Codazzi tensor fields in reductive homogeneous spaces

We extend the results about left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics obtained by d'Atri in 1985 to the setting of reductive homogeneous spaces $G/H$, where the curvature of the canonical connection of second kind associated with the fixed reductive decomposition $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$ enters the picture. In particular, we show that invariant Codazzi tensor fields on a naturally reductive homogeneous space are parallel.


Introduction
Whenever M is a smooth manifold equipped with a connection ∇, a twicecovariant symmetric tensor field A on M is called a Codazzi tensor field if d ∇ A = 0, where d ∇ is the exterior derivative operator (defined with the aid of ∇) acting on tensor bundles over M, and we regard A as a T * M-valued 1-form.When ∇ is torsionfree, A is a Codazzi tensor field if and only if ( †) (∇ X A)(Y, Z) = (∇ Y A)(X, Z), for all X, Y, Z ∈ X(M), which is to say that the covariant differential ∇A, a three-times covariant tensor field on M, is totally symmetric.
Codazzi tensors are ubiquitous in geometry, with the most prominent examples being the second fundamental form of a non-degenerate hypersurface in a pseudo-Riemannian manifold with constant sectional curvature (due to the Codazzi-Mainardi compatibility equation), and the Ricci or Schouten tensors of a pseudo-Riemannian manifold with harmonic curvature or harmonic Weyl curvature (due to the relations div R = d ∇ Ric and div W = d ∇ Sch).Whenever a Riemannian manifold (M, g) has constant sectional curvature K, every Codazzi tensor field locally has the form Hess f + K f g for some smooth function f , cf. [4].
Both topological and geometric consequences of the existence of a nontrivial Codazzi tensor field on a Riemannian manifold have been studied in [4,7], and the local structure of a Riemannian manifold carrying a Codazzi tensor field satisfying additional multiplicity assumptions on its spectra and eigendistributions is obtained in [9].Many such results are compiled in [3, §16.6- §16.22], which then led to further work [5,14].
In a different and more specific direction, left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics have been discussed in [6], with the goal of better understanding the harmonic curvature condition in this setting.New results have been recently obtained in [1], where it is shown that solvable Lie groups equipped with left-invariant Riemannian metrics having harmonic curvature must necessarily be Ricci-parallel.
In this paper, we extend the results in [6] to the more general class of invariant Codazzi tensor fields on reductive homogeneous spaces equipped with invariant Riemannian metrics.Our approach to achieve this is straightforward: once a reductive decomposition g = h ⊕ m for the homogeneous space G/H is fixed, we run the computations done in [6] in the reductive complement m (a non-associative algebra) instead of in the Lie algebra g.However, unlike in some results in [6] which involve positivity and negativity of sectional and scalar curvatures, the curvatures of (G/H, •, • ) are now compared with curvatures of the canonical connection of second kind associated with the decomposition g = h ⊕ m -with its flatness when h = {0} and m = g explaining its absence in [6].Full proofs are included for the sake of completeness.

Organization of the text
We work in the smooth category and all manifolds considered are connected.
In Section 1, we gather some well-known standard facts regarding reductive homogeneous spaces needed for the rest of the text, the most important ones being Nomizu's Theorem [15] on invariant connections and Lemma 1.In particular, we conclude that every invariant Codazzi tensor field on a naturally reductive homogeneous space is parallel.

Preliminaries
The material in this section is standard and it is included for the convenience of the reader.We refer to [13,  Following [8, Section 5.2], a G-invariant connection ∇ on G/H and an Ad(H)-equivariant multiplication α in m related via (1.4) determine each other by the relation Here, we are using that every X ∈ g determines its corresponding action field X # ∈ X(G/H), with X # eH = X m and whose complete flow is explicitly given by (t, aH) → τ exp(tX) (aH).Note that the right-invariant vector field on G generated by X is π-related to X # .For future reference, we also observe that this implies that as the flow of X # leaves Θ invariant.The torsion and curvature of ∇ are given in for all X, Y, Z ∈ m, cf.[15, formulas (9.1) and (9.6)] or [8, formula (22)].
LEMMA 1.1.For a G-invariant connection ∇ and a G-invariant k-times covariant tensor field Θ on G/H, corresponding to α and θ on m under (1.4)-(1.5)and (1.2), the covariant differential ∇Θ is also G-invariant and corresponds under (1.2) to α(•, θ) on m given by PROOF.We will establish (1.8) when k = 1, with the general case being an exercise in notation.The identity (∇

The Codazzi compatibility condition in m
In this section, let G/H be a homogeneous space admitting a reductive decomposition g = h ⊕ m be equipped with a G-invariant Riemannian metric •, • and its Levi-Civita product α.By Lemma 1.1 and ( †) in the Introduction, a twice-covariant for all X, Y, Z ∈ m.As A is symmetric and g is positive-definite, the spectral theorem allows us to write an orthogonal direct sum decomposition , where r ≥ 1 and each m i is the eigenspace of A associated with the eigenvalue λ i , ordered so that We will also write (•) i : m → m i for the corresponding direct sum projections.
A subalgebra of (m, [ holds for all X, Y, Z ∈ m and i, j, k ∈ {1, . . ., r}.Conversely, if a direct sum decompo- In addi- tion, ∇A = 0 if and only if there exists a triple (i, j, k) of mutually distinct indices with follows for all X, Y, Z ∈ m.The Codazzi condition (2.1) now reads Using (1.9) twice and rearranging terms, (2.5) becomes Permuting elements, we also have Conversely, to verify that A = r i=1 λ i •, • | m i ×m i defines a Codazzi tensor field whenever (2.3) holds, it suffices to note that it implies (2.6) (and hence (2.5), due to (1.9)).Indeed: (2.3) becomes (2.6) when i = k = j while, if i = j, adding to (2.3) the expression obtained from it after permuting (i, j, k) → (j, k, i) yields (2.7) (and hence (2.6)).
Finally, (2.3) also implies whenever i = j.Substituting (2.8) into (1.9) and simplifying it with the aid of (2.4), we obtain which directly implies the last assertions regarding ∇A.(2.10) if a self-adjoint endomorphism T of a pseudo-Euclidean space (V, •, • ) with dim V ≥ 3 satisfies that Tv, v = 0 for every null v ∈ V {0}, then T is diagonalizable in an orthonormal basis of V.
To justify (2.10), it suffices to choose Φ = As pointed out in [6], there is a simple interpretation for the compatibility relation (2.3).For each k ∈ {1, . . ., r}, considering the inner product •, • k on m defined by 1 Here, the representation is a representation of the vector space m k , not of the non-associative algebra (m k , [•, •] k ).As a consequence: (2.11) for each Z k ∈ m k , the chararacteristic roots of the Recall that a non-associative algebra a is: (a) nilpotent [16, p. 18] if there is a positive integer t such that the product of t elements in a, no matter how associated, equals zero.
Following [6], we call a G-invariant Codazzi tensor field A on G/H essential if ∇A = 0 and none of the eigenspaces m i is an ideal of (m, [ the above, we obtain: PROOF.As in the Lie category, one may define a 'Killing form' holds, where β k stands for the Killing form of (m k , [•, •] k ).

Codazzi tensors versus difference curvatures
In this section, we continue to work with a homogeneous space G/H equipped with a reductive decomposition g = h ⊕ m, G-invariant Riemannian metric •, • , and Levi-Civita product α.
We will also need the canonical connection of second kind induced by given reductive decomposition, that is, the affine connection ∇ 0 on G/H corresponding under (1.4)-(1.5) to the zero product in m.By (1.7-ii), the curvature tensor and Ad(H)-invariance of •, • that: The Ricci tensor Ric 0 of ∇ 0 is defined by Ric with no reference to the metric •, • , and it is only guaranteed to be symmetric if R 0 satisfies the Bianchi identity.We also consider the sectional and scalar curvature functions K 0 and s 0 associated with ∇ 0 and •, • : for any plane Π ⊆ m we let , where {X, Y} is any orthonormal basis for Π (with its choice being immaterial due to (ii) above), and s 0 = tr •,• Ric 0 .The results in this section are most conveniently stated and proved in terms of (3.1) the difference curvature tensor R d = R − R 0 and the corresponding notions of sectional, Ricci, and scalar curvatures: they are respectively defined by As setup for the next result, observe that whenever A is a G-invariant Codazzi tensor field on G/H and m is decomposed as in (2.2), an equivalent formulation to (2.9) is Applying (3.2) to separately compute each term in the curvature relation (1.7-ii) for (X, Y, Z) = (X i , Y j , Y j ), with i = j, we obtain α(X i , α(Y j , Y j )), X i = 0 and ] k and switching the roles of X and Y in (2.3) leads to which, when combined with (3.3), implies that We are ready to generalize [6, Proposition 3]: If either (3.5-a) or (3.5-b) fails to hold, then [X i , Y j ] m , Z k = 0 whenever i, j, k are mutually distinct, so that ∇A = 0 by Proposition 2.1.Indeed, if (a) fails then is orthogonal to m k , and we may apply (2.3).If (b) fails instead, then again [m i , m j ] m ⊆ m i ⊕ m j is orthogonal to m k whenever i, j, k are mutually distinct.This proves (3.5).
For ρ as in (3.5-a), minimality of ρ implies that [m i , m j ] ρ = 0 whenever i, j < ρ, and so [m i , m ρ ] j = 0 for distinct i, j < ρ by (2.3) with k = ρ.Hence, (2.2) and (3.4) yield Lastly, for µ, ν as in (3.5-b) chosen so that the difference ν − µ is maximal, we have that 3) with (µ, ν) = (i, j).Choosing unit vectors X µ and Y ν with [X µ , Y ν ] ℓ = 0, for some ℓ = µ, ν, it follows from (2.2) and (3.4) that PROOF.First, observe that the cyclic identity holds for all X, Y, Z ∈ m whenever i, j and k are mutually distinct, as a direct consequence of (2.8).Now, writing d i = dim m i and letting {E i,a } d i a=1 be an orthonormal basis for m i , for each i = 1, . . ., r, it follows from the definition of Ric Using (2.2) and the fact that (λ k − λ j )(λ i − λ j ) is a product of positive (or, negative) factors when j = 1 (or, j = r) for all i and k, (i) follows.Finally, setting Y j = E j,c in

1 . Section 2 generalizes [ 6 ,
Proposition 1] to Proposition 2.1: the same compatibility condition (2.3) ensures that a symmetric bilinear form on m reconstructed from prescribed eigenspaces gives rise to a Codazzi tensor field on G/H.Section 3 explores the effects of the existence of an invariant Codazzi tensor field on curvature, generalizing [6, Propositions 3 and 4] and expressing the new conclusions, Propositions 3.1 and 3.3, with the aid of the difference curvature tensor introduced in (3.1).

7 )(
Ric d (Y j ) = Ric d j (Y j ) + 2 λ i − λ k )(λ j − λ k ) (λ i − λ j ) 2 [E i,a , Y j ] m , E k,bfor every Y j ∈ m j .Here, we write Ric d (Y j ) as a shorthand for Ric d (Y j , Y j ), and similarly for Ric d j .The summand in the right side of (3.7) vanishes when k = i and, relabeling dummy indices (i, a) ⇌ (k, b) in one of the two copies of such summation, we see that (3.6) leads to (3.8) Ric d (Y j ) = Ric d j (Y j ) −

( 3 . 8 )
and summing over 1 ≤ c ≤ d j and 1 ≤ j ≤ r, we conclude that (ii) holds: the difference s d 1 + • • • + s d r − s d equals the sum over mutually distinct indices i, j, k of terms appearing in (3.6), and therefore it must vanish.A last consequence of Proposition 3.3 is the counterpart to [6, Proposition 5]: COROLLARY 3.4.Suppose that Ric d itself is a Codazzi tensor field on G/H, with ∇Ricd = 0.If s d i ≥ 0 for 1 ≤ i ≤ r − 1, then s d r = 0.In particular, not all eigenspaces of Ric d can be Abelian subalgebras of (m, [•, •] m ).PROOF.Item (i) of Proposition 3.3 for A = Ric d reads Ric d (Y r , Y r ) ≥ λ r for all unit vectors Y r ∈ m r , so averaging over an orthonormal basis yields s d r /d r ≥ λ r .If it were to be s d r = 0, (2.2) would imply that λ 1 < • • • < λ r ≤ 0, and hence s d = d 1 λ 1 + • • • + d r λ r < 0. However, it is clear from s d i ≥ 0, for 1 ≤ i ≤ r − 1, and item (ii) of Proposition 3.3, that s d ≥ 0. The last claim now follows as R d i = 0 (and thus s d i = 0) whenever m i is Abelian, as α| m i ×m i = 0 in view of (1.9) and Proposition 2.1.
The group G acts transitively on G/H via the "left translations" τ g : G/H → G/H given by τ g (aH) = (ga)H.Writing g and h for the Lie algebras of G and H, we assume that G/H is reductive: there is a vector space direct sum decomposition g = h ⊕ m such that m is Ad(H)-invariant.We write (•) h : g → h and (•) m : g → m for the direct sum projections, and so (m, [•, •] m ) becomes a non-associative algebra.The derivative dπ e restricts to an m → m.
PROPOSITION 2.1.Whenever A is a G-invariant Codazzi tensor field on G/H, all the factors in decomposition (2.2) are Ad(H)-invariant totally geodesic subalgebras of (m, [•, •] m ), and the compatibility condition Z k ∈ m k be arbitrary, and assume that (m, [•, •] m ) is nilpotent.It follows that both operators ad m (Z k ) and ad m (Z k )| m k are nilpotent, and so both β(Z k , Z k ) and β k (Z k , Z k ) vanish.In particular, (2.12) leads to tr (7)ware of the typo in [6, formula(7)]: the formula there has X, Y instead of X j , Y j . Lt By [12, Corollary 1.30], whose 'necessity' implication does not rely on the Jacobi identity, the characteristic roots of each ad m (Z k ), for Z k ∈ m k , are real.Combined with (2.11), it follows that π as required.EXAMPLE 3.2.Recall that a homogeneous space G/H with a G-invariant Riemannian metric •, • is called naturally reductive if it admits a reductive decompo-sition g = h ⊕ m with the additional property that [X, Y] m , Z + Y, [X, Z] m = 0, for all X, Y, Z ∈ m.Rearranging the formula in [2, Proposition 5.7] we see that, in this case, K d (Π) = [X, Y] m 2 /4 ≥ 0,where {X, Y} is any orthonormal basis for Π.By Proposition 3.1, every G-invariant Codazzi tensor field on such a naturally reductive homogeneous space is necessarily parallel.∈ m i , the endomorphism X → R d (X, Y i )Z i of m restricts to an endomorphism of m i , whose trace is Ric d i (Y i , Z i ).Then, the trace of Ric d i computed with •, • | m i ×m i is s d i .
[6, the next result, which generalizes[6, Proposition 4], we let M i be the leaf passing through eH of the eigendistribution of A associated with λ i , so thatT eH M i = m i .Each M i isa totally geodesic submanifold of G/H equipped either with the Levi-Civita connection of •, • (by Proposition 2.1), or with the canonical connection ∇ 0 .This allows us to consider the difference Ricci and scalar curvatures Ric d i and s d i in (3.1) for each M i .More precisely, given Y i , Z i PROPOSITION 3.3.If G/H has a G-invariant Codazzi tensor field, then: