformal factor f i does not have to be constant along any M j . Within a multiconformal class, we may (1) conformally change the representative metrics g 1 , . . . , g l on M 1 , . . . , M l , respectively, (2) conformally deform an arbitrary metricg ∈ [[g]], and (3) consider warped product type metrics of all kinds. Our main result is the following trichotomy. See Remark 6.2 regarding its dimensional assumption.
] of metrics multiconformal to g such that the scalar curvature ofg (n) is constantly equal to −1 on M for every n ≥ 1 and that Vol M,g (n) → ∞ as n → ∞. Theorem 1.3. Assume l = 2, R g 1 , R g 2 ≥ 0. If f i is constant along M i for i = 1, 2, then the scalar curvature of f 2 1 g 1 ⊕ f 2 2 g 2 cannot be nonpositive everywhere and negative somewhere at the same time.
This article is organized as follows. In Sect. 2, we characterize criticality with respect to the normalized Einstein-Hilbert functional restricted to a multiconformal class. In Sect. 3, we introduce some differential operators and observe their behavior under a change of dependent variables. In Sect. 4, we compute the scalar curvature of a multiconformally related metric. In Sect. 5, we derive integral formulas which play a crucial role in the proof of Theorem 1.1. We prove the trichotomy theorem (Theorem 6.1) in Sect. 6, which is equivalent to Theorem 1.1. The proofs of Theorems 1.2, 1.3 are in Sects. 7 and 8, respectively.

The normalized Einstein-Hilbert functional
Let E : Met → R be the normalized Einstein-Hilbert functional defined on the space Met of all Riemannian metrics on a closed connected manifold M m by Here, R g is the scalar curvature of g, p m = 2m/(m − 2), and m ≥ 3. For every h ∈ Γ(Sym 2 T M * ), at a metric g of constant scalar curvature. We introduce two intermediate notions of criticality between the constant scalar curvature and Einstein conditions. Fix a direct sum decomposition T M = E 1 + · · ·+E l of the tangent bundle where each E i is a vector subbundle of fiber dimension m i ≥ 1, and let P i ∈ Γ(End T M) be the corresponding projection onto E i . We say g ∈ Met is compatible with this decomposition if E i ⊥ E j for all i = j with respect to g. We denote by Met ⊥ the subspace of all compatible Riemannian metrics. Note that g ∈ Met ⊥ can be written uniquely as g = g 1 ⊕ · · · ⊕ g l using fiberwise inner products g 1 , . . . , g l on E 1 , . . . , E l , respectively. Two compatible metricsg =g 1 ⊕ · · · ⊕g l and g = g 1 ⊕ · · · ⊕ g l are said to be multiconformal to each other if there exist functions f 1 , . . . , f l : For a compatible metric g, define Ric g i ∈ Γ(Sym 2 T M * ) and R g i ∈ C ∞ (M) by for all X, Y ∈ Γ(T M). We note that always holds while Ric g = Ric g 1 + · · ·+Ric g l holds if and only if Ric g (P i X, P j Y ) = 0 for all X, Y ∈ Γ(T M) whenever i = j.
(1) g is critical with respect to the functional E restricted to [[g]] if and only if there exists a real number c independent of i such that R g i /m i = c for all i ∈ {1, . . . , l}.
(2) g is critical with respect to the functional E restricted to Met ⊥ if and only if there exists a constant c independent of i such that Ric g i = cg i for all i ∈ {1, . . . , l}.

Proof. A section h of Sym 2 T M * is tangent to Met ⊥ if and only if h can be written as the sum
. . , l}. By (2.2) and (2.3), we may assume without loss of generality that g has constant scalar curvature.
The invariants σ(M, [[g]]) and σ ⊥ (M) have a resemblance in spirit to the equivariant Yamabe constant or invariant (cf. Bérard-Bergery [4], Hebey-Vaugon [23]) defined for manifolds with group actions. However, since an equivariant conformal class is smaller than the ordinary conformal class, one cannot expect an inequality like (2.4) for the equivariant ones (cf.
We set a := a 1 · · · a l T , take an orthogonal matrix P = (p αj ), and define Taking the trace of both sides, we obtain the following formula for the change of dependent variables.
In particular, when

Scalar curvature computationà la Karcher
Let M m = M m 1 1 × · · · × M m l l be a direct product of smooth manifolds. We denote by E i → M the vector subbundle of T M defined as the pullback of T M i via the projection M → M i , so that T M = E 1 + · · · + E l and E i ∩ E j is trivial if i = j and P i : T M → E i are the corresponding projections. We adopt the notation and terminology from Section 2 in this special case. Following (3.1), we define grad g i := grad g E i , Hess g i := Hess g E i , ∆ g i := ∆ g E i for an arbitrary compatible metric g, so that In this section, we derive the formula for the scalar curvature under a multiconformal change (Theorem 4.4). The strategy is the same as that of Karcher [25]. Hence, we fix a compatible g = g 1 ⊕ . . . ⊕ g l on M = M 1 × . . . × M l and a multiconformal changeg = f 2 1 g 1 ⊕ . . . ⊕ f 2 l g l . In what follows, we writeg = ·, · , g = ·, · whenever convenient, with the respective norms || · ||, | · |. In the proof, we also compute the differences Ricg i − Ric g i , see Eq. (4.11), and Rg i − R g i /f 2 i , see Eq. (4.12).
Remark 4.1. When g is a direct product metric, the multiconformally related metricg is also called a twisted product 1 . If in addition l = 2, it is more common to say thatg is biconformal 2 to g. Moreover, for a direct product metric g, a formula without proof for the curvature tensor ofg can be found in Meumertzheim-Reckziegel-Schaaf [29, Proposition 1]. We present here the detailed computation, which works for an arbitrary compatible metric g on M = M 1 × · · · × M l .
Before we start computing the scalar curvature under a multiconformal change, we first relate the Levi-Civita connections ∇ g , ∇g of g,g, respectively. As the derivatives of the multiconformal factors f 1 , . . . , f l will be involved it is reasonable to first compare the gradients taken with respect to the metrics g andg. Proof. For every X ∈ Γ(T M), we can express the derivative X(ϕ) either with respect to g or with respect tog. This leads to, X(ϕ) = gradg ϕ, X , X(ϕ) = grad g ϕ, X = l a=1 f −2 a P a · grad g ϕ, X .
As X(ϕ) is independent of the Riemannian metric, (4.1) holds. (4.2) Proof. Since T is tensorial, we assume without loss of generality that X ∈ Γ(T M i ), Y ∈ Γ(T M j ). Comparing the Koszul formulas forg and g, we observe that if Z ∈ Γ(T M a ) then Hence, To conclude the claimed formula it remains to show that for X, Y ∈ Γ(T M). The term − 1 2 gradg X, Y + 1 2 grad g X, Y is tensorial, since it is the difference of two tensors T X Y and l a=1 X, grad g f a 1 fa P a Y + l a=1 Y, grad g f a 1 fa P a X. Thus, it suffices to check (4.3) for an g-orthonormal frame {e α } m α=1 such that for every α ∈ {1, . . . , m} there exists some i ∈ {1, . . . , l} with e α ∈ Γ(E i ). Since {e α } m α=1 remains orthogonal with respect tog, (4.3) holds for X = e α , Y = e β if α = β as both sides evaluate to 0. If α = β, we derive for the left hand side and for the right hand side − l a,b=1

Now (4.3) follows by combining these two identities.
Using the tensor T = ∇g − ∇ g and the above lemma we are now able to derive a formula for the scalar curvature under a multiconformal change.
Proof of Theorem 4.4. To begin with, we recall the general formula for X, Y, Z ∈ Γ(T M), which can be shown by summing up the following three identities Here and henceforth in the proof, quantities without any superscript such as R and ∇ are understood to be the ones with respect to g. We want to express all these identities in terms of the functions f 1 , . . . , f l and their derivatives.
Up to interchanging the roles of X and Y there are two terms that we need to take care of. Namely, (∇ X T ) Y Z and T X T Y Z On the one hand, (4.7) and (4.2) yields Taking the inner product with W ∈ Γ(E h ) leads to, On the other hand, plug (4.7) into (4.6) to get Taking the inner product with W , Therefore, (4.5), (4.8), and (4.9) yields a formula for the difference and by linearity, it extends to an identity for all vector fields on M. However, the resulting formula is a very long expression. As we are interested in a formula for the scalar curvature for a multiconformal change we only consider the difference Rg(X, Y )Y −R(X, Y )Y, X for X ∈ Γ(E i ) and Y ∈ Γ(E j ). In that case we obtain (4.10) Taking an g-orthonormal frame {e α } m α=1 so that for each α ∈ {1, . . . , m} there exists some i = i(α) ∈ {1, . . . , l} with e α ∈ Γ(E i ), we define the associatedgorthonormal frame via {ẽ α = f −1 i(α) e α } m α=1 . With respect to these orthonormal frames we conclude Inserting (4.10) leads to We thus obtain (4.11) Taking the trace with respect to g in (4.11) yields (4.12) Since Rg = l j=1 Rg j we sum (4.12) over j ∈ {1, . . . , l} and derive This is equivalent to the claimed formula (4.4) for the scalar curvature under a multiconformal change.

Integral and pointwise formulas
We summarize some formulas which will be necessary in the following sections. Let (M, g) = (M 1 , g 1 ) × · · · × (M l , g l ) be a direct product Riemannian manifold andg = f 2 1 g 1 ⊕ · · · ⊕ f 2 l g l multiconformal to g. Since we would like to apply the results of Section 3 to the identity for the scalar curvature of (M,g) derived in Theorem 4.4, it is convenient to write 2) it follows that for each (q 1 , . . . , q l ) ∈ R l the symmetric (l × l)-matrix B i (q 1 , . . . , q l ) defined by Now we can apply the results of Section 3 to conclude that the right hand side is nonpositive (resp. nonnegative) if for all 1 ≤ i ≤ l the matrices B i are negative (resp. positive) definite.
In Section 8 we consider multiconformal metrics of permutation type, see Definition 8.1 and metrics of warped product type. Since in these cases the multiconformally changed metricg = f 2 1 g 1 ⊕ · · · ⊕ f 2 l g l is such that the functions f i are constants along some of the factors of the product manifold M = M 1 × · · · × M l the integral formula 5.5 simplifies.
Let us assume that the metricg = f 2 1 g 1 ⊕ .
Since (4.4) can be written as (5.6) Hence, integral formula (5.2) with (q 1 , . . . , q l ) = (m 1 , . . . , m l ) simplifies to   Proof. Without loss of generality we assume that g i is a constant scalar curvature metric for all 1 ≤ i ≤ l. By Theorem 4.4 the scalar curvature ofg = f 2 1 g 1 ⊕ · · · ⊕ f 2 l g l ∈ [[g]] is given by where ρ i is the scale-invariant function defined in (5.1) for any 1 ≤ i ≤ l. Considering (5.5) for (q 1 , . . . , q l ) = (0, . . . , 0) we obtain where (b i jk ) jk are the entries of the symmetric matrix Since m i ≥ 2 for all 1 ≤ i ≤ l the matrix B i is negative definite for any 1 ≤ i ≤ l. To see this, let x ∈ R l . Then We note that the first summand is nonpositive and 0 if and only if (x 1 , . . . , x l ) ⊥ (m 1 , . . . , m l ) while the second summand is nonpositive and 0 if and only if m i = 2 and x = (0, . . . , 0, x i , 0, . . . , 0). As these both sets are disjoint it follows that B i is negative definite. Applying Lemma 3.1 we conclude that the right-hand side of (6.1) is nonpositive and 0 if and only if grad g f i = 0 for all 1 ≤ i ≤ l. In particular, where equality holds if and only if f 1 , . . . , f l are all constant.
Using this inequality we can now prove the statements of Theorem 6.1: (1) If [[g]] contains a metricg of positive scalar curvature, then it follows from (6.2) that (2) If [[g]] does not contain a metric of positive scalar curvature but a scalar flat metricg, then ≤ 0 for all 1 ≤ i ≤ l as otherwise there would be a metric of positive scalar curvature in [[g]]. Thus, it follows that the above inequality is satisfied if and only if R g i = 0, i.e. µ(M i , [g i ]) = 0 for all 1 ≤ i ≤ l. As in this case the above inequality is in fact an equality the functions f 1 , . . . , f l have to be constant. In particular, the scalar flat metric g is a product metric.

Multiconformal metrics of permutation type
In Sect. 7, we saw that the scalar curvature of a multiconformal metricg ∈ [[g]] can be negative everywhere even in the case (1) of Theorem 6.1. In this section, we show that such negative scalar curvature metrics cannot be of permutation type in the sense of Definition 8.1. Let (M, g) = (M 1 , g 1 ) × · · · × (M l , g l ) be a direct product of Riemannian manifolds . For functions f 1 , . . . , f l ∈ C ∞ + (M), we may associate the (l × l)-matrix of 1-forms on M. In view of this matrix, we impose the following conditions on the multiconformal factors.
We remark that a multiconformal metricg = f 2 1 g 1 ⊕ · · · ⊕ f 2 l g l has off-diagonal and permutation type, respectively, if and only if (8.1) has zero diagonal entries and is a generalized permutation matrix.
Remark 8.2. Related terminology is the following. The notion of warped products in the sense of Bishop-O'Neill (cf. [6, §7], [32, §7]) has been generalized to various situations. We remark that doubly warped products can have two different meanings; some authors 3 deal only with two factors while others 4 need three factors to define them. The term multiply warped products seems to be unambiguous 5 , but it conflicts with the first meaning of doubly warped products.
Twisted products in the sense of Chen [9, p. 66], also called umbilic products in earlier work of Bishop [5, p. 27], are defined on direct product manifolds, which are topologically not twisted. Note that Bishop-O'Neill [6, p. 29] used the term warped bundles for the generalization of warped products to (possibly topologically twisted) bundles. These notions are generalized, depending on the authors' preferences, to umbilic products 6 , twisted products 7 , and doubly or multiply twisted products 8 etc.
Yang [54, Theorem 1] observed that if (M 1 × M 2 ,g = f 2 1 g 1 ⊕ f 2 2 g 2 ) is of warped product type and has constant scalar curvature and if f i is nonconstant for i = 1, 2, then Rg must be indeed zero. He then asked whether there exist such scalar flat metrics of nontrivial warped product type. Before providing an answer to his question, we generalize Yang's Theorem slightly as follows.
for some c i ∈ R. Indeed, the right hand side of (8.4) is constant along M 1 × · · · × M σ(i)−1 × M σ(i)+1 × · · · × M l , while its left hand side is the multiple of X 1 f 2 i , which is constant along the same space and is not identically equal to zero for a good choice of X, and the rest which is constant along M σ(i) . Going back to (8.3), we obtain .
Differentiation with respect to a vector field X tangent to M σ(i) then yields .
A similar separation of variables then shows c i Following Yang, we ask whether there exist such scalar flat metrics. Theorems 6.1, 8.3 imply the following partial answer to his question. If either µ(M i , [g i ]) ≤ 0 for i = 1, 2 or R g i ≥ 0 for i = 1 and 2, then such scalar flat metrics of nontrivial warped product type do not exist.