The Weitzenböck Type Curvature Operator for Singular Distributions

mathematics

Article The Weitzenböck Type Curvature Operator for Singular Distributions

Paul Popescu 1,* , Vladimir Rovenski 2 and Sergey Stepanov 3

1 Department of Applied Mathematics, University of Craiova, Str. Al. Cuza, No, 13, 200585 Craiova, Romania 2 Department of Mathematics, University of Haifa, Mount Carmel, 31905 Haifa, Israel; [email protected] 3 Russian Institute for Scientiﬁc and Technical Information of the Russian Academy of Sciences, 125190 Moscow, Russia; [email protected] * Correspondence: [email protected]

Received: 16 February 2020; Accepted: 3 March 2020; Published: 6 March 2020

Abstract: We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we deﬁne the modiﬁed covariant and exterior derivatives and their L2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f -manifolds, including almost Hermitian and almost contact ones.

Keywords: Riemannian manifold; singular distribution; Weitzenböck curvature operator; Hodge Laplacian; almost Lie algebroid

MSC: 53C15; 53C21

1. Introduction Distributions, as subbundles of the tangent bundle on a manifold, arise in such topics of mathematics and physics as fiber bundles, Lie groups actions, almost contact, Poisson and sub-Riemannian manifolds, e.g., [1–5]. Lie algebroids are generalizations of Lie algebras and integrable regular distributions, a more general concept of an almost Lie algebroid, i.e., an anchored vector bundle with an almost Lie bracket, is involved in nonholonomic geometry, e.g., [6–8]. Singularities play a crucial role in mathematics (e.g., in real and complex analysis, algebraic geometry and differential topology, theory of dynamical systems), and its applications in natural and technical sciences (e.g., in theories of nonlinear control systems and relativity). There is definite interest of pure and applied mathematicians, e.g., [3,9], to singular distributions and foliations, i.e., having varying dimension: just mention (singular) Riemannian foliations, i.e., every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets; an example is the orbital decomposition of the isometric actions of a Lie group. A singular distribution D on a manifold M assigns to each point x ∈ M a linear subspace Dx of the tangent space Tx M in such a way that, for any v ∈ Dx, there exists a smooth vector field V defined in a neighborhood U of x and such that V(x) = v and V(y) ∈ Dy for all y of U. A priori, the dimension dim Dx depends on x ∈ M. If dim Dx = const, then we obtain a regular distribution. Singular foliations are defined as families of maximal integral submanifolds (leaves) of integrable generalized distributions (certainly, regular foliations correspond to integrable regular distributions). The study of singular distributions is important also because there

Mathematics 2020, 8, 365; doi:10.3390/math8030365 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 365 2 of 18 are plenty of manifolds that exist that do not admit smooth (codimension-one) distributions, while all of them admit such distributions defined outside some “set of singularities”. Let M be a connected smooth n-dimensional manifold, g = h·, ·i is a Riemannian metric on M, TM is the tangent bundle, XM is the Lie algebra of smooth vector fields on M, and End(TM) are smooth endomorphisms of TM onto itself, i.e., linear maps on the fibers of TM. Here, we determine singular distributions as images of TM with smooth endomorphisms.

Deﬁnition 1 (see [10]). An image D = P(TM) of a smooth endomorphism P ∈ End(TM) will be called a generalized vector subbundle of TM or a singular distribution.

Example 1. (a) Consider P ∈ End(TRn) having as image D = P(TRn) the singular distribution tangent to spheres centered at the origin and the origin itself as a singular point. Namely, D is spanned by the j i vector ﬁelds Zij = x ∂i − x ∂j for 1 ≤ i < j ≤ n. For n = 2 and coordinates (x, y), set P(x, y; X, Y) = (x, y; y(yX − xY), −x(yX − xY)). Then, the vector ﬁelds P(x, y; 1, 0) and P(x, y; 0, 1) generate D. The case of n > 2 is similar. (b) Let P ∈ End(TM) on (M, g) be of constant rank and satisfying conditions

P2 = P, P∗ = P, (1) where P∗ is adjoint endomorphism to P, i.e., hP∗X, Yi = hX, PYi, then we have an almost-product structure on (M, g), see [11]. In this case, P and P˜ = id −P (that can be orthoprojectors) determine complementary orthogonal regular distributions P(TM) = D and P˜(TM) = De, neither of which is in general integrable. Twisted (and warped) products give us popular examples of foliations (integrable regular distributions), e.g., [12,13]. Furthermore, given a Riemannian metric g and a singular distribution D on M, there are self-adjoint endomorphisms P and P˜ of TM onto itself such that P(TM) = D and P˜(TM) = De are smooth orthogonal distributions and pointwise orthogonal sum decomposition TM = P(TM) ⊕ P˜(TM) holds on a dense subset of M (see [14]).

The Hodge Laplacian ∆H = dδ + δd, where d is the exterior differential operator, and δ is its adjoint operator with respect to the L2 inner product, can be decomposed into two terms,

∆H = ∆¯ + Ric. (2)

One is the connection Laplacian ∆¯ = ∇∗∇, and the second term (depends linearly on the Riemannian curvature tensor) is called the Weitzenböck curvature operator on (0, k)-tensors S over (M, g),

R ( )( ) = k n ( )( ) ic S X1,..., Xk ∑ = ∑ = R ei,Xα S X1,..., ei,..., Xk α 1 i 1 | {z } α

= −2 ∑ < R(ei, Xa, em, Xb) · S(X1,..., em,..., ei,..., Xk) i,m,a;b a | {z } | {z } b a−b + ∑ Ric(ei, Xa) · S(X1,..., ei,..., Xk), (3) i,a | {z } a

j p j or, in coordinates, Ric (S) = −2 ∑ R S + ∑ Ric S , e.g., [15]. Note that i1,...,ik adifferential geometry of a Riemannian manifold (M, g) endowed with a singular (or regular) distribution P(TM) and continues our study [7,8,10,14,20]. In Section2, we recall the construction of an almost Lie algebroid on a vector bundle. Following this construction, in Sections3 and4, we deﬁne the derivatives ∇P and dP and their L2 adjoint operators on tensors. The assumptions that we use in the results are reasonable, as illustrated in Section3 by examples with metric f -manifolds, including almost Hermitian and almost contact manifolds. In Section5, using ∇P and making some assumptions about P (which are trivial when P = id TM), we deﬁne the curvature operator RP and the Weitzenböck type curvature operator on tensors. Then, we prove the Weitzenböck type decomposition formula and derive the Bochner–Weitzenböck type formula. In Section7 (see also Section4), we obtain vanishing results, Theorem3 and Corollary1, about the null space of the Hodge type Laplacian. AppendixA contains the proof of Proposition6.

2. The Almost Lie Algebroid Structure

Let πE : E → M be a smooth vector bundle of rank k over M (i.e., a smooth fibre bundle with fibre Rk). A Riemannian bundle (E, g) over M has a symmetric positive definite (0,2)-tensor field g = h·, ·i, i.e., gx is an inner product in each fiber Ex smoothly depending on x ∈ M. The main example is E = TM with the Riemannian structure (metric) on M, see more examples in [21]. Recall some facts about another important structure on E, see, e.g., [6,7,20].

Deﬁnition 2. An anchor on E is a morphism ρ : E → TM of vector bundles. A skew-symmetric bracket on E is a map [·, ·]ρ : XE × XE → XE such that

[Y, X]ρ = −[X, Y]ρ, [X, f Y]ρ = ρ(X)( f )Y + f [X, Y]ρ, ρ([X, Y]ρ) = [ρ(X), ρ(Y)] (4)

∞ for all X, Y ∈ XE and f ∈ C (M). The anchor and the skew-symmetric bracket give an almost Lie algebroid structure on a vector bundle E. The tensor map Jρ : XE × XE × XE → XE, given by

J ( ) = [ [ ] ] ρ X, Y, Z ∑ circ. X, Y, Z ρ ρ, (5) is called the Jacobiator of the bracket; using (4)3, we get ρJρ = 0. An almost Lie algebroid is a Lie algebroid provided that the Jacobiator of the bracket [·, ·]ρ vanishes.

Note that axiom (4)3 means vanishing of the following operator:

ρ D (X, Y) = [ρX, ρY] − ρ([X, Y]ρ). (6)

There is a bijective correspondence between almost Lie algebroid structures on E and the exterior L k ∗ k differentials of the exterior algebra Λ(E) = k∈N Λ (E) of the dual bundle E , see [8]; here Λ (E) is the set of k-forms over E. The exterior differential dρ, corresponding to the almost Lie algebroid structure (E, ρ, [·, ·]ρ), is given by

ρ ( ) = k (− )i( )( ( )) d ω X0,..., Xk ∑ i=0 1 ρXi ω X0,..., cXi,..., Xk + (− )i+j ([ ] ) ∑ 0≤i

k where X0,..., Xk ∈ XE and ω ∈ Λ (E) for k ≥ 0. For k = 0, we have

ρ ∞ 0 d f (X) = (ρX)( f ), X ∈ XE, f ∈ C (M) = Λ (E). Mathematics 2020, 8, 365 4 of 18

Recall that a skew-symmetric algebroid deﬁnes uniquely an exterior differential dρ on Λ(TM), and it gives rise to (see also Proposition5) – a skew-symmetric algebroid if and only if (dρ)2 f = 0 for f ∈ C∞(M); – a Lie algebroid if and only if (dρ)2 f = 0 and (dρ)2 ω = 0 for f ∈ C∞(M) and ω ∈ Λ1(TM).

ρ Deﬁnition 3. A ρ-connection on (E, ρ) is a map ∇ : XE × XE → XE satisfying Koszul conditions

ρ ρ ρ ρ ρ ρ ∇X ( f Y + Z) = ρ(X)( f )Y + f ∇X Y + ∇X Z, ∇ f X+Z Y = f ∇X Y + ∇Z Y.

ρ ρ ρ For a ρ-connection ∇ on E, they deﬁne torsion T : XE × XE → XE and curvature R : XE × XE × XE → XE by “usual" formulas

ρ ρ ρ T (X, Y) = ∇X Y − ∇Y X − [X, Y]ρ, (7) ρ ρ ρ ρ ρ ρ R Z = ∇ ∇ Z − ∇ ∇ Z − ∇ Z. (8) X,Y X Y Y X [X,Y]ρ

The following equality holds, see [7]:

ρ = [(∇ρ ρ)( ) + ρ( ρ( ) )] + J ( ) ∑ circ. R X,Y Z ∑ circ. X T Y, Z T T X, Y , Z P X, Y, Z . (9)

We will apply the almost Lie algebroid structure to singular distributions on M, namely, in the rest of paper, we assume E = TM and ρ = P ∈ End(TM), see Deﬁnition1.

3. The Modified Covariant Derivative and Its L2-Adjoint The L2-scalar product on compactly supported C∞-sections of TM and the Hilbert space L2(TM) R 2 are defined by (u, v) = M hu, vi d volg, where d vol is the volume form of g. Using the L -structure on C∞(TM), they define the connection Laplacian by ∆¯ = ∇∗∇, where ∇∗ is the L2-adjoint of the Levi–Civita connection ∇. The covariant derivative of a (m, k)-tensor S on TM is a (m, k + 1)-tensor, e.g., [15],

(∇ S)(Y, X1,..., Xk) = (∇Y S)(X1,..., Xk) = ∇ ( ( )) − ( ∇ ) Y S X1,..., Xk ∑i≤k S X1,..., Y Xi,..., Xk ,

2 where Y, Xi ∈ XM. Its L -adjoint is given by

(∇∗ )( ) = − (∇ )( ) S X2,..., Xk ∑i ei S ei, X2,..., Xk .

The divergence of a (0, k + 1)-tensor S is a (0, k)-tensor div S = −∇∗S. Using a metric g on j a manifold, they identify (1, k)-tensors with (0, k + 1)-tensors: S = gmjS . Thus, for a · i1,...ik m,i1,...ik (1, k)-tensor S, we can equivalently write div S = trace(Y → ∇YS), that is,

( )( ) = h(∇ )( ) i div S X1,..., Xk ∑ i i S X1,..., Xk , ei ,

j or (div S) = ∇ S in coordinates. For example, if S is a (1,1)-tensor, then ∇∗S is a vector i1,...ik j · i1,...ik field and div S is a 1-form. For the divergence div X = trace(Y → ∇Y X) of a vector field X ∈ XM, we have div X = −∇∗X[, where X[ is the 1-form dual to X. The divergence of a vector field can also be defined by d(ιX d vol) = (divX) d vol, (10) If either X has compact support or M is closed, then the Divergence Theorem reads as Z (div X) d volg = 0. (11) M Mathematics 2020, 8, 365 5 of 18

P Given P ∈ End(TM), deﬁne a P-connection ∇ : XM × XM → XM (see Deﬁnition3, which generally is not a linear connection on TM) by

P ∇X Y := ∇PX Y (X, Y ∈ XM), (12)

P ∞ P and set ∇X f = (PX) f for f ∈ C (M). We will use ∇ to deﬁne the modiﬁed covariant derivative on tensors and its L2-adjoint on (M, g) endowed with a singular distribution.

Deﬁnition 4. The P-covariant derivative on tensors is deﬁned by the following:

∞ P P P – If f ∈ C (M), then (∇ f )(X) = ∇X f , and ∇ f is called the P-gradient of f . P P – If Y ∈ XM, then (∇ Y)(X) = ∇X Y, see (12). – If S is a (s, k)-tensor, where s, k = 0, 1, then

(∇P )( ) = ∇P( ( )) − k ( ∇P ) S Y, X1,..., Xk Y S X1,..., Xk ∑i=1 S X1,..., Y Xi,..., Xk . (13)

– This makes sense for all s, k ≥ 0, using the product rule with respect to tensors:

P P P ∇X(S1 ⊗ S2) = ∇XS1 ⊗ S2 + S1 ⊗ ∇XS2.

If ∇PS = 0, then the tensor S is called P-parallel.

Notice that the metric and volume form are P-parallel, since ∇P has the metric property:

P P P ∇XhY, Zi = h∇PXY, Zi + hY, ∇PX Zi = h∇XY, Zi + hY, ∇X Zi

P P for all X, Y, Z ∈ XM, and commutes with contractions for tensors: ∇X(Tr S) = Tr (∇XS). The following pointwise inner products and norms for (0, k)-tensors will be used: q hS1, S2i = S1(ei ,..., ei ) S2(ei ,..., ei ), kSk = hS, Si ∑ i1,..., ik 1 k 1 k with respect to any local orthonormal frame {ei} while, for k-forms, we set

hω1, ω2i = ω1(ei ,..., ei ) ω2(ei ,..., ei ). ∑ i1<...

The next proposition generalizes ([15], Proposition 2.2.8) and shows that, under certain conditions, the L2-adjoint to the P-covariant derivative on tensors, denoted here by ∇∗P, is given by

(∇∗PS)(X ,..., X ) = − (∇P S)(e , X ,..., X ). (14) 2 k ∑i ei i 2 k

Lemma 1 (see Proposition 2.4 in [10]). Given P ∈ End(TM), condition

div(PP∗) = 0 (15) is equivalent to the following one:

∗ divP X = div(PP X), X ∈ XM. (16)

Condition (16) means that (divPX) d vol is an exact form: using (10), we have

∗ (divP X) d vol = (div(PP X)) d vol = d(ιPP∗X d vol). Mathematics 2020, 8, 365 6 of 18

Thus, Lemma1 allows us to extend the Divergence Theorem, see (11): if (15) holds and either X has compact support or M is closed, then Z (divP X) d volg = 0. (17) M

Proposition 1. If condition (15) holds, then ∇∗P is L2-adjoint to ∇P on tensors; namely, for any compactly supported (s, t)-tensor S and (s, t + 1)-tensor T, we have

∗P P (∇ S2, S1) = (S2, ∇ S1). (18)

Proof. Deﬁne a compactly supported 1-form ω by

ω(Y) = hι Y T, Si, Y ∈ XM.

Take an orthonormal frame (ei) such that ∇Yei = 0 for all Y ∈ Tx M. To simplify calculations, assume that s = t = 1, then (similarly to ([15], Proposition 2.2.8)) at x ∈ M,

−∇∗P = h i (∇ )( ) = h i ∇ h ( ) ( ) i ω ∑i,j Pej, ei ei ω ej ∑i,j,c Pej, ei ei S2 ej, ec , S1 ec = h ih∇ ( ) ( ) i + h ( ) ∇ ( ) i ∑i,j,c Pej, ei ei S2 ej, ec , S1 ec S2 ej, ec , ei S1 ec = h h i∇ ( ) ( )i + h ( ) h i∇ ( ) i ∑i,j,c Pej, ei ei S2 ej, ec , S1 ec S2 ej, ec , Pej, ei ei S1 ec = h∇P S (e , e ), S (e ) i + hS (e , e ), ∇P S (e ) i ∑j,c ej 2 j c 1 c 2 j c ej 1 c ∗P P = −h∇ S2, S1i + hS2, ∇ S1i.

∗P P The ∇ is related to the P-divergence divP X = trace(Y→ ∇Y X) of a vector ﬁeld X ∈ XM (deﬁned equivalently in [10]) by ∗P [ divP X = −∇ X . (19) Indeed, at x ∈ M, we calculate

div X = h∇P X, e i = hPe , e i h∇ X, e i P ∑j ej j ∑i,j j i ei j = h i h i = h i ( [( )) ∑i,j Pej, ei ei X, ej ∑i,j Pej, ei ei X ej = hPe e i(∇ X[)(e ) = (∇ X[)(e ) ∑i,j j, i ei j ∑j Pej j = (∇P X[)(e ) = −∇∗PX[. ∑j ej j

By (19) with X[ = ω and extended Divergence Theorem, see (17), we obtain (18).

Deﬁne the modiﬁed connection Laplacian by

∆¯ P = ∇∗P∇P. (20)

The next maximum principle generalizes ones used in the past.

Proposition 2. Let condition (15) hold for P ∈ End(TM) on a closed Riemannian manifold (M, g) and let S be a smooth tensor ﬁeld such that h∆¯ P S, Si ≤ 0. Then, S is P-parallel.

Proof. We apply Formulas (18) and (20),

0 ≥ (∆¯ P S, S) = (∇∗P∇P S, S) = (∇P S, ∇P S) ≥ 0; hence, ∇P S = 0. Mathematics 2020, 8, 365 7 of 18

Example 2 (see [10]). We use popular geometrical structures that do not satisfy (1) to clarify the property (15). 2 (a) An almost complex manifold (M, J) (i.e., J = − id TM) admits a Hermitian metric: hJX, JYi = ∗ hX, Yi for X, Y ∈ XM, e.g., [2]. If P is an almost complex structure on (M, g), then PP = id TM and condition (15) obviously holds. This observation can be developed as follows. (b) An f -structure (due to Yano, [22]) on a manifold M is a non-null tensor f ∈ End(TM) of constant rank such that f 3 + f = 0. Such f generalizes the almost complex and the almost contact structures. The restriction of f to D = f (TM) determines a complex structure on it. An interesting case of f -structure on M2n+p occurs when ker f is parallelizable for which there exist global vector ﬁelds ξi, i ∈ {1, ... , p}, with their dual 1-forms ηi, satisfying (see [23]) the following relations:

2 = − + i ⊗ i( ) = i f id TM ∑i η ξi, η ξj δj. (21)

It is known that f ξi = 0, ηi ◦ f = 0 and f has rank 2n. A Riemannian metric g = h·, ·i is compatible, if ∗ i ∗ f f = id TM − ∑i η ⊗ ξi. We have f = − f , and for P = f , we get

( PP∗)(X) = − h∇ + ( ) Xi div ∑j ξj ξj div ξj ξj, .

Thus, (15) holds if and only if the distributions f (TM) and ker f are both harmonic. The metric f -structure for p = 1 reduces to an almost contact metric structure, e.g., [2]. The condition (15) holds for an almost contact metric manifold (M, φ, ξ, η) and P = φ if and only if ξ is a geodesic vector ﬁeld (∇ξ ξ = 0) and the distribution φ(TM) is harmonic (div ξ = 0).

Deﬁnition 5. Deﬁne a skew-symmetric P-bracket [·, ·]P : XM × XM → XM by

P P [X, Y]P = ∇XY − ∇Y X. (22)

Proposition 3. Condition DP = 0, see deﬁnition (6) with ρ = P, is equivalent to the following symmetry (on covariant components) of the (1, 2)-tensor ∇PP:

P P (∇ P)(X, Y) = (∇ P)(Y, X), X, Y ∈ XM. (23)

Proof. Generally, we have

P (∇X P)(Y) = (∇PX P)(Y) = ∇PX PY − P∇PX Y,

[PX, PY] − P[X, Y]P = (∇PX P)(Y) − (∇PY P)(X).

Thus, P P P D (X, Y) = (∇X P)(Y) − (∇Y P)(X), X, Y ∈ XM, and the claim follows.

If condition (23) holds, then, by deﬁnitions (22) and (7), the P-derivative is torsion free; thus, [·, ·]P deﬁnes a skew-symmetric algebroid structure with E = TM and ρ = P.

Example 3. One can use structures of Example2 to clarify the property (23). (a) For an almost complex structure P = J on TM, see Example2(a), the property (23) describes a class of 2 almost Hermitian manifolds which includes Kählerian manifolds, i.e., ∇J = 0. Differentiating J = − id TM, we obtain (∇X J)J = −J(∇X J), X ∈ XM. (24) Mathematics 2020, 8, 365 8 of 18

Using (24), we can show that our class contains a wider class of nearly Kählerian manifolds, which are deﬁned by (∇X J)X = 0, see [24]. Indeed, if (M, J, g) is nearly Kählerian, then

0 = (∇J(X+Y) J)J(X + Y) = (∇JX J)JX + (∇JX J)JY + (∇JY J)JX + (∇JY J)JY = (∇JX J)JY + (∇JY J)JX = −J (∇JX J)Y + (∇JY J)X from which (23) follows. There are many nearly Kählerian manifolds that are not Kählerian. (b) The integrability tensor of a singular distribution D = P(TM) is deﬁned by

1 T (X, Y) = P˜(∇ PY − ∇ PX), 2 PX PY where P˜ : TM → D⊥ is the orthoprojector onto complementary orthogonal to D distribution, see [10]. Since ˜ 1 ˜ P P ◦ P = 0, we get T (X, Y) = 2 P ◦ D (X, Y). Hence, (23) yields (but is not equivalent to) T = 0, in particular, for P of constant rank, (23) yields integrability of P(TM). Consequently, (23) is not satisﬁed for non-integrable regular distributions in Example2(b). On the other hand, there exist integrable regular distributions (foliations) not satisfying condition (23). Indeed, taking a unit vector Y ∈ D⊥, since PY = 0, we obtain

P D (X, Y) = −P(∇PX Y) = AY(PX),

> where AY is the shape operator of D: AY(X) = −(∇X Y) . For example, if AY(X) = λX (i.e., D is totally umbilical) and P|D is not identically zero, then AY(PX) 6= 0 for some X. (c) For a metric f -structure on M, see Example2(b), differentiating (21)1, we get

i (∇X f ) f = − f (∇X f ) + ∇X(η ⊗ ξi), i i i ∇X(η ⊗ ξi)(Y) = (∇X η )(Y) ξi + η (Y) ∇X ξi.

Consider a nearly Kähler f -structure on M, i.e., (∇ f X f ) f X = 0, see, e.g., [25]. Then,

0 = (∇ f (X+Y) f ) f (X + Y) = (∇ f X f ) f X + (∇ f X f ) f Y + (∇ f Y f ) f X + (∇ f Y f ) f Y = (∇ f X f ) f Y + (∇ f Y f ) f X = − f (∇ f X f )Y + (∇ f Y f )X i i i i −(∇ f X η )(Y) ξi − (∇ f Y η )(X) ξi − η (X)∇ f Y ξi − η (Y)∇ f X ξi .

∗ Taking D-component of this, using f = − f and assuming ∇ ξi = 0 (in this case, f (TM) is tangent to a totally geodesic foliation), we obtain (23). Thus, our class (23) contains nearly Kähler f -manifolds with parallel distribution ker f . Remark that the class of nearly Kähler f -manifolds contains the class of Killing f -manifolds, i.e., (∇X f )X = 0, which is often deﬁned by the condition that the fundamental form F(X, Y) = hX, f Yi is a Killing form. For a metric f -structure with p = 1, we have two appropriate cases. The class of almost contact metric manifolds (M, g, φ, ξ, η) with the condition (23) for P = φ includes:

• approximately cosymplectic manifolds (deﬁned by (∇X φ)X = 0 for X ∈ XM, see [26]; such non-cosymplectic structure exists, e.g., on S7) with parallel Reeb vector ﬁeld, ∇ξ = 0.

• nearly Sasakian manifolds (deﬁned by (∇X φ)X = hX, Xiξ − η(X)X, see [2,27]) with parallel ξ. Unfortunately, any nearly Sasakian manifold of dimension greater than 5 is Sasakian.

4. The Modified Hodge and Beltrami Laplacians First, we define the modified exterior derivative and its L2-adjoint. The P-exterior derivative of ω ∈ Λk(TM) as a (k + 1)-form

dPω(X ,..., X ) = (−1)i(∇P ω)(X ,..., X ,... X ). 0 k ∑i Xi 0 ci k Mathematics 2020, 8, 365 9 of 18

P P For a differential form ωp, the form ∇ ω given in (13) is not skew-symmetric, but the form d ω is skew-symmetric. For a function f on M, we have dP f = ∇P f , see Deﬁnition4. Put δP = ∇∗P : Λk+1(TM) → Λk(TM) for differential forms. By (14), we have

(δPω)(X ,..., X ) = − (∇P ω)(e , X ,..., X ). 2 k ∑i ei i 2 k

Similarly to Proposition1, we have the following.

Proposition 4. If condition (15) holds on a closed (M, g) then δP is L2-adjoint to dP, i.e.,

P P (δ ωk+1, ωk) = (ωk+1, d ωk),

k k+1 for any two differential forms ωk ∈ Λ (TM) and ωk+1 ∈ Λ (TM).

P P Proof. We use (∇ ωk, ωk+1) = (ωk, δ ωk+1), which requires (15), and

k P i P i0 j0 ik jk hd ωk, ωk+ i = (−1) ∇ ωk(∂i ,..., b∂i ,..., ∂i )g ... g ωk+ (∂i ,..., ∂i ) 1 ∑ u=0 ∂iu 1 u k 1 1 k = ∇P ( ) i0 j0 ik jk ( ) = h∇P i ∂ ωk ∂i1 ,..., ∂ik g ... g ωk+1 ∂j0 ,..., ∂jk ωk, ωk+1 , i0

P P as in the classical case. Thus, (d ωk, ωk+1) = (∇ ωk, ωk+1). Using condition (15), by (18), we get P P∗ P (∇ ωk, ωk+1) = (ωk, ∇ ωk+1) = (ωk+1, d ωk).

Proposition 5. The dP : Λk(TM) → Λk+1(TM)(1 ≤ k ≤ n) is an exterior derivation, see the deﬁnition in Section2, that is,

P ( ) = k (− )i ( ( )) d ω X0,..., Xk ∑i=0 1 PXi ω X0,..., cXi,..., Xk + (− )i+j [ ] ∑0≤i

Proof. We have

k dP (X X ) = (− )i(∇ )(X X X ) ω 0,..., k ∑i=0 1 PXi ω 0,..., ci,..., k = k (− )i ( ( )) ∑i=0 1 PXi ω X0,..., cXi,..., Xk k i−1 + (− )i (X ∇ X X X ) ∑i=0 1 ∑j=0 ω 0,..., PXi j,..., ci,..., k k + (X X ∇ X X ) ∑j=i+1 ω 0,..., ci,..., PXi j,..., k = k (− )i ( ( )) ∑i=0 1 PXi ω X0,..., cXi,..., Xk + (− )i+j ∇ X − ∇ X X X X X ∑0≤i

The use of (12) and (22) completes the proof.

k k Next, we extend the deﬁnition of the Hodge Laplacian ∆H = dδ + δd : Λ (TM) → Λ (TM).

P Deﬁnition 6. Deﬁne the Hodge type Laplacian ∆H for differential forms by

P P P P P ∆H = d δ + δ d . (25)

P We say that a differential form ω is P-harmonic if ∆H ω = 0. Mathematics 2020, 8, 365 10 of 18

Remark 1. The P-harmonic forms have similar properties as in the classical case, see, e.g., ([15], Lemma 9.1.1). Let condition (15) hold on a closed (M, g). For ω ∈ Λk(TM), using Proposition4 and (25), we have

P P P P P (∆H ω, ω) = (d ω, d ω) + (δ ω, δ ω), thus, a differential form ω is P-harmonic if and only if

dPω = 0, δPω = 0.

P P P P P Observe that, if ∆H ω = 0 and ω = d θ, then δ d θ = δ ω = 0. It follows that (ω, ω) = (dPθ, dPθ) = (θ, δPdPθ) = (θ, δPω) = 0.

Thus, if ω is P-harmonic and ω = dP θ, then ω = 0.

P P P For functions, ∆B f = div (∇ f ) is the Beltrami type Laplacian. For any x ∈ M, we have

P ( ) = ( )2( )( ) ∆B f x ∑ i Pei f x , (26)

∞ where (ei) is an orthonormal frame such that ∇Y ei = 0 for all Y ∈ Tx M. A function f ∈ C (M) is P P called P-subharmonic if ∆B f ≤ 0. If ∆B f ≥ 0, then f is called P-superharmonic.

Lemma 2 (see Proposition 2.8 in [10]). Let (M, g) be a complete open Riemannian manifold endowed with a vector ﬁeld X such that divP X ≥ 0 (or divP X ≤ 0), where P ∈ End(TM) such that (15) and ∗ 1 kPP (X)kg ∈ L (M, g) hold. Then, divP X ≡ 0.

Consider the following system of singular distributions: D1 = D, D2 = D1 + [D, D1], etc. The distribution D is said to be bracket-generating of the step r ∈ N if Dr = TM, see, e.g., [4]. Note that integrable distributions, i.e., [X, Y] ∈ XD (X, Y ∈ XD), are not bracket-generating. The condition ∇P f = 0 means that f is constant along the (integral curves of) P(TM); moreover, f = const on M when D = P(TM) is bracket-generating. The next theorem for P = id TM reduces to the classical result on subharmonic functions.

2 P P Theorem 1. Let condition (15) hold and let f ∈ C (M) satisfy ∆B f ≥ 0 or ∆B f ≤ 0. Suppose that any of the following conditions hold: a) (M, g) is closed; b) (M, g) is open complete, kPP∗∇P f k ∈ L1(M, g) and k f PP∗∇P f k ∈ L1(M, g). Then, ∇P f = 0; moreover, if P(TM) is bracket-generating, then f = const.

P P Proof. Set X = ∇ f , then ∆B f = divP X. a) Using the extended Divergence Theorem, see (17), we P get ∆B f ≡ 0. Using the equality

P divP( f · Y) = f · divP Y + h∇ f , Yi (27) with Y = ∇P f and again the extended Divergence Theorem with X = f ∇P f , we get (∇P f , ∇P f ) = 0, P P ∗ P 1 P hence ∇ f = 0. b) By Lemma2 with X = ∇ f and condition kPP ∇ f k ∈ L (M, g), we get ∆B f ≡ 0. Using the Formula (27), with Y = ∇P f , Lemma2 with X = f ∇P f and condition k f PP∗∇P f k ∈ L1(M, g), we get (∇P f , ∇P f ) = 0, hence ∇P f = 0. If the distribution P(TM) is bracket-generating, then using Chow’s theorem [28] completes the proof for both cases.

5. The Modiﬁed Curvature Tensor We will use ∇P to construct the second P-derivative on tensors and then the P-curvature. Mathematics 2020, 8, 365 11 of 18

Deﬁnition 7. Deﬁne the second P-derivative of an (s, k)-tensor S by

P 2 P P P (∇ )X,Y S = ∇X(∇Y S) − ∇ P S . ∇XY

Deﬁne the P-curvature of connection ∇P by

ρ RP Z = (∇P)2 Z − (∇P)2 Z = ∇P ∇ Z − ∇P∇P Z − ∇P Z. X,Y X,Y Y,X X Y Y X [X,Y]P

See (8) with ρ = P, and set

P P R (X, Y, Z, W) = hRX,Y Z, Wi, X, Y, Z, W ∈ XM. (28)

P P The P-Ricci tensor is deﬁned by the standard way: Ric (X, Y) = ∑ i R (X, ei, ei, Y).

P Set JP(X, Y, Z) = ∑ circ.[X, [Y, Z]P ]P, see (5). By (9) and since ∇ is torsionless, we get

P = J ( ) ∑ circ. R X,Y Z P X, Y, Z .

The RP acts on tensor ﬁelds as follows:

P P ∞ • RX,Y( f ) = D (X, Y)( f ) on functions f ∈ C (M), P • RX,Y Z given by Formula (28) on vector ﬁelds Z ∈ XM, P P P • (RX,Y ω)(Z) = D (X, Y)(ω(Z)) − ω(RX,Y Z) on differential 1-forms, • and similarly on (0, k)-tensors:

( P )( ) = DP( )( ( )) − ( P ) RX,Y S X1,..., Xk X, Y S X1,..., Xk ∑ i S X1,... RX,Y Xi,..., Xk .

P ∞ P By the above calculations, the action of RX,Y on tensors is C (M)-linear if D = 0, i.e., (23) holds; ∞ P in this case, we conclude the same about the C (M)-homogeneity of RX,Y ω in ω. In other words, P the action of the endomorphism RX,Y on differential forms is deﬁned as in the classical case (see [15], Section 2.3.1): as a one degree derivation acting trivially on functions, as an endomorphism on vector ﬁelds and as an adjoint endomorphism on forms. Based on the above arguments, we assume (15) and (23) in the rest of the article. We will show that the P-curvature has some symmetry properties, similar to the classical variants, when P = id TM and according to ([15], Section 3.1.1).

1 ∞ Proposition 6. For X, Y, Z, W ∈ XM, ω ∈ Λ (TM) and f ∈ C (M), we have

P P 1.R X,Y f = 0;RX,Y g = 0; P P P 2.R X,Y Z = RPX,PY Z; (RX,Y ω)(Z) = −ω(RX,Y Z); P 3.R X,Y T = RPX,PY T for every tensor T; P P 4.PR X,Y Z = RX,Y PZ; 5.R P(X, Y, Z, (P∗P)W) = RP(X, Y, PZ, PW) = R(PX, PY, PZ, PW);

6.R P(X, Y, Z, W) = −RP(Y, X, Z, W) = −RP(X, Y, W, Z);

P P 7. hRX,Y Z, Wi = −hRX,YW, Zi. Mathematics 2020, 8, 365 12 of 18

The endomorphism P of TM induces endomorphisms of Λ2(TM): P and its adjoint P ∗,

P(X ∧ Y) = PX ∧ PY, P ∗(X ∧ Y) = P∗X ∧ P∗Y.

Indeed,

h i h i PX, Z PX, W hP(X ∧ Y), Z ∧ Wi = hPY, Zi hPY, Wi

h ∗ i h ∗ i X, P Z X, P W ∗ = ∗ ∗ = hX ∧ Y, P (Z ∧ W)i. hY, P Zi hY, P Wi

2 The curvature tensor RX,Y can be seen as a self-adjoint linear operator R on the space Λ (TM) of bivectors, called the curvature operator (note the reversal of Z and W):

hR(X ∧ Y), Z ∧ Wi = R(X, Y, W, Z), R∗ = R.

P P Similarly, consider RX,Y = RPX,PY as a linear operator R = R ◦ P or as a corresponding bilinear form on Λ2(TM), that is,

RP(X ∧ Y) = R ◦ P(X ∧ Y) = R(PX ∧ PY), RP(X ∧ Y, Z ∧ W) = hR P(X ∧ Y), Z ∧ Wi.

Using known properties of R and RP, we have

h RP(X ∧ Y), Z ∧ Wi = h R(PX ∧ PY), Z ∧ Wi = RP(X, Y, W, Z). (29)

Remark 2. The RP on Λ2(TM) generally is not self-adjoint:

(RP)∗ = (R ◦ P)∗ = P ∗ ◦ R 6= R ◦ P = RP.

6. The Weitzenböck Type Curvature Operator The next deﬁnition generalizes the Weitzenböck curvature operator (3).

Deﬁnition 8. Deﬁne the P-Weitzenböck curvature operator on (0, k)-tensors S over (M, g) by

k n RicP(S)(X ,..., X ) = (RP S)(X ,..., e ,..., X ) 1 k ∑ = ∑ = ei,Xα 1 i k α 1 i 1 | {z } α P = −2 ∑ < R (ei, Xa, em, Xb) · S(X1,..., em,..., ei,..., Xk) i,m,a;b a | {z } | {z } b a−b P + ∑ Ric (ei, Xa) · S(X1,..., ei,..., Xk), (30) i,a | {z } a

j p j or, in coordinates, RicP(S) = −2 ∑ RP S + ∑ RicP S . i1,...,ik a

For a differential form ω, the RicP(ω) is skew-symmetric. For vector ﬁelds and 1-forms, the RicP is similar to the classical Ricci tensor. The following result extends a Weitzenböck theorem (see [15], Theorem 9.4.1).

Theorem 2. The following Weitzenböck type decomposition formula holds:

P ¯ P P ∆H = ∆ + Ric . (31) Mathematics 2020, 8, 365 13 of 18

Proof. We follow the same lines as in the proof of ([15], Theorem 9.4.1): for ω ∈ Λk(TM),

P P α+1 P,2 d δ ω(X1,..., Xk) = − ∑ (−1) (∇X ,e ω)(ei, X1,..., Xcα,..., Xk) i,α α i | {z } i P,2 = − ∑ (∇X ,e ω)(X1,..., ei,..., Xk), i,α α i | {z } α δPdPω(X ,..., X ) = − (∇P,2 ω)(X ,..., X ) 1 k ∑ i ei,ei 1 k α P,2 − ∑ (−1) (∇e ,X ω)(ei, X1 ..., Xcα,..., Xk) i,α i α | {z } i ∗P P P,2 = (∇ ∇ ω)(X1,..., Xk) + ∑ (∇e ,X ω)(X1 ..., ei,..., Xk), i,α i α | {z } α where 1 ≤ i ≤ n and 1 ≤ α ≤ k. Substituting the above into (25) yields (31).

From ∆H S = ∆¯ S + Ric(S), see (2), we obtain the Bochner–Weitzenböck formula

1 ∆ (kSk2) = −h∆ S, Si + k∇ Sk2 + hRic(S), Si, (32) 2 B H

∞ where ∆B = div ◦ grad is the Beltrami Laplacian on C (M), e.g., [15–18]. In the next proposition, we generalize (32) for distributions.

Proposition 7. The modiﬁed Bochner–Weitzenböck formula is valid:

1 ∆P(kSk2) = −h∆P S, Si + k∇PSk2 + hRicP(S), Si. (33) 2 B H

Proof. First, we ﬁnd 1 ∆P(kSk2) = −h∆¯ PS, Si + k ∇PSk2. 2 B

Indeed, take an orthonormal frame (ei) such that ∇Yei = 0 for all Y ∈ Tx M and, in order to simplify calculations, assume that s = t = 1. By (26), and taking into account the properties of normal coordinates, we have at x ∈ M,

1 1 ∆P(kSk2) = (Pe )2hS, Si = (Pe )h∇ S, Si 2 B 2 ∑ i i ∑ i i Pei = h∇ ∇ i + h∇ ∇ i ∑ i Pei Pei S, S Pei S, Pei S = −h∆¯ P S, Si + k∇P Sk2.

Then, using (31), we get (33).

7. Applications of the Weitzenböck Type Curvature Operator

Here, rewrite the P-Weitzenböck curvature operator using an orthonormal basis (ξa) of skew-symmetric transformations so(TM) and give some applications of RicP. 2 P For every bivector X ∧ Y ∈ Λ (TM), we build a map R (X ∧ Y) : XM → XM, given by

hRP(X ∧ Y)Z, Wi = hRP(X ∧ Y), W ∧ Zi = R(PX, PY, Z, W).

Since bivectors are generators of the vector space Λ2(TM), we obtain in this way a map RP(ξ) : XM → XM (similarly to algebraic curvature operator R(ξ)). Mathematics 2020, 8, 365 14 of 18

Lemma 3. The map RP(ξ), where ξ ∈ Λ2(TM), is skew-symmetric: hRP(ξ)W, Zi = −hRP(ξ)Z, Wi.

Proof. It sufﬁces to check the statement for the generators. We have, using Proposition6,

hRP(X ∧ Y)Z, Wi = R(PX, PY, Z, W) = −R(PX, PY, W, Z) = −hRP(X ∧ Y)W, Zi.

Thus, the statement follows.

The associated P-curvature operator is given by hRP(X ∧ Y), Z ∧ Wi = R(PX, PY, W, Z). We are based on the fact that, if X and Y are orthonormal, then X ∧ Y is a unit bivector, while the corresponding skew-symmetric√ operator (a counterclockwise rotation of π/2 in the plane span(X, Y)) has Euclidean norm 2. To simplify calculations, we assume that so(TM) is endowed with metric induced from Λ2(TM), see, e.g., [19]. If L ∈ so(TM), then

( )( ) = − ( ( ) ) LS X1,..., Xk ∑ i S X1,..., L Xi ,..., Xk . (34)

Let {ξa} be an orthonormal base of skew-symmetric transformations such that (ξa)x ∈ so(Tx M) for x in an open set U ⊂ M. By (34), for any (0, k)-tensor S,

( )( ) = − ( ( ) ) ξαS X1,..., Xk ∑ i S X1,..., ξα Xi ,..., Xk ;

P 2 The R (X ∧ Y) on Λ (TM) can be decomposed using {ξa}.

Lemma 4. We have

RP( ∧ ) = − hP ∗ ◦ R( ) i = − hR( ) i X Y ∑ α ξα X, Y ξα ∑ α ξα PX, PY ξα.

Proof. Using that (RP)∗ = P ∗ ◦ R and Lemma3, we have:

RP( ∧ ) = hRP( ∧ ) i X Y ∑ α X Y , ξα ξα = hP ∗ ◦ R( ) ∧ i = − hR( ) i ∑ α ξα , X Y ξα ∑ α ξα PX, PY ξα.

Lemma4 allows us to rewrite the Weitzenböck type curvature operator (30).

Proposition 8. If S is a (0, k)-tensor on (M, g), then

∗ R P( ) = − RP( )( ) (R P( ))∗ = R P ( ) ic S ∑ α ξa ξaS , ic S ic S .

In particular, if P is self-adjoint, then RicP is self-adjoint too.

Proof. We follow similar arguments as in the proof of ([15], Lemma 9.3.3):

R P( )( ) = (RP( ∧ ) )( ) ic S X1,..., Xk ∑ i,j ej Xi S X1,..., ej,..., Xk | {z } i = − hP ∗ ◦ R( ) i( )( ) ∑ i,j,α ξα ej, Xi ξαS X1,..., ej,..., Xk = − ( )( hP ∗ ◦ R( ) i ) ∑ i,j,α ξαS X1,..., ξα ej, Xi ej,..., Xk = − ( )( h RP( ) i ) ∑ i,j,α ξαS X1,..., ej, ξα Xi ej,..., Xk = − ( )( RP( ) ) ∑ i,α ξαS X1,..., ξα Xi,..., Xk = − (RP( )( ))( ) ∑ α ξα ξαS X1,..., Xk . Mathematics 2020, 8, 365 15 of 18

Thus, the ﬁrst claim follows. Since R : Λ2(TM) → Λ2(TM) is self-adjoint, there is a local orthonormal 2 base {ξa} of Λ (TM) such that R(ξa) = λa ξa. Using this base, for any (0, k)-tensors S1 and S2, we have

hR P( ) i = − hRP( )( ) i = − h (RP)∗( ) i ic S2 , S1 ∑ α ξα ξαS2 , S1 ∑ α ξαS2, ξα S1 = h P ∗ ◦ R( )( )i = h P( ) i ∑ α ξαS2, ξα S1 ∑ α λα ξαS2 , ξαS1 , (35) and, similarly,

∗ h R P ( )i = h P ∗( )i = h P( ) i S2, ic S1 ∑ α λα ξαS2, ξαS1 ∑ α λα ξαS2 , ξαS1 .

Thus, the second claim follows.

For a (0, k)-tensor S on M, deﬁne a (0, k)-tensor Sb with values in Λ2(TM) implicitly by

h L, Sb(X1,..., Xk) i = (LS)(X1,..., Xk), L ∈ so(TM), see [19]. The following result generalizes ([19], Lemma 3.5).

Proposition 9. If S1, S2 are (0, k)-tensors, then

P P h Ric (S1), S2 i = h Sb1, R (Sb2) i.

2 Proof. Using a local orthonormal base {ξα} of Λ (TM), we get

h R P( ) i = h RP( )( ) i = h RP∗( ) i ic S1 , S2 ∑ α ξα ξαS1 , S2 ∑ α ξαS1, ξα S2 = (ξ S )(e e )(RP∗(ξ )S )(e e ) ∑ α,i α 1 i1 ,..., ik α 2 i1 ,..., ik = h ξ S (e e ) ih RP∗(ξ ) S (e e ) i ∑ α,i α, b1 i1 ,..., ik α , b2 i1 ,..., ik = h RP∗(h ξ S (e e ) i ξ ) S (e e ) i ∑ α,i α, b1 i1 ,..., ik α , b2 i1 ,..., ik = h RP∗(S (e e )) S (e e ) i ∑ i b1 i1 ,..., ik , b2 i1 ,..., ik = h S (e e ) RP(S (e e )) i ∑ i b1 i1 ,..., ik , b2 i1 ,..., ik .

Thus, the claim follows.

P P P Example 4. For vector ﬁelds and 1-forms, Ric reduces to the kind of usual Ricci curvature, Ric = Tr 1,3 R . [ Let X ∈ XM and ω = X be the dual 1-form. Recall ([19], Proposition 3.6) that ωb(Z) = Z ∧ X. Thus, using Proposition8 and (29), we obtain

hR P( ) i = − h RP( )( ) i = h RP( ) i ic ω , ω ∑ α ξa ξaω , ω ∑ α ξaω, ξa ω = h ( ) RP( ( ))i = h ( ) RP( ( ))i ∑ i ωb ei , ωb ei ∑ i ωb ei , ωb ei = h RP( ∧ ) ∧ i = h P( i = P( ) ∑ i ei X , ei X ∑ i R ei, X, X, ei Ric X, X .

Now, we will extend ([15], Corollary 9.3.4).

Proposition 10. If hRP(S), Si ≥ 0 for any (0, k)-tensor S, then hRicP(S), Si ≥ 0. Moreover, if hRP(S), Si ≥ k kSk2 for any (0, k)-tensor S, where k < 0, then hRic P(S), Si ≥ k C kSk2, where a constant C depends only on the type of S.

2 Proof. Using (35) and a local orthonormal base {ξα} of Λ (TM) such that R(ξα) = λαξα, we get

hR P( ) i = h P( ) i = h P( ) R( )i = h RP( ) i ic S , S ∑ α λα ξαS , ξαS ∑ α ξαS , ξαS ∑ α ξαS , ξαS . Mathematics 2020, 8, 365 16 of 18

P P If h R (ξαS), ξαSi ≥ 0 for all α, then hRic (S), Si ≥ 0, and the ﬁrst claim follows. There is a constant C > 0 depending only on the type of the tensor and dim M such that

k k2 ≥ k k2 C S ∑ α ξαS ,

P 2 see ([15], Corollary 9.3.4). By conditions, h R (ξαS), ξαS i ≥ kkξαSk for all α. The above, for k < 0, P 2 yields h R (ξαS), ξαSi ≥ k C kSk —thus the second claim.

Theorem 3. Let hRP(ω), ωi ≥ 0 for any k-form ω on an open complete (M, g). If kPP∗∇P(kωk2)k ∈ L1(M, g) hold for a P-harmonic form ω, then ω is P-parallel.

Proof. By conditions and Proposition 10, hRicP(ω), ωi ≥ 0. By the modiﬁed Bochner–Weitzenböck P ¯ P 2 P 2 formula (33) with S = ω, since ∆H ω = 0, we obtain ∆ (kωk ) ≥ 0. By Lemma2 with X = ∇ (kωk ), we get ∆¯ P(kωk2) = 0. Hence, by (33) again, ∇Pω = 0.

P Corollary 1 (For P = id TM, see Theorem 3.3 in [19]). If hR (ω), ωi ≥ 0 for any k-form ω on a closed (M, g), then any P-harmonic differential form on M is P-parallel.

Proof. We give a short proof. By conditions and Proposition 10, hRicP(ω), ωi ≥ 0. By (31), since P ¯ P P ∆H ω = 0, we obtain h∆ ω, ωi ≤ 0. By Proposition2 with S = ω, we get ∇ ω = 0. Notice that, if the distribution P(TM) in Theorem3 and Corollary1 is bracket-generating, then kωk = const.

Remark 3. One may deﬁne also a linear operator R¯ P = P ∗ ◦ R ◦ P on Λ2(TM) that is

R¯ P(X ∧ Y) = P ∗ ◦ RP(X ∧ Y) = P ∗ ◦ R ◦ P(X ∧ Y).

Then, using known properties of R and Proposition6, we have

hR¯ P(X ∧ Y), Z ∧ Wi = hR(PX ∧ PY), P(Z ∧ W)i = R(PX, PY, PW, PZ).

The R¯ P on Λ2(TM) is self-adjoint (we can also consider it as a bilinear form on Λ2(TM)):

hR¯ P(X ∧ Y), Z ∧ Wi = R(PX, PY, PW, PZ) = R(PZ, PW, PY, PX) = hR(PZ ∧ PW), PX ∧ PYi = hR¯ P(Z ∧ W), X ∧ Yi.

P 2 One may deﬁne a map R¯ (ξ) : XM → XM for ξ ∈ Λ (TM),

hR¯ P(X ∧ Y)Z, Wi = hR¯ P(X ∧ Y), W ∧ Zi = R(PX, PY, PZ, PW).

Unlike R P, the map R¯ P is skew-symmetric: hR¯ P(ξ)W, Zi = −hR¯ P(ξ)Z, Wi. P ¯ P ¯ P It allows similar to R (in Lemma4) decomposition R (X ∧ Y) = − ∑ αhR (ξα)X, Yiξα . Next, we deﬁne another Weitzenböck type curvature operator on a (0, k)-tensor S,

P ¯ P Ric (S)(X1,..., Xk) = ∑ (R (ei ∧ Xα) S)(X1,..., ei,..., Xk), i,α | {z } α

P P ¯ P which unlike Ric is self-adjoint, and rewrite it as Ric (S) = − ∑ α R (ξa)(ξaS), see Proposition8. Thus, if S1 and S2 are (0, k)-tensors on (M, g), then (compare with Corollary9)

P P h Ric (S1), S2 i = h R¯ (Sb1), Sb2 i. Mathematics 2020, 8, 365 17 of 18

Similarly to Proposition 10, we get the following: If R¯ P ≥ 0, then Ric P ≥ 0; moreover, if R¯ P ≥ k, where k < 0, then h Ric P(S), Si ≥ k CkSk2, where a constant C depends only on the type of the tensor S. We conclude by the following analog of Corollary1: If R¯ P ≥ 0 on a closed (M, g), then all P-harmonic forms on M are P-parallel. We omit results similar to Theorem3.

Author Contributions: Investigation, P.P., V.R. and S.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A. Proof of Proposition6

P P By (23), we have for further use P∇PXY = P∇XY = ∇X PY = ∇PX PY. 1. We calculate

∇P ∇P f − ∇P∇P f − ∇P f = PX(PY( f )) − PY(PX( f )) − (P[X, Y] ) f = DP(X, Y) f = 0. X Y Y X [X,Y]P P

Then,

hRP Z, Wi = h∇P ∇P Z − ∇P∇P Z − ∇P Z, Wi X,Y X Y Y X [X,Y]P = h∇ ∇ − ∇ ∇ − ∇ i PX PY Z PY PX Z P[X,Y]P Z, W

= h∇PX∇PY Z − ∇PY∇PX Z − ∇[PX,PY]Z, Wi = hRPX,PY Z, Wi.

P Similarly, hRX,YW, Zi = hRPX,PYW, Zi. Thus,

P P P P (RX,Y g)(Z, W) = RX,YhZ, Wi − hRX,Y Z, Wi − hZ, RX,YWi

= −hRPX,PY Z, Wi − hZ, RPX,PYWi = (RPX,PY g)(Z, W) = 0.

2. Since P[X, Y]P = [PX, PY], we have

P = ∇ ∇ − ∇ ∇ − ∇ = RX,Y Z PX PY Z PY PX Z P[X,Y]P Z R PX,PY Z, P P P (RX,Y ω)Z = RX,Y(ω(Z)) − ω(RX,Y Z) = −ω(RPX,PY Z).

P 3. This follows from 1 and 2, since the actions of RX,Y and RPX,PY on generators (functions, vector and covector ﬁelds) are the same. 4. We have

PRP Z = P(∇P ∇P Z − ∇P∇P Z − ∇P Z) = ∇P P∇P Z − ∇P P∇P Z − ∇P PZ X,Y X Y Y X [X,Y]P X Y Y X [X,Y]P = ∇P ∇P PZ − ∇P∇P PZ − ∇P PZ = RP PZ. X Y Y X [X,Y]P X,Y

5. We have, using 4,

P∗ P P ∗ P R (X, Y, Z, W) = hRX,Y Z, (P P)Wi = hPRX,Y Z, PWi P P = hRX,Y PZ, PWi = R (X, Y, PZ, PW) = R(PX, PY, PZ, PW).

P P 6. Since RX,Y Z = −RY,X Z, the ﬁrst equality follows. For the second one, we use 1:

P P 0 = RX,Y g(Z, Z) = 2hRX,Y Z, Zi;

P thus, we obtain 6 by polarizing the identity hRX,Y(Z + W), Z + Wi = 0. P P P 7. By ∇g = 0, we get RX,Y g = 0, then hRX,Y Z, Wi + hZ, RX,YWi = 0. Mathematics 2020, 8, 365 18 of 18

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