Product Inequalities for T ⋊ -Stabilized Scalar Curvature

. We study metric invariants of Riemannian manifolds X defined via the T ⋊ - stabilized scalar curvatures of manifolds Y mapped to X and prove in some cases additivity of these invariants under Riemannian products X 1 × X 2 .

To the 75th birthday of Jean-Pierre Bourguignon

T ⋊ -stabilization
A "warped" T N -extension, N = 0, 1, . . ., of a Riemannian manifold X = (X, g), possibly with a boundary, is where T N is the (flat split) N -torus and where X ⋊ N is endowed with a warped metric, where φ i (x) ≥ 0 are smooth positive functions, which are strictly positive (> 0) in the interior X \ ∂X of X. Assume g is smooth and let Sc ⋊ {φ i } (X) be the scalar curvature of g ⋊ , that is, where Sc ⋊ {φ i } (X) = Sc ⋊ {φ i } (X, x) is a function on X, since g ⋊ N is invariant under the obvious action of T N on X ⋊ N = X × T N .Let Sc ⋊ (X), X = (X, g), be the supremum of the numbers σ such that Sc ⋊ N,{φ i } (X) > σ for some N and φ i .
for all smooth domains Y ⊂ X.
Recall at this point that if X is a compact connected manifold without boundary, then, for all functions σ(x), where the latter inequality follows from positivity of −∆ and this inequality is strict (>) with φ = φ 1 , unless σ(x) is constant.Also, the strict inequality λ 1 > inf x∈X Sc(X, x) holds for the Dirichlet eigenvalue λ 1 on compact connected manifolds with boundaries, since the above relations are satisfied for functions φ(x), which vanish on the boundary.
This shows that Sc g + N i=1 φ 2 i dt 2 i increases under replacing all φ i by their geometric mean, φ i ; ϕ = N i φ i , i.e., Sc(g) for Ψ = log ϕ N , where the equality holds only if all ∇ log φ i are mutually equal.Hence, sup where this "sup" increases with N ; thus, by letting N → ∞, we see that Sc ⋊ (X) = sup Rewrite this equation with Ψ = 2Θ as follows: Therefore, if X is compact, then Sc ⋊ (X) ≥ 4λ ⋊ 1 (X), where λ ⋊ 1 (X) is the lowest eigenvalue of the operator −∆ + 1 4 Sc on X with the Dirichlet (vanishing on the boundary) condition.
(If a connected manifold X has no boundary and the scalar curvature of X is constant, then Sc ⋊ (X) = Sc(X); otherwise for all β > 1/4, since the operator −∆ is strictly positive on non constant functions on X.) 1.B. 1 4 -Proposition.Let X = (X, g) be a compact Riemannian manifold with a boundary.Then Sc ⋊ (X) = 4λ ⋊ 1 (X).
Finally, by ⊃-monotonicity of Sc ⋊ and the continuity of the first eigenvalue λ 1 (−∆ g +Sc(g)/4) in (the space of C 2 -metrics) g, this majorization holds for functions θ which is strictly positive only in the interior of X. 2 Then the proof of the 1 4 -proposition follows.Sc via the Kato inequality for the squared Dirac operator on X ⋊ = X × T N , g ⋊ , which, to make it index-wise more interesting, may be twisted with the canonical N -parametric family of flat unitary complex line bundles over X ⋊ .(Probably, there is much of what we do not understand about the relations between the two 1 4 .) Corollaries/Examples For instance, the rectangular solids satisfy 1.B 3 .Manifolds X with constant scalar curvature σ satisfy Sc ⋊ (X) = 4λ 1 (X) + σ for the first eigenvalue λ 1 of the Laplace operator on X.
1.B 4 .Ricci comparison inequality.Let X be a (metrically) complete Riemannian manifold with a boundary such that Ricci(X) ≥ (n − 1)κ and mean.curv(∂x)≥ µ.Then where B n κ,µ is the ball in the complete simply connected n-space S n κ with sectional curvature κ, and where the mean curvature of the boundary ∂B n κ,µ is equal to µ.3For instance, if Ricci(X) ≥ 0 and mean.curv(∂X) In fact, let and the proof follows.
for a bounded positive function c(r) such that c(r) → 1 for r → ∞, and Thus, 1.C.General torical stabilizations.The most permissive torical "extension" of a Riemannian manifold X is a Riemannian manifold X ♮ with an isometric T N -action and an isometry X ♮ /T N ↔ X.Here, as earlier, one defines the number Sc ♮ (X), which is clearly ≥ Sc ⋊ (X).
It seems, however -I did not honestly checked this -that the curvature formulas for Riemannian submersions [35] imply that Sc ♮ (X) ≤ Sc ⋊ (X).
Alternatively, if the fibration X ♮ → X ♮ /T N = X admits a section, then the ⋊-rendition of the Schoen-Yau argument4 implies the equality Sc ♮ (X) = Sc ⋊ (X) for dim(X) = n ≤ 8, 5 while for all n this may follow from [43], where both arguments apply not only to Riemannian submersions but to all distance non-increasing maps X × T N , G → (X, g).
The geometric meaning of other β, as well as of the higher eigenvalues λ i (X, β) of −∆ + β Sc is unclear. 6 Sc ⋊ ↓ , Sc ⋊ ↓ sp , . . ., Sc ⋊ ↓ * on homology Let X be a metric space, e.g., a Riemannin manifold Sc ⋊ ↓ (h) = Sc Smoothness remark.If X is a smooth Riemannian manifold, then an obvious approximation argument shows that requiring maps f to be smooth does not change the value of Sc (However, smoothness of a distance non-increasing map f in the extremal case, where Sc ⋊ (Y ) = Sc ⋊ ↓ (h) is a delicate matter, see [5].) Below the are several versions of this definition with the generic notation Sc ⋊ ↓ * .I. Restrict/relax the topology of Y , e.g., by requiring that the second homotopy group of Y is zero; • st.par Y is stably parallelizable; • ⊙ allow representation of h by quasi-proper maps from complete manifolds Y to X, where "quasi-proper" means locally constant at infinity.
and Sc II.Assuming X is a Riemannian manifold, relax the distance decreasing condition on f by the following • area the map f decreases the areas of all surfaces in Y .9 Clearly, Sc area ≥ Sc ⋊ ↓ and we show in §2.E below that the ratio Sc ⋊ ↓ area

Sc
⋊ ↓ can be arbitrarily large.III.Replace the integer homology by the rational homology H * (X; Q), which is essentially (but not quite) the same as allowing maps f : Y → X, where f * sends the fundamental class of Y to a non-zero multiple of h.
IV. Instead of homology, use a bordism group of X, e.g., the spin bordism group, which is well adapted to Sc > 0.
V. Remarks on singular Y .
for a smooth manifold Y , it may be interesting to try pseudomanifolds Y with suitably defined singular Riemannian metrics g with Sc(g) ≥ σ.(b) Conical example.Here Y has an isolated singularity at a point y 0 ∈ Y , where g is a smooth Riemannian metric on the complement to y 0 such that Sc(g) ≥ σ and such that g is (approximately) conical at y 0 .This means that there exists an ε-neighbourhood (ball) U = U ε ⊂ Y of y 0 , which topologically splits away from y 0 , where Z = (Z, g Z ) is a compact smooth Riemannian manifold such that the metric g restricted to U \ {y 0 } is related to g Z as follows: where a 2 (t) > 0 is a smooth positive function on the (now closed) interval [0, ε].(One may assume, if one wishes, that a(t) is constant near t = 0.) (c) One may additionally assume that Sc(g Z ) ≥ Sc S m−1 = (m − 1)(m − 2) for m = dim(Y ) and, to make the metric truly conical, to require a(t) to be constant near t = 0.But this is not truly needed, since it can always be achieved by a small deformation of our g near y 0 .
(d) Iterated conical singularities.Next, following [8], define m-dimensional (roughly) conesingular spaces Y with Sc(Y ) ≥ σ by induction on m, where (as in the above conical case) the metric (i.e., the distance function) on Y is defined by a smooth Riemannian metric g on the non-singular part Y 0 ⊂ Y , where the following conditions are satisfied: (ii) Sc(g, y) ≥ σ for all y ∈ Y 0 .
(iii) Each y 0 ∈ Y admits a neighbourhood U 0 , which is topologically (but not metrically) cylindrically splits away from y 0 , where Z 0 = (Z 0 , g Z 0 ), is a compact (m − 1)-dimensional cone-singular space and where the Riemannian metric g on the non-singular part of where a 2 0 (t) > 0 is a smooth positive function on [0, ε 0 ] and where δ 0 = δ 0 (u) is a (small) smooth quadratic differential form on U 0 , which converges to zero for u → y 0 .
(e) Due to h 0 , the above "conical" is slightly more general than how it is defined in (b) for an isolated singularity y 0 .
(f) Similarly to the isolated singularity case, the requirement Sc(Z 0 ) ≥ m − 1)(m − 2) does not significantly change the definition of Sc(Y ) ≥ σ.
(g) One may also insist on the split-conical geometry at all points y 0 : if y 0 is contained in the interior of an l-dimensional strata S ⊂ Σ, then a small neighbourhood U 0 ⊂ Y of y 0 metrically splits: U 0 = S 0 × N 0 , where S 0 = U 0 ∩ S, and where N 0 is a con-singular manifold with an (m − l − 1)-dimensional base.
(h) Probably, as in the isolated singularity case, this additional condition can be achieved by a small deformation of g near Σ.
Question.How does the resulting Sc ⋊ ↓ (h), h ∈ H m (X), depend on the topology of the singular locus Σ ⊂ Y ?
For instance, Y may be iterated conical space with initial cones based on products of complex projective spaces and/or other generators of the oriented bordisms groups with Sc > 0 metrics (e.g., as in [21]), where we allow cones over l-dimensional Y only if they admit metrics g with "suitably defined" Sc(g) > 0 and/or, which is probably equivalent with Sc ⋊ (g) > 0.
(Probably, stable minimal hypersurfaces and µ-bubbles in such Y , similarly to how it is in a smooth Y , enjoy necessary properties required for the study of the scalar curvature and this is also conceivable for the Dirac theoretic approach (compare with [2]).) VI.If X is non-compact, allow classes h with infinite supports10 and use proper (and quasiproper) maps f : Y → X.
Remark on Completeness of Y .Regardless of X being complete or not, the value Sc ⋊ ↓ area [X] defined with complete Y mapped to X may be very different without this completeness.
For instance, Sc .
Indeed, the inequality Sc follows from the existence of a measure preserving diffeomorphism from the 2-sphere with constant scalar curvature σ = 8π area(X) onto X; the opposite inequality follows from Zhu's lemma (see [46] and [17,Section 2.8]).
Similarly, one shows that Sc for closed surfaces X of positive genera.On the opposite end of the spectrum, non-compact connected surfaces X satisfy Sc ⋊ ↓ area [X] = ∞, since all the surfaces X admit Riemannian metrics with Sc = 1, and with the given areas (including area = ∞ for non-compact X) and since connected mutually diffeomorphic Riemann surfaces X 1 and X 2 of equal areas admit area preserving diffeomorphisms area , the geometric meaning of Sc ⋊ ↓ [X] for spherical surfaces X remains obscure.All one knows besides Zhu's lemma for general X (see [17,Section 2.8]) is that 2.B.H 2m (CP n )-example.Let the complex projective space CP n be endowed with the U (n + 1) invariant (Fubini-Study)-metric such that the projective lines have scalar curvatures equal 2 and let CP m → CP n be an m-plane.
If m is odd, then the manifold CP m is spin, and both homology and the spin-bordism class of and the same holds for the multiples k In fact, if an oriented manifold Y 2m contains a smooth hypersurface H such that the m-fold self-intersection index H ⌢ • • • ⌢ H m is odd, then the (m − 1)-fold intersection is an orientable surface Σ ⊂ Y , which for even m has non-trivial normal bundle; hence for all m and n ≥ m, since the quotient space m by the permutation group Π(m) admits a natural biholomorphic map ψ : S 2 m → CP m , where const m > 0 is the squared reciprocal to the minimal Lipschitz constant of maps in the homotopy class of this ψ.
Question.What are Sc ⋊ ↓ S 2 m /Π(m) and of the symmetric powers [(X) m /Π(m)] for more general manifolds X? 112.C.Upper bounds and equalities.The (T ⋊ -stabilized and sp-generalized) rigidity theorem by Min-Oo [34] and (the spin cobordims version of) Goette-Semmelmann's theorem from [11] imply that the class h 2m = [CP m ] ∈ H 2m (CP m ) satisfy the following relations: for all m and n ≥ m, where "sp.brd" indicates that this Sc ⋊ ↓ area defined with smooth maps Y → X which are spinbordant to the embedding for odd m, for all m and n ≥ 2m − 1.
2.D. Homological homogeneity conjecture.Let X be a compact symmetric space and H ⊂ H m (X, Q) be the linear subspace generated by the fundamental classes [Y i ] ∈ H m (X) of homogeneous (not necessarily totally geodesic) m-submanifolds Y i ⊂ X. 13 Then all classes h m ∈ H can be represented by linear combinations of homogeneous Y i such that Sc (This maybe overoptimistic in general, but the sp-version of this can be, probably, proved with available means for products of spheres, complex and quaternionic projective spaces.) 2.E.Equivalence conjecture.All rational h ∈ H m (X) for compact Riemannian manifolds X without boundaries satisfy: 14

Sc
and Sc where A = A n , B = B n and C = C n are universal constants, and where the same relations are expected for the area version of these five Sc ⋊ ↓ .
2.F.Positivity.Unlike Sc ⋊ , the values of all Sc ⋊ ↓ = Sc ⋊ ↓ dist -invariants are non-negative, since all (compact or not) manifolds Y admit arbitrarily large Riemannian metrics with Sc > ε.
Moreover, the fundamental classes of compact connected manifolds X with non-empty boundaries are strictly positive since such manifolds admit metrics with Sc > 0. (For instance, the r-balls with hyperbolic metrics g, sect.curv(g)= −1, admit (obvious radial) metrics g + ≥ g with Sc(g 152.H. Finiteness.Sc ⋊ ↓ (h) may be, a priori, infinite.However, the finiteness of Sc ⋊ ↓ = Sc ⋊ ↓ dist easily follows from the □ m -inequality (3.8) in [17], where the proof for m ≥ 9 relies on Theorem 4.6 in [39] and where the finiteness of Sc ⋊ ↓ sp (h) for all m follows from [43].(Probably, the arguments used in [43] generalize to Sc ⋊ ↓ sp .)2.I. ∄ Sc > 0-Problem.Does non-vanishing of a rational h ∈ H m (X; Q) under the above classifying map B(π 1 (X)) imply that Sc ⋊ ↓ = 0? This is known for m = 3, and also for Sc ⋊ ↓ sp (h) and all m if the spinorial curvature Sp.curv ↓ (β * (h) ∈ H m (B(π 1 (X)) defined in Section 7 vanishes, e.g., if our B(π 1 ) admits a complete metric with sect.curv≤ 0, see [17] and references therein.(I am not certain if there are examples of non-zero rational homology classes h in aspherical, say, compact finite-dimensional spaces such that Sp.curv ↓ (h) ̸ = 0.) 2.J. Sc < ∞ for all compact Riemannian manifolds X without boundaries?
Remarks.(a) All metrics g + on a compact Riemannian manifold (X, g) such that area g + (Σ) ≥ area g (Σ) for all surfaces Σ ⊂ X satisfy In fact, this inequality holds for all (Y, g + ) what admit area decreasing spin maps 16 f : Y → X with non-zero degrees.
(b) Let X = X 0 × Y , where Y is enlargeable 17 and dim(X 0 ) = 2. Then the finiteness of Sc ⋊ ↓ area (X) for X follows from [47], where for dim(X) ≥ 8 one needs a version of Theorem 4.6 from [39].
Also the ⋊-stabilized version of the area slicing theorem from [28] (this stabilization is likely to be true) delivers an effective finite bound on Sc But the principal case, where X = S n remains problematic for all n ≥ 4 and neither can one prove or disprove the existence of (necessarily non-spin) complete orientable n-manifolds Y , n ≥ 4, with Sc ≥ σ > 0, which admit smooth proper area decreasing maps to R n , n ≥ 4, with non-zero degrees.

2.K. Outline of construction for Sc
Let g 0 be a metric on a manifold Y such that Sc(g 0 ) > 0, then there exists metrics g on Y with arbitrarily large ratios Sc ⋊↓ area (g)/ Sc ⋊↓ dist (g).In fact, let Y = (Y, g 0 ) be an arbitrary Riemannian manifold and U ⊂ Y an open subset.Then for all ε > 0 and δ > 0, there exists a Riemannian metric g ε,δ on Y such that Sc(g ε,δ ) ≥ Sc(g 0 ) − ε and • Y \U the metric g ε,δ is equal to g 0 outside U ; • area the metric g ε,δ is area-wise smaller than g 0 , area g ε,δ (S) ≤ area g 0 (S) for all smooth surfaces S ⊂ Y ; • dist all Riemannian manifolds X, which 1-Lipschitz dominate 18 (Y, g ε,δ ), have Sc ⋊ (X) ≤ δ.
The only non-trivial condition here is • dist , which is achieved with the follow ing one.
) is isometric to the product where T 1 (ε) is the circle of length ε and T n−2 (2π) is the standard flat torus. 16A continuous map between orientable manifolds, f : Y → X, is spin if f * (w2(X)) = w2(Y ), where w2 is the second Stiefel-Whitney class. 17A compact Riemannian n-manifold X is enlargeable if there exists a sequence of oriented coverings Xi → X and distance decreasing maps fi : Xi → S n (Ri), Ri → ∞, which are constant at infinity and which have non-zero degrees, compare with [4,20,23,38], in [17, §4.7], [18, §2.A].
The construction of g ε,δ , which satisfies • Y \U , • area and • D is elementary (left to the reader 19 ) while the implication • D ⇒ • dist follows the 2π n -inequality (see [17, §3.6] and references therein). 20 Thus, we see that if a non-torsion homology class h in a compact manifold X satisfies Sc ⋊ ↓ dist (h) > 0, then the ratio Sc ⋊ ↓ area,sp (h)/ Sc dist,sp (h) ⋊ ↓ can be made arbitrarily large with some Riemannian metric on X.
Probably, if X is simply connected, β ≤ β m > 0 and m ≥ 3, then an integer multiples lh for some l ̸ = 0 and all h ∈ H m (X) are representable by a distance decreasing maps Y → X, where λ 1 (Y, β) ≥ C for a given C > 0.
Exercises 2.M.Let g be a Riemannian metric on an open manifold 21 X of dimension dim(X) = n ≥ 2. Show that there exists a Riemannian metric g + on X such that Sc(g + ) = 1 and area g + (Σ) ≥ area g (Σ) for all smooth surfaces Σ ⊂ X.
Hint: Observe that [0, 1] × R n−1 admits an area decreasing diffeomorphism onto R n and use products of surfaces with constant curvatures by R n−2 as building blocks for (X, g + ).
Remark.If Y is a complete spin n-manifold with Sc ⋊ (Y ) ≥ σ > 0, then it admits no proper area decreasing map to R n with non-zero degree [21].
2.N.Show that non-zero multiples of homology classes h in simply connected manifolds X have Sc ⋊ ↓ st.par (ih) > 0, for some i ̸ = 0. Hint.Recall the Serre-Thom theorem on framed bordisms and apply Stolz' theorem on spin manifolds [41].

3.A. Homological Sc
⋊ ↓ * -problems.Let X be a Riemannian manifold and h ∈ H m (X) a homology class, e.g., m = n = dim(X), and let h be the fundamental class [X] of X, where X is assumed oriented.
Evaluate Sc ⋊ ↓ * and/or find relations between Sc ⋊ ↓ * and more accessible metric invariants of X.

Decide if Sc
where f is the distance or the area decreasing depending on " * " and where, ideally, f is an isometric immersion.
For instance, given a submanifold Y → X, e.g., Y = X decide if it is Sc Find examples of h, where there is no extremal manifold Y → X with f * [Y ] = h, but such a generalized Y , e.g., a singular extremal one does exist.(We saw some potential examples of such singular Y , and stable minimal singular hypersurfaces suggest further examples.) 19To get an insight, start with Y = S 2 , then look at Y = S 2 × T n−2 . 20The proof of 2π n -inequality for n ≥ 9 relies on Theorem 4.6 in [39], and if one is satisfied with Sc ⋊↓ area (g)/ Sc ⋊↓ dist,sp (g) → ∞, then one can use the spinorial version of 2π n from [45]. 21A manifold X is open if it contains no closed manifold connected component.
In general, the existence of a metric g 0 on X with Ricci(g 0 ) ≥ 0 might be necessary for the existence of an extremal metric g on X.
Examples 3.B S n .Complete manifolds with constant sectional curvatures, e.g., unit spheres, flat tori and Euclidean spaces are Sc ⋊ ↓ area, sp -extremal.This follows from the ⋊-stabilized Llarull's theorem (see [17] and references therein).3.B R>0 .A compact spin manifold X with non-negative curvature operator, R(X) ≥ 0, e.g., a compact symmetric space is Sc ⋊ ↓ area, sp -extremal, provided scalar curvature Sc(X) is constant 22and the Euler characteristic of the universal covering X does not vanish.
This follows by an elaboration on the proof of the Goette-Semmelmann extremality theorem [12].(We say a few words about it in Section 5.) Probably, the corresponding rigidity arguments (see [29] and references therein) also admit ⋊-stabilization, but I did not check this carefully.
3.B ×× .Riemannian products of the manifolds from the above examples, e.g., As above, this follows by a simple generalization of argument from [43] combined with the basic (algebraic) inequality in [12] for twisted Dirac operators on manifolds with R ≥ 0.
But the ⋊ ↓ sp -extremality remains problematic even for n ≤ 8.For instance, if k ≤ 4 and n ≤ 8 (probably n ≤ 10 will do), then the □ ∃∃ (n, m, N )-inequality combined with the warped product splitting argument in [17, §5.5] yield ⋊ ↓ sp -extremality of Yet, there is no approach so far to non-spin extremality of the spheres S k for k ≥ 5. 233.B warp .There are several classes of log-concave warped product manifolds, e.g., S n minus a point, where the ⋊ sp -extremality (and ⋊-extremality for n = 4) follow by ⋊-stabilization of the arguments in [17, § §5.5-5.7] and [6].In fact, the ⋊-extremality for warped manifolds is more common then non-stabilized extremality.
For instance, geodesic balls in spheres and in R n are not non-stably extremal: one can increase their metrics without diminishing the scalar curvatures.But, probably, they are ⋊ ↓extremal. 3.C.Questions.
(i) Which convex subsets in R n are ⋊ ↓ -extremal?
3.D.About rigidity.The proofs of extremality of the manifolds X in the above examples can be upgraded to rigidity that says in the present case that if a smooth distance non-increasing positive degree map f : Y → X satisfies Sc ⋊ * (Y ) ≥ Sc ⋊↓ * (X) (where Sc ⋊↓ * (X) = Sc ⋊ * (X) by extremality), then f is homotopic to a local isometry, where one can drop "homotopic to" if X has no local scalar flat factors.
This follows by combining the ⋊-stabilized rigidity arguments in [12] and [29] with those in [17, §5.7] but to be honest, I did not check this in full generality.
4 Sc ⋊↓ -product inequalities, conjectures and problems 4.A.Additivity for cylinders.Since, obviously, then, for all Riemannian manifolds X 1 and X 2 , the inequality is equivalent to the equality Thus, in particular, the □ ∃∃ (n, m, N )-inequality from [18] and/or equivariant separation theorem for stable µ-bubbles 24 along with the equality imply the following.
Proposition.The fundamental homology classes of oriented Riemannian cylindrical manifolds 4.B.The spin case.This additivity formula remains problematic for n ≥ 9, 25 but the spin cube inequality from [43] (proved with an index theorem for deformed Dirac operators on manifolds with boundaries) implies, as we stated earlier, that 24 See [17, §5.4] and compare with [22,37] and with [18, the proof of §2.B]. 25 The dimensions n = 9, 10, probably, can be taken care by the argument in [9].
Yet, as far as I can see, the present day Dirac theoretic argument does not yield the general Sc sp -inequality However, this argument does apply, if Y is a special (extremal) manifold as in §3.B ×× , e.g., a product of spheres.
4.C.[sect.curv≤ 0]-Remark.Let Y and Z be compact Riemannian manifolds, where Z has no boundary and the sectional curvature sect.curv(Y ) ≤ 0.Then, similarly as above, one can prove additivity in the following two cases: for all n.

4.D. Riemannian additivity conjecture. Riemannian products of all oriented Riemannian manifolds satisfy
In fact, the following stronger inequality might be true.4.E.Sup-metric product conjecture.Let X i , i = 1, . . ., k, be metric spaces (e.g., closed oriented Riemannian manifolds) and let be their product endowed with the sup-metric Then rational homology classes h i ∈ H m i (X i ; Q) (e.g., the rational fundamental classes [X i ] 27 ) satisfy where the opposite inequality follows from additivity of the scalar curvature; hence, (4.1) implies the equality 26 In view of [9], the inequality n ≤ 10 may suffice. 27"Rational" in the case of compact locally contractible spaces means "a non-zero integer multiple of", that is, The above indicated proofs of §4.A and §4.B actually show that the rectangular solids × n 1 [a i , b i ] with the Riemannian product and the sup-product metrics have the same Sc ⋊ ↓ for n ≤ 8 and have the same Sc ⋊ ↓ sp for all n.This confirms the validity of (4.1) for rectangular solids.
4.F ′ .× [0, d i ]-Sub-Example.Let Y ⊂ R n be a diffeomorphic image of the n-cube and let d i , i = 1, . . ., n, be the distances between the images in Y of the pairs of the opposite (n − 1)faces of the cube.Then the first Dirichlet eigenvalue of the Laplacian −∆ Y is bounded by that of the solid × i [0, d i ], Exercise.Find a direct elementary proof of this inequality. 28 Sup-distance, sup-area and Sc is greater than the sup-metric but only by a factor √ k, The situation is somewhat different with areas.Namely, let X = × i X i be the product of Riemannian manifolds and let sup i -area(Σ) for a smooth surface Σ ⊂ X be the maximum of the areas of the projections Σ → X i .Here again sup i -area(Σ) ≤ area(Σ) but now, unlike to how it is with the distances, the ratio area(Σ) sup i -area(Σ) may be infinite.Accordingly, the corresponding Sc area decreasing, can be significantly greater than Sc ⋊ ↓ sup.area (h), where the maps f must be area decreasing themselves.
Thus, the area version of (4.1), Sc e.g., Sc is qualitatively stronger than corresponding inequality for Sc Although we have no known means for bounding Sc ⋊ ↓ area and even less for Sc sup.area in most cases, we shall do this in the next section for Sc ⋊ ↓ sup.area, sp and thus prove the sp-version of (4.2) in some cases.
4.G.Semiadditivity problem.Let X = X n and Z = Z k , k ≤ n − 2, be compact Riemannian manifolds, possibly with boundaries, and let f : X → Z be a smooth distance decreasing map such that ∂X f → ∂Z, and let h m = f −1 (z) ∈ H m (X), m = n − k, be the homology class of the pullback of a generic z ∈ Z.
Identify the cases, where at least for "simple" manifolds Z, e.g., compact convex domains in R k and in S k and, in general, evaluate the difference in terms of the geometry of Z, for instance, where Z is the product of balls , a satisfactory lower bound on Sc ⋊↓ (h m ) for rectangular solids Z follows from §4.A. Also [18, §2.B] yields similar bounds for products of 2-discs and 2-spheres (compare [22]).But it is unclear, for instance, how large the difference Sc ⋊↓

Additivity of the twisted SLWB-formula and applications
Let Y be a Riemannian spin n-manifold and V → X be a complex vector bundle with a unitary connection ∇ and let D ⊗V denote the Dirac operator on spinors S on Y tensored with V .Then the square of D ⊗V satisfies the following Schrödinger-Lichnerowicz-Weitzenböck-Bochner formula (see [27]) where where e i ∈ T y (Y ), i = 1, . . ., m = dim(Y ), are orthonormal tangent vectors at y ∈ Y , where • is the Clifford multiplication and R V e i ∧e j : V → V is the curvature operator of ∇.Next, recall that the curvature of the tensor product of two bundles with connections satisfies where 1 V : L → V is the identity operator, and observe that the operators on and by have the same spectra up to multiplicity as and correspondingly.Therefore, the lowest eigenvalue λ ⊗1⊗2 (often negative) of the (self-adjoint) operator is bounded from below by the sum of these for R ⊗V 1 and R ⊗V 2 ,29 This yields the following.

5.
A. Theorem. 30Let X = × k X k , k = 1, . . . ,l, be an orientable Riemannian n-manifold split into Riemannian product, where the factors X k = (X k , g k ) are either (a) compact n k -manifolds with non-negative curvature operators, R X k ≥ 0 (e.g., closed convex hypersurfaces in R n k +1 ) and with non-vanishing Euler characteristics χ(X k ) ̸ = 0 (hence of even dimensions n k ), or (b) spheres S n k with constant sectional curvatures (possibly of odd dimension n k ). Let ) be a smooth complete orientable Riemannian (n + N )manifold with Sc(g) > 0, and let g ♮ = Sc(g) • g.Let Z be an orientable enlargeable N -manifold, e.g., Z = R N , and let f : Y → X × Z be a smooth proper (quasi-proper will do) map such that the corresponding maps f k : Y → X k are strictly sum-wise area decreasing with respect to g ♮ in Y and g ♮ k in X k . 31his means that the norms of the exterior squares of the differentials of f k with respect the ♮-metrics satisfies (5.1) and f is the identity map.)If either the map f is spin or the universal covering of Y is spin, then the topological degree of f is zero.
Proof.First, let X and Y be spin, let f : Y → X be a smooth map and let V → Y be the f -pullback of the spin bundle S(X) → X to Y .Then, if R X ≥ 0 and f : Y → X is ♮-area decreasing at point y ∈ Y , i.e., ∥∧ 2 df (y)∥ ≤ 1, then according to [12] (also see [29]) the lowest eigenvalue of the operator R ⊗V at y ∈ Y satisfies where this inequality is strict if f is strictly ♮-area decreasing at y.
Next, let X k be spin, let X = × k X k and let V → Y be the tensor product Then, if the maps f k are ♮-area decreasing and at least one of f k is strictly ♮-area decreasing at a point y ∈ Y then, assuming Y is connected and spin, the Dirac operator D ⊗V on Y has index zero.
On the other hand, if χ(X k ) ̸ = 0, if dim(Y ) = dim(X) and deg(f ) ̸ = 0,32 then ind(D ⊗V ) ̸ = 0 by the Atiyah-Singer theorem (compare with [12,29,30]).This proves §5.A in the case where the manifold X is spin and it contains neither a Z-factor, nor an odd spherical factor.
To pass to the general case we argue as follows: 1. Odd dimensional spheres are suspended to even dimensional ones S n k ; S n k +1 , where these suspensions are accompanied by multiplying Y by a long circles and a suspending [f k : Y → S n k ; [Y × S 1 → S n k +1 ] as in [30], also see [17, §3.4.1] and [18].
2. If a Z, which may be assumed even-dimensional, is enlargeable, it supports an almost flat bundle, say W → Z with non-zero top-dimensional Chern class and the above V → Y is tensored by the pullback If neither X nor Y are spin but the map f is spin, then the Dirac operator D ⊗V is defined (this is explained in the present context in [34] and in [12]) and the above applies.
5.B.Spherical trace and symplectic remarks.The || ∧ 2 df k || contribution of each spherical factor X k with constant sectional curvature can be replaced in the formula (5.1) by an a priori smaller entity, that is, 1) , where and where the numbers λ j,k (y) ≥ 0 are defined by diagonalizing the differential df k : T y (Y ) → T f k (X k ) with an orthonormal frame e i,k ∈ T (y)(Y ), which is sent by df k to an orthogonal frame in T f k (X k ) with the vectors of lengths λ j,k (y).
The S 2 factors in X contribute to complex line bundles as ⊗-factors in V → Y .This, in view of Schrödinger-Hitchin (see [25]) formula for D ⊗L allows one to replace the product of these S 2 by a single (quasi)symplectic manifold (compare with [17, §2.7 and §3.4.4(4)]).

5.C. Sc
⋊ ↓ area, sp -additivity corollary.Let X k be manifolds as in 5.A, where we additionally assume that they are spin and have constant scalar curvatures.Then the fundamental classes [X k ] satisfy the sp-version of the Sc ⋊ ↓ sup.area -additivity (4.2) in §4.F:

5.D. Questions. (i) Does vanishing of Sc
There are examples of manifolds X k , where Sc ⋊ ↓ [X i ] = 0 and where their products admit metrics with Sc > 0; hence, Sc ⋊ ↓ [ × i X i ] > 0 for these X i , see [19].
(ii) Do products of spheres X = S n 1 × S n 2 , n 1 , n 2 ≥ 2, admit Riemannian metrics g ε , for all ε > 0, with Sc(g ε ) ≥ 1 and such that all non-zero homology classes h in H n 1 (X) and in H n 2 (X) satisfy Sc The existence of such a g ε , for n 1 = n 2 = 2, would imply the absence of the lower bounds on the 2-systoles of manifolds (X, g) in terms of σ(g) = inf x∈X Sc(X, g, x) > 0, 33   sup Recall, that the 2-systole is the infimum of the areas of all non-zero classes h ∈ H 2 (X), for area(h) = inf [c]∈h area(c) for the 2-cycles c ⊂ Y that represent h. 34 (iii) Let X be a compact symmetric space.What is the minimal seminorm on the linear maps λ 2 d : ∧ 2 R n → ∧ 2 T (X), say ||λ 2 d|| min such that the ♮-normalized inequality || ∧ 2 df || min < 1 for smooth equidimensional spin maps f : Y → X would imply that deg(f ) = 0? 35  (If X is the products of spheres, this seminorm is equal to the sum of the mean trace norms (as in §5.B) for maps R n d k → X k = S n k and for all symmetric spaces X of dimension ≥ 4 with χ(X) ̸ = 0 this norm is, probably, strictly smaller then the sup-norm || ∧ 2 d|| from the Goette-Semmelmann theorem.)

P -families of maps to product of spheres
Let Y = (Y, g) be an n-dimensional Riemannian manifold with Sc(X) > 0, where as earlier g ♮ = Sc(Y ) • g, let h m ∈ H m (Y ) be a homology class and let P be a locally contractible topological space, e.g., a manifold and h K ∈ H K (P ) be a homology class.
Let X be a product of spheres of variable radii, where dim(X) = k n k = m + K, and where the spheres are endowed with the usual metrics with sectional curvatures 1/R 2 k .Let F : Y × P → X be a continuous map such that the maps F p = F |Y ×p : Y → X are smooth and C 1 -continuous in p ∈ P .
Let the universal covering of Y be spin and let h m be equal to the homology class of the pullback of a genetic point under a smooth map ϕ : Y → Z, where Z is a smooth enlargeable manifold of dimension dim(Y ) − m.For instance, m = n and h 6.A. Theorem.Let the norms of the exterior squares of the differentials of the maps Then, in the following two cases, the (2) The dimension of Y is bounded by n ≤ 8. 33 Such a counter example would undermine (but not disprove) the conjectural bound waist2(X) ≤ constn σ for compact Riemannian n-manifolds with Sc(X) ≥ σ > 0. Thus, it may be safer to assume n1, n2 ≥ 3. 34 There are bounds on the 2-systoles of manifolds X with Sc ⋊ (X) ≥ σ in terms of their □⊥ -spreads (see [37,44]) which are proved as □ ∃∃ (n, m)-inequality in §2.B with a use of minimal hypersurfaces and µ-bubbles.Also there are similar bounds on the stable systoles of spin manifolds obtained with Dirac operators twisted with line bundles, where, recall, st.syst 2 (X) = lim infN→∞ area(N h) N . 35This norm must be invariant under isometries of X.
Proof.Case 1.If ranks(f p ) ≤ m, then the Llarull (Listing) trace inequality (4.6) in [30] together withe the above λ ⊗V -additivity show that index of the family of the Dirac operators on Y , twisted with the pullbacks of k S k as in §5.A, vanishes and the Atiyah-Singer theorem for families shows that F * (h m ⊗ h K ) = 0. (See in [17, §4] and references therein.)Case 2. If dim(Y ) ≤ 8, then, at last generically, the homology class h m can be realized by an m-submanifold Y 0 ⊂ Y such that the product Y 0 × T dim(Y )−m admits a warped product metric g ⋊ such that Sc(g ⋊ , y) ≥ Sc(Y, y) for all y ∈ Y 0 (see [18, §3] and references therein).Now the case 1 applies to Y 0 × T dim(Y )−m and the proof follows.
Remarks/Problems.(a) For all we know, the spin and dim(Y ) ≤ 8 condition are redundant and there is a fair chance that a further study of singularities of minimal hypersurfaces in he spirit of [39] and/or [31,32] will allow one to remove the latter.But removing the spin condition needs a new idea.
The argument in Case 1 can be extended to maps of foliated manifolds to × k S n k as in [42], but a foliated version of Case 2 is problematic.

Spinorial curvature
Given a closed orientable even dimensional Riemannian manifold Y let Sp.curv(Y ) be the infimum of the numbers κ ≥ 0 such that there exist a complex vector bundle V → X with a unitary connection such that (b) The Kazdan-Warner conformal change theorem [26] and conformal invariance of harmonic spinors [25] show that if λ 1 (X, β) > 0 for β = (n − 2)(4(n − 2), then X supports no non-zero harmonic spinors.However, it is unclear how to extract further geometric, rather than topological information from the inequality λ 1 (X, β) > σ for β < (n − 1)/4n and σ > 0.
Next, let X be a Riemannian manifold, let h m ∈ H m (X) be a homology class and let Y be a class of smooth closed orientable m-manifolds Y along with maps f : Y → X.
Define Sp.curv ↓ Y (X) via smooth maps F : Y × T N → X × T N and Riemannian metrics G on Y × T N as the infimum where the infimum is taken over N such that m + N is even, where F is area decreasing with respect to the metric G and where Clearly, by the above, if the universal coverings of manifolds Y ∈ Y are spin, 38 then area,Y (h m ) .
Remark.If the universal coverings of the manifolds Y ∈ Y are spin, then the fundamental classes [X] of compact symmetric spaces X with χ(X) ̸ = 0, satisfy the equally by the T ⋊ -stabilized Goette-Semmelmann theorem and this equally applies to products X = × X i , where X i are as in §5.A.
Possibly, (a version of) this equality holds true for all symmetric spaces but it seems unlikely in general, even for rational homology classes h, that the Dirac operator is the only source of bounds on Sc ⋊ ↓ area,Y (h).Power stabilization.Let Questions.I. What are further instances (besides the above h = [X]) of the equality and what are examples where this fails to be true?II.Can one pass to the limit, set M = ∞ and prove scalar curvature bounds for "Riemannian metrics" G on infinite dimensional manifolds X, e.g., where such a G differs from the infinite sum of Riemannian metrics, ∞ Remarks.(a) If ∆ = ∆ i,j decays very fast, for i and/or j tending to infinity, then finite products X M = × M 1 X i embed to X = × ∞ 1 (X i , g i ) with small relative curvatures and a bound on "Sc(X)" may be derived in some cases from such a bound on X M , but it would be more interesting to develop a truly infinite dimensional argument for bounds on "Sc(X)" and/or to find applications of such bounds.
Test question.Let X = {x i } i x 2 i ≤∞ be the Hilbert space and G = G ij be a smooth Riemannian metric on X, which is greater than the background Hilbertian metric, G(τ, τ ) ≥ ||τ || 2 for all tangent vectors τ ∈ T (X) and let M = 3, 4, . . .be an integer.
Can the M -scalar curvature of G (defined below) be strictly positive, say Sc M (G) ≥ 1?Here Sc M is the function on the tangent M -planes P M ∈ T x (X), x ∈ X, which is equal to the scalar curvature at zero in P M = R M of the Riemannian metric induced by the exponential map exp : P M → X from G. (It may be worthwhile to compare Sc M with with the m-intermediate curvature from [3].)(b) A natural approach to these problems is by a finite-dimensional approximation as in (a) but this seems that uncomfortably restrictive conditions on G are needed (compare with [14]).
(c) Basic features of positive scalar curvature have their counterparts for mean convex hypersurfaces (see [16]), where the infinite dimensional geometry is a bit more transparent than that of the scalar curvature.

1 )
The ranks of the (differentials of the) maps f p : Y → X are everywhere ≤ m, e.g., dim(Y ) = m and h m = [Y ].

∂X), denote the supremum of the numbers σ such that the homology class h is representable by a distance decreasing map f from an oriented Riemannian m-manifold Y with Sc ⋊
is zero or infinity without the spin assumption on Y .)2.A. Surface examples.Closed connected simply connected, i.e., spherical Riemann surfaces X satisfy Sc