CR-structures of codimension 2 on tangent bundles in Riemann–Finsler geometry

Abstract

We determine a 2-codimensional CR-structure on the slit tangent bundle $T_0M$ of a Finsler manifold (M, F) by imposing a condition on the almost complex structure $\Psi$ associated to F when restricted to the structural distribution of a framed f-structure. This condition is satisfied when (M, F) is of scalar flag curvature (particularly flat). In the Riemannian case (M, g) this last condition means that g is of constant curvature. This CR-structure is finally generalized by using one positive parameter but under more difficult conditions.

1 Introduction

Finsler geometry is very rich in remarkable tensor fields $\varphi$ of (1, 1)-type and associated structures. More precisely, there are: an (almost) tangent structure ($\varphi ^2=0$), an almost complex one ($\varphi ^2=-I$) and also an almost product structure $(\varphi ^2=I)$. In [1] another well-known type of structures, namely an f-structure ($\varphi ^3+\varphi =0$) is obtained in this geometry. In fact, this f-structure belongs to a very interesting particular case which is called framed f-structure and has, in addition to $\varphi$, a set of vector fields and differential 1-forms interrelated. Moreover, a conformal deformation of the Sasaki type metric can be added in order to obtain a metric framed f-structure. This metric framed f-structure of M. Anastasiei was recently generalized in [8, 15].

The present note concerns yet another kind of structures, namely the CR-structures, with an important rôle at the border between differential geometry and complex analysis, as it is pointed out in [7]. We restrict ourselves to the real case; more precisely, based on a relationship between framed f-structures and CR-structure established in [2, p. 130] we found a CR-structure on the slit tangent bundle $T_0M$ of a Finsler manifold (M, F). This CR-structure is constructed with the above almost complex structure denoted by $\Psi _F$ in Sect. 3 and its existence is constrained by one condition expressing the vanishing of the Nijenhuis tensor of $\Psi _F$ on the structural distribution of the framed f-structure from [1]. The above condition is expressed as a relation between the curvature of the Cartan nonlinear connection and the Jacobi endomorphism and is satisfied in dimension two or if (M, F) is of scalar flag curvature which in the particular case of Riemannian geometry (M, g) means that the metric g has a constant curvature. Several important classes of Finsler manifolds with scalar flag curvature are discussed in Chapter 7 of [5].

Inspired by [15] we generalize this CR-structure using a real parameter $\beta >\frac{1}{2}$ but with more difficult conditions. More precisely, we take into account the same vector fields and 1-forms as in the previous framed f-structure but deform the metric and the almost complex structure on both horizontal and vertical directions. At $\beta =1$ we recover the previous CR-structure.

Finally, let us note that our CR-structures are of codimension 2 and the (complex) geometry of these structures was studied in [11, 12] and recently in [9, 10]. But for the Riemannian case the only studies until now are on hypersurfaces of Sasakian manifolds [13, 14] and not on (slit) tangent bundle. The para-CR version of this study is the paper [6].

2 CR-structures from framed f-structures

Framed f-structures constitute a particular case of f-structures. A detailed study of this class of tensor fields of (1, 1)-type, especially from a local point of view, can be found in [16].

Let N be a smooth $(2n+s)$-dimensional manifold with $n, s\ge 1$ and fix a distribution D of dimension 2n on N. Considering D as a vector bundle over N let $\Gamma (D)$ be the module of its sections. Supposing D is endowed with a morphism $J:D\rightarrow D$ of vector bundles satisfying $J^2=-I$ where I is the identity (Kronecker) morphism on D, the pair (D, J) is called almost complex distribution.

The first main notion is given in [2, p.128].

Definition 2.1

If for all $X, Y\in \Gamma (D)$ we have

$$\begin{aligned} \left\{ \begin{array}{ll} [JX, JY]-[X, Y]\in \Gamma (D) \\ N_J(X, Y):=[JX, JY]-[X, Y]-J([X, JY]+[JX, Y])=0, \end{array} \right. \end{aligned}$$

(2.1)

then (D, J) is a CR-structure on N and the triple (N, D, J) is a CR-manifold.

A second main notion is that of a framed f-structure.

Definition 2.2

Let $\varphi$ be a tensor field of (1, 1)-type and s pairs $(\xi _a, \eta ^a)$, $1\le a\le s$ of vector fields and 1-forms on N. If

• (i) $\varphi ^3+\varphi =0$, $rank \ \varphi =2n$,

• (ii) $\varphi ^2=-I+\sum _{a=1}^s\eta ^a\otimes \xi _a$, $\varphi (\xi _a)=0$, $\eta ^a(\xi _b)=\delta ^a_b$, $\eta ^a\circ \varphi =0$, then the data $(\varphi , \xi _a, \eta ^a)$ is called a framed f-structure.

Following [2, p. 130] we associate to a framed f-structure

• (1) the (1, 2)-type torsion tensor field

  $$\begin{aligned} S=N_{\varphi }+2\sum _{a=1}^sd\eta ^a\otimes \xi _a, \end{aligned}$$

  (2.2)

• (2) the structural distribution

  $$\begin{aligned} D=\{X\in \Gamma (TM); \eta ^1(X)=...=\eta ^s(X)=0\}=\cap _{a=1}^s\ker \eta ^a . \end{aligned}$$

  (2.3)

For a 1-form $\eta$ we use the differential

$$\begin{aligned} 2d\eta (X, Y)=X(\eta (Y))-Y(\eta (X))-\eta ([X, Y]). \end{aligned}$$

(2.4)

These notions lead to

Definition 2.3

The framed f-structure is called D-normal if S vanishes on D i.e. $S(X, Y)=0$ for all $X, Y\in \Gamma (D)$.

The relationship between the above structures was pointed out by A. Bejancu in Proposition 1.1 of [2, p. 130].

Proposition 2.4

If $(\varphi , \xi _a, \eta ^a)$ is a D-normal framed f-structure, then $(D, J=\varphi |_D)$ is a CR-structure.

Proof

The restriction J of $\varphi$ to D is obviously an almost complex structure. Conditions (2.1) result from the fact that for $X, Y\in \Gamma (D)$ we have

$$\begin{aligned} S(X, Y)=0=[JX, JY]+\varphi ^2([X, Y])-\varphi ([X, JY]+[JX, Y])-\sum _{a=1}^s\eta ^a([X, Y])\xi _a.\nonumber \\ \end{aligned}$$

(2.5)

For other details see the cited reference. $\square$

3 A metric framed f-structure on the tangent bundle of a Finsler manifold

Let M be now a smooth m-dimensional manifold with $m\ge 2$ and $\pi : TM\rightarrow M$ its tangent bundle. Let $x=(x^i)=(x^1,..., x^m)$ be local coordinates on M and $(x, y)=(x^i, y^i)=(x^1,...,x^m, y^1,...., y^m)$ the induced local coordinates on TM. Denote by O the null-section of $\pi$.

Recall after [5] that a Finsler fundamental function on M is a map $F:TM\rightarrow \mathbb {R}_{+}$ with the following properties:

• (F1) F is smooth on the slit tangent bundle $T_0M:=TM\setminus O$ and continuous on O,

• (F2) F is positive homogeneous of degree 1: $F(x, \lambda y)=\lambda F(x, y)$ for every $\lambda >0$,

• (F3) the matrix $(g_{ij})=\left( \frac{1}{2}\frac{\partial ^2F^2}{\partial y^i\partial y^j}\right)$ is invertible and its associated quadratic form is positive definite.

The tensor field $g=\{g_{ij}(x, y); 1\le i, j\le m\}$ is called the Finsler metric and the homogeneity of F implies:

$$\begin{aligned} F^2(x, y)=g_{ij}y^iy^j=y_iy^i, \end{aligned}$$

(3.1)

where $y_i=g_{ij}y^j$. The pair (M, F) is called Finsler manifold.

On $T_0M$ we have two distributions:

• (i) $V(TM):=\ker \pi _*$, called the vertical distribution and not depending of F. It is integrable and has the basis $\left\{ \frac{\partial }{\partial y^i}; 1\le i\le m\right\}$. A remarkable section of it is the Liouville vector field $\Gamma =y^i\frac{\partial }{\partial y^i}$.

• (ii) H(TM) with the basis $\left\{ \frac{\delta }{\delta x^i}:=\frac{\partial }{\partial y^i}-N^j_i\frac{\partial }{\partial y^j}\right\}$, where

  $$\begin{aligned} N^i_j=\frac{1}{2}\frac{\gamma ^i_{00}}{\partial y^j} \end{aligned}$$

  (3.2)

  with $\gamma ^i_{00}=\gamma ^i_{jk}y^jy^k$ built from the usual Christoffel symbols

  $$\begin{aligned} \gamma ^i_{jk}=\frac{1}{2}g^{ia}\left( \frac{\partial g_{ak}}{\partial x^j}+\frac{\partial g_{ja}}{\partial x^k}-\frac{\partial g_{jk}}{\partial x^a}\right) . \end{aligned}$$

  (3.3)

  H(TM) is often called the Cartan (or canonical) nonlinear connection of the geometry (M, F) and a remarkable section of it is the geodesic spray

  $$\begin{aligned} S_F=y^i\frac{\delta }{\delta x^i}. \end{aligned}$$

  (3.4)

  In particular, if g does not depend on y, we recover Riemannian geometry.

The dual basis of the above local basis $\{\frac{\delta }{\delta x^i}, \frac{\partial }{\partial y^i}\}$ of $\Gamma (T_0M)$ is $(dx^i, \delta y^i=dy^i+N^i_jdx^j)$. On $T_0M$ we have a Riemannian metric of Sasaki type

$$\begin{aligned} G_F=g_{ij}dx^i\otimes dx^j+g_{ij}\delta y^i\otimes \delta y^j. \end{aligned}$$

(3.5)

Another Finslerian object is the tensor field of (1, 1)-type $\Psi _F:\Gamma (T_0M)\rightarrow \Gamma (T_0M)$

$$\begin{aligned} \Psi _F\left( \frac{\delta }{\delta x^i}\right) =-\frac{\partial }{\partial y^i}, \quad \Psi _F\left( \frac{\partial }{\partial y^i}\right) =\frac{\delta }{\delta x^i}. \end{aligned}$$

(3.6)

It results that $\Psi _F$ is an almost complex structure and the pair $(\Psi _F, G_F)$ is an almost Kähler structure on $T_0M$.

In order to obtain a framed f-structure on $T_0M$ associated to the Finslerian function F, the following objects are considered in [1]

$$\begin{aligned} \left\{ \begin{array}{llll} \xi _1=S_F, \xi _2=\Gamma , \\ \eta ^1=\frac{1}{F^2}y_idx^i, \quad \eta ^2=\frac{1}{F^2}y_i\delta y^i , \\ \varphi =\Psi _F+\eta ^1\otimes \xi _2-\eta ^2\otimes \xi _1 , \\ G=\frac{1}{F^2}G_F. \end{array} \right. \end{aligned}$$

(3.7)

Then the main result of [1] is that the data $(\varphi , \xi _1, \xi _2, \eta ^1, \eta ^2)$ is a framed f-structure on $T_0M$ with $\eta ^a$ the G-dual of $\xi _a$, $1\le a\le 2$ and, moreover

$$\begin{aligned} G(\varphi \cdot , \varphi \cdot )=G-\eta ^1\otimes \eta ^1-\eta ^2\otimes \eta ^2. \end{aligned}$$

(3.8)

Also, $\xi _a$ are unitary vector fields with respect to G and $(G, \varphi , \xi _a, \eta ^a)$ is a metric framed f-structure.

4 Putting all together

The last paragraph of the previous section provides the ingredients of Sect. 2 with $N=T_0M$, $s=2$ and $n=m-1$, which motivates our choice $m\ge 2$. Then the structural distribution is

$$\begin{aligned} D_F=\ker \eta ^1\cap \ker \eta ^2=\{\xi _1\}^{\bot G}\cap \{\xi _2\}^{\bot G}=\{\xi _1\}^{\bot G_F}\cap \{\xi _2\}^{\bot G_F}, \end{aligned}$$

(4.1)

where $\{X\}^{\bot G}$ is the G-orthogonal complement of $span\{X\}$. We have $D_F=(span\{\xi _1, \xi _2\})^{\bot G_F}$ and this implies that $D_F$ has dimension $2m-2$. For a geometrical meaning of the distribution $span\{\xi _1, \xi _2\}$ in [1] is defined the differential 2-form $\omega _F$, naturally associated to the metric framed f-structure

$$\begin{aligned} \omega _F=G(\cdot , \varphi \cdot ), \end{aligned}$$

(4.2)

and it follows that $span\{\xi _1, \xi _2\}$ is the kernel of $\omega _F$. Also, the homogeneity of F implies the homogeneity of $S_F=\xi _1$, which means

$$\begin{aligned}{}[\Gamma , S_F]=[\xi _2, \xi _1]=\xi _1, \end{aligned}$$

(4.3)

and thus $span\{\xi _1, \xi _2\}$ is an integrable distribution; see also Theorem 3.15 of [3, p. 236].

A concrete expression of $D_F$ appears in [4, p. 11]. More precisely, consider after the cited paper

• (i) the horizontal vector fields

  $$\begin{aligned} h_i=\frac{\delta }{\delta x^i}-\frac{1}{F^2}y_iS_F, \end{aligned}$$

  (4.4)

  and the corresponding $(m-1)$-distribution $\mathcal {H}_{m-1}=span\{h_i; 1\le i\le m\}$,

• (ii) the vertical vector fields

  $$\begin{aligned} v_i=\frac{\partial }{\partial y^i}-\frac{1}{F^2}y_i\Gamma , \end{aligned}$$

  (4.5)

  and also the corresponding $(m-1)$-distribution $\mathcal {V}_{m-1}=span\{v_i; 1\le i\le m\}$.

We have

$$\begin{aligned} D_F=\mathcal {H}_{m-1}\oplus \mathcal {V}_{m-1} , \end{aligned}$$

(4.6)

and the same Theorem 3.15 of [3, p. 236] proves the integrability of $\mathcal {V}_{m-1}$; see also [4, p. 12].

Regarding the integrability of the nonlinear connection H(TM) we have

$$\begin{aligned} \left[ \frac{\delta }{\delta x^j}, \frac{\delta }{\delta x^k}\right] =R^i_{jk}\frac{\partial }{\partial y^i} , \end{aligned}$$

(4.7)

where

$$\begin{aligned} R^i_{jk}=\frac{\delta N^i_j}{\delta x^k}-\frac{\delta N^i_k}{\delta x^j}. \end{aligned}$$

(4.8)

The tensor field $R=\{R^i_{jk}(x, y); 1\le i, j, k\le m\}$ is called the curvature of the Cartan nonlinear connection and

$$\begin{aligned} R^i_j:=R^i_{kj}y^k \end{aligned}$$

(4.9)

are the components of the Jacobi endomorphism $\Phi =R^i_{j}\frac{\partial }{\partial y^i}\otimes dx^j$, [4, p. 5]. Now we are ready for the first main result:

Theorem 4.1

If the curvature tensor of (M, F) has the form

$$\begin{aligned} R^i_{jk}=\lambda \left( X^i_ky_j-X^i_jy_k\right) \end{aligned}$$

(4.10)

with $\lambda$ a smooth function on $T_0M$ and the tensor field $\{X^i_j(x, y); 1\le i, j\le m\}$ satisfying

$$\begin{aligned} y_iX^i_j=y_j \end{aligned}$$

(4.11)

for all $i, j\in \{1,...,m\}$, then the pair $(D_F, J_F=\Psi _F|_{D_F})$ is a CR-structure on $T_0M$.

Proof

We express the Nijenhuis tensor field of $\Psi _F$ as

$$\begin{aligned} N_{\Psi _F}(X, Y)= & {} [\Psi _FX, \Psi _FY]-[X, Y]-\Psi _F(A(X, Y))=B(X, Y)\nonumber \\&-\Psi _F(A(X, Y)) \end{aligned}$$

(4.12)

with $A(X, Y):=[X, \Psi _FY]+[\Psi _FX, Y]$ and $B(X, Y)=[\Psi _FX, \Psi _FY]-[X, Y]$. It follows that $B(X, Y)=A(\Psi _FX, Y)$ and then

$$\begin{aligned} N_{\Psi _F}(X, Y)=A(\Psi _FX, Y)-\Psi _F\circ A(X, Y). \end{aligned}$$

(4.13)

We prove firstly that A is a $D_F$-valued (0, 2)-tensor field. From (4.7) and

$$\begin{aligned} \left[ \frac{\delta }{\delta x^j}, \frac{\partial }{\partial y^k}\right] =\frac{\partial N^i_j}{\partial y^k}\frac{\partial }{\partial y^i}= \frac{\partial ^2 \gamma ^i_{00}}{\partial y^j\partial y^k}\frac{\partial }{\partial y^i} \end{aligned}$$

(4.14)

we obtain

$$\begin{aligned} A\left( \frac{\delta }{\delta x^j}, \frac{\delta }{\delta x^k}\right) =A\left( \frac{\partial }{\partial y^j}, \frac{\partial }{\partial y^k}\right) =0, \quad A\left( \frac{\delta }{\delta x^j}, \frac{\partial }{\partial y^k}\right) =R^i_{jk}\frac{\partial }{\partial y^i} , \end{aligned}$$

(4.15)

which means that $\eta ^1\circ A=0$ and

$$\begin{aligned} A=R^i_{jk}dx^j\wedge \delta y^k\otimes \frac{\partial }{\partial y^i}. \end{aligned}$$

(4.16)

A main identity in Finsler geometry is

$$\begin{aligned} y_iR^i_{ab}=0 , \end{aligned}$$

(4.17)

and then $\eta ^2\circ A=0$, which conclude the first part of the proof.

Secondly, we search for the framework of Proposition 2.4. The torsion tensor S on $D_F$ is

$$\begin{aligned} S(X, Y)=N_{\varphi }(X, Y)-\eta ^1([X, Y])\xi _1-\eta ^2([X, Y])\xi _2 \end{aligned}$$

with

$$\begin{aligned} N_{\varphi }(X, Y)=[\Psi _FX, \Psi _FY]+\varphi ^2([X, Y])-\varphi \circ A(X, Y). \end{aligned}$$

Since $\varphi$ is an element of a framed f-structure, we get

$$\begin{aligned} N_{\varphi }(X, Y)=[\Psi _FX, \Psi _FY]-[X, Y]+\eta ^1([X, Y])\xi _1+\eta ^2([X, Y])\xi _2-\varphi \circ A(X, Y) \end{aligned}$$

and from the definition $(3.7_3)$ of $\varphi$ it follows

$$\begin{aligned} S(X, Y)= & {} [\Psi _FX, \Psi _FY]-[X, Y]-(\Psi _F+\eta ^1\otimes \xi _2-\eta ^2\otimes \xi _1)\circ A(X, Y)\nonumber \\= & {} N_{\Psi _F}(X, Y). \end{aligned}$$

(4.18)

In local coordinates we have

$$\begin{aligned} N_{\Psi _F}=R^i_{jk}\delta y^j\wedge \delta y^k\otimes \frac{\partial }{\partial y^i}, \end{aligned}$$

(4.19)

and then $N_{\Psi _F}$ has components only when applied on the pair $(v_a, v_b)$. A long but straightforward computation yields

$$\begin{aligned} N_{\Psi _F}(v_a, v_b)=2\left[ R^i_{ab}+\frac{1}{F^2}(R^i_ay_b-R^i_by_a)\right] \frac{\partial }{\partial y^i}, \end{aligned}$$

(4.20)

and therefore the normality condition is

$$\begin{aligned} F^2R^i_{ab}=R^i_by_a-R^i_ay_b, \end{aligned}$$

(4.21)

which can be expressed as

$$\begin{aligned} N_{\Psi _F}=\eta ^2\wedge \left( R^i_k\delta y^k\otimes \frac{\partial }{\partial y^i}\right) . \end{aligned}$$

(4.22)

Relation (4.10) yields

$$\begin{aligned} R^i_k=\lambda \left( F^2X^i_k-y^aX^i_ay_k\right) \end{aligned}$$

(4.23)

and then both sides of (4.21) are equal to $\lambda F^2(X^i_ky_j-X^i_jy_k)$, which gives the final conclusion. Condition (4.11) corresponds to relation (4.17).

Let us also point out that condition (4.10) gives the following expression for the Nijenhuis tensor

$$\begin{aligned} N_{\Psi _F}=2\lambda F^2\eta ^2\wedge \left( X^i_j\delta y^j\otimes \frac{\partial }{\partial y^i}\right) , \end{aligned}$$

(4.24)

which yields again the vanishing of $N_{\Psi _F}$ on $D_F$ due to the presence of $\eta ^2$. Concerning the tensor field A we have

$$\begin{aligned} A=\lambda F^2\left[ \eta ^1\wedge \left( X^i_j\delta y^j\otimes \frac{\partial }{\partial y^i}\right) -\left( X^i_jdx^j\otimes \frac{\partial }{\partial y^i}\right) \wedge \eta ^2\right] , \end{aligned}$$

(4.25)

which proves the relations $\eta ^1\circ A=\eta ^2\circ A=0$. $\square$

Example 4.2

Recall that in dimension 2 the Nijenhuis tensor field of any almost complex structure vanishes. Then every 2-dimensional Finsler manifold $(M^2, F)$ satisfies the condition of Theorem 4.1. Let V(TM) be spanned by the vector fields $\Gamma$ and V respectively, H(TM) be spanned by the vector fields $S_F$ and H. Then $D_F$ is spanned by V and H and

$$\begin{aligned} J_F(H)=-V, \quad J_F(V)=H. \end{aligned}$$

(4.26)

We have that H is a linear combination of $h_1$ and $h_2$ while V is a linear combination of $v_1$ and $v_2$. $\square$

In order to consider examples in any dimension we remark that a solution of condition (4.11) is

$$\begin{aligned} X^i_j=\mu \delta ^i_j+(1-\mu )\frac{y^iy_j}{F^2} \end{aligned}$$

(4.27)

again with $\mu$ a smooth function on $T_0M$.

Example 4.3

If $\mu =1$ then $X^i_j=\delta ^i_j$ and the Finsler manifold (M, F) is of scalar flag curvature $\lambda$ since

$$\begin{aligned} R^i_{jk}=\lambda \left( \delta ^i_ky_j-\delta ^i_jy_k\right) , \end{aligned}$$

(4.28)

and then

$$\begin{aligned} R^i_k=\lambda \left( \delta ^i_kF^2-y^iy_k\right) . \end{aligned}$$

(4.29)

Corollary 4.4

If (M, F) is of scalar flag curvature, then $(D_F=(span\{S_F, \Gamma \})^{\bot G_F}, J_F)$ is a CR-structure on $T_0M$.

Remark also that the hypothesis of scalar flag curvature yields

$$\begin{aligned} N_{\Psi _F}=2\lambda F^2\eta ^2\wedge \pi _{V(TM)} , \end{aligned}$$

(4.30)

where $\pi _{V(TM)}$ is the projector on the vertical part in the $G_F$-orthogonal decomposition $T(T_0M)=H(TM)\oplus V(TM)$ i.e $\pi _{V(TM)}=\delta y^i\otimes \frac{\partial }{\partial y^i}$. However, $\Psi _F$ is integrable only in the flat case (i.e. $\lambda =0$) since $N_{\Psi _F}(\Gamma , v_a)=2\lambda F^2v_a$. The integrability of $\Psi _F$ as a tensor field of (1, 1)-type is equivalent with the integrability of the Cartan nonlinear connection of (M, F) and then $(T_0M, \Psi _F, G_F)$ is a Kähler manifold.

Particular case 4.5

(Riemannian space) Let $g=(g_{ij}(x))$ be a Riemannian metric on M. Then $\gamma ^i_{jk}(x, y)=\Gamma ^i_{jk}(x)$ are the Riemannian Christoffel symbols and

$$\begin{aligned} R ^i_{jk}(x, y)=R^i_{jka}(x)y^a \end{aligned}$$

(4.31)

where $R_g=(R^i_{jka})$ is the Riemannian curvature tensor of g. It results that a Riemannian geometry $(M, F=(g_{ij}(x)y^iy^j)^{\frac{1}{2}})$ is of scalar flag curvature if and only if g is of constant curvature. Therefore on the slit tangent bundle of a space form (M, g) there exists a CR-structure on the distribution complementary (with respect to the Sasaki lift of g) to the distribution generated by the Liouville vector field and the geodesic spray $S_g$. $\square$

Example 4.6

Returning to the general non-Riemannian case (4.27) with $\mu =0$ we get

$$\begin{aligned} X^i_j=\frac{y^iy_j}{F^2}, \end{aligned}$$

(4.32)

and then $R^i_{jk}=0$, which means that (M, F) is flat, a situation belonging also to Example 4.3 for vanishing scalar curvature. $\square$

For the general $\mu$ we have

$$\begin{aligned} N_{\Psi _F}=2\lambda F^2\eta ^2\wedge \left[ \mu \pi _{V(TM)}+(1-\mu )\eta ^2\otimes \Gamma \right] =2\lambda \mu F^2\eta ^2\wedge \mu \pi _{V(TM)}. \end{aligned}$$

(4.33)

5 A 1-parametric generalization

Let $\alpha >0$ and $\beta >0$ be two positive numbers. Following the approach of [15], let $v:TM\rightarrow \mathbb {R}$ be a function of the form $v=\bar{v}\circ \tau$ where $\tau =F^2$ and $\bar{v}:[0, +\infty )\rightarrow \mathbb {R}$ is a smooth function. Supposing that

$$\begin{aligned} \alpha +2t\bar{v}(t)>0 \end{aligned}$$

(5.1)

for any $t\in (0, +\infty )$, in the cited paper, the smooth functions $\bar{w}:[0, +\infty )\rightarrow \mathbb {R}$, $w:TM\rightarrow \mathbb {R}$

$$\begin{aligned} \bar{w}(t)=-\frac{\beta \bar{v}(t)}{\alpha +t\bar{v}(t)} \quad \text { and } \quad w=\bar{w}\circ \tau , \end{aligned}$$

(5.2)

and the Riemannian metric on $T_0M$

$$\begin{aligned} \bar{G}=G_{ij}dx^i\otimes dx^j+H_{ij}\delta y^i\otimes \delta y^j \end{aligned}$$

(5.3)

are defined, where

$$\begin{aligned} \left\{ \begin{array}{ll} G_{ij}=\frac{1}{\beta }g_{ij}+\frac{v}{\alpha \beta }y_iy_j \\ H_{ij}=\beta g_{ij}+w\circ \tau y_iy_j. \end{array} \right. \end{aligned}$$

(5.4)

Inspired by [15] we define also

$$\begin{aligned} \left\{ \begin{array}{ll} \bar{\xi }_1=(\beta +w\tau )S_F, \quad \quad \,\, \bar{\xi }_2=\Gamma =\xi _2 , \\ \bar{\eta }^1=\frac{1}{\tau }y_idx^i=\eta ^1, \quad \quad \bar{\eta }^2=(\frac{\beta }{\tau }+w)y_i\delta y^i , \\ \bar{\Psi }_F(\frac{\delta }{\delta x^i})=-G^a_i\frac{\partial }{\partial y^a}, \quad \bar{\Psi }_F(\frac{\partial }{\partial y^i})=H^a_i\frac{\delta }{\delta x^a}, \end{array} \right. \end{aligned}$$

(5.5)

where the lift of indices in the third line is constructed with $g^{-1}=(g^{ab})$. In fact, the only difference between us and [15] is with respect to 1-form $\bar{\eta }^i$; in order to reobtain that of Sect. 3 we divide with $\tau$ the 1-forms of Peyghan–Zhong. With a computation similar to that of Theorem 4.8 of Peyghan–Zhong we derive that $(\bar{G}, \bar{\varphi }, \bar{\xi }_a, \bar{\eta }^a)$ with

$$\begin{aligned} \bar{\varphi }=\bar{\Psi }_F+\bar{\eta }^1\otimes \bar{\xi }_2-\bar{\eta }^2\otimes \bar{\xi }_1 \end{aligned}$$

(5.6)

is a metric framed f-structure on $T_0M$ if and only if

$$\begin{aligned} \beta +t\bar{w}(t)=1. \end{aligned}$$

(5.7)

From this condition we get that $\bar{\xi }_a=\xi _a$ and $\bar{\eta }^a=\eta ^a$. From (5.2) and (5.7) we obtain

$$\begin{aligned} \bar{v}(t)=\frac{\alpha (\beta -1)}{t}, \quad \bar{w}(t)=\frac{1-\beta }{t}. \end{aligned}$$

(5.8)

In the particular case $\alpha =\beta =1$ we recover the metric framed f-structure of Anastasiei since $\bar{v}=\bar{w}\equiv 0$.

Now, under condition (5.7) we have the same structural distribution $D_F$ but the expression of the tensor field

$$\begin{aligned} \bar{A}(X, Y):=[X, \bar{\Psi }_FY]+[\bar{\Psi }_FX, Y] \end{aligned}$$

(5.9)

is more complicated. More detailed

$$\begin{aligned} \left\{ \begin{array}{ll} \bar{A}(\frac{\delta }{\delta x^j}, \frac{\delta }{\delta x^k})=\left( \frac{\delta G^v_j}{\delta x^k}-\frac{\delta G^v_k}{\delta x^j}+G^u_j\frac{\partial N^v_k}{\partial y^u}-G^u_k\frac{\partial N^v_j}{\partial y^u}\right) \frac{\partial }{\partial y^v} \\ \bar{A}(\frac{\partial }{\partial y^j}, \frac{\partial }{\partial y^k})=\left( \frac{\partial H^v_k}{\partial y^j}-\frac{\partial H^v_j}{\partial y^k}\right) \frac{\delta }{\delta x^v}+\left( H^u_j\frac{\partial N^v_u}{\partial y^k}-H^u_k\frac{\partial N^v_u}{\partial y^j}\right) \frac{\partial }{\partial y^v}, \\ \bar{A}(\frac{\delta }{\delta x^j}, \frac{\partial }{\partial y^k})=\frac{\delta H^v_k}{\delta x^j}\frac{\delta }{\delta x^v}+\left( H^u_kR^v_{ju}+\frac{\partial G^v_j}{\partial y^k}\right) \frac{\partial }{\partial y^v}, \end{array} \right. \end{aligned}$$

(5.10)

where, with (5.7)

$$\begin{aligned} \left\{ \begin{array}{ll} G_{ij}=\frac{1}{\beta }g_{ij}+\frac{\beta -1}{\beta \tau }y_iy_j, \qquad \qquad \; H_{ij}=\beta g_{ij}+\frac{1-\beta }{\tau }y_iy_j \\ G^a_j=\frac{1}{\beta }\delta ^a_j+\frac{\beta -1}{\beta \tau }y^ay_j, \qquad \qquad \, \; H^a_j=\beta \delta ^a_j+\frac{1-\beta }{\tau }y^ay_j \\ \bar{\Psi }_F(\frac{\delta }{\delta x^i})=-\frac{1}{\beta }\frac{\partial }{\partial y^i}+\frac{1-\beta }{\beta \tau }y_i\Gamma , \quad \bar{\Psi }_F(\frac{\partial }{\partial y^i})=\beta \frac{\delta }{\delta x^i}+\frac{1-\beta }{\tau }y_iS_F. \end{array} \right. \end{aligned}$$

(5.11)

It results that $\alpha$ disappears and this motivates the title of this section, namely 1-parametric generalization and not 2-parametric. Note that $\bar{\Psi }_F\left( h_i\right) =-\frac{1}{\beta }v_i$ and $\bar{\Psi }_F(v_i)=\beta h_i$.

Then

$$\begin{aligned} \left\{ \begin{array}{ll} \bar{A}(\frac{\delta }{\delta x^j}, \frac{\delta }{\delta x^k})=\frac{\beta -1}{\beta \tau }\left[ \frac{\delta }{\delta x^k}\left( y_jy^v\right) -\frac{\delta }{\delta x^j}\left( y_ky^v\right) \right] \frac{\partial }{\partial y^v} \\ \bar{A}(\frac{\partial }{\partial y^j}, \frac{\partial }{\partial y^k})=(1-\beta )\left[ \frac{\partial }{\partial y^j}(\frac{y_ky^v}{\tau })-\frac{\partial }{\partial y^k}(\frac{y_jy^v}{\tau })\right] \frac{\delta }{\delta x^v} \\ \bar{A}(\frac{\delta }{\delta x^j}, \frac{\partial }{\partial y^k})=\frac{1-\beta }{\tau }\frac{\delta }{\delta x^j}\left( y_ky^v\right) \frac{\delta }{\delta x^v}+\left[ \beta R^v_{jk}+\frac{1-\beta }{\tau }y_ky^uR^v_{ju}+\frac{\beta -1}{\beta }\frac{\partial }{\partial y^k}\left( \frac{y_jy^v}{\tau }\right) \right] \frac{\partial }{\partial y^v}. \end{array} \right. \end{aligned}$$

(5.12)

Choosing $\alpha =1$ the second main result is

Theorem 5.1

Let $\beta >\frac{1}{2}$ and the smooth functions $\bar{v}(t)=-\bar{w}(t)=\frac{\beta -1}{t}$. If for any $X, Y\in D_F$ we have

• (1) $\bar{A}(X, Y)\in D_F$,

• (2) $N_{\bar{\Psi }_F}(X, Y)=0$, then $(D_F, \bar{J}_F=\bar{\Psi }_F|_{D_F})$ is a CR-structure on $T_0M$.

Proof

The condition in $\beta$ follows from (5.1). Exactly as in the proof of Theorem 4.1 we have

$$\begin{aligned} S(X, Y)=N_{\bar{\Psi }_F}(X, Y)-\eta ^1(\bar{A}(X, Y))\xi _2+\eta ^2(\bar{A}(X, Y))\xi _1. \end{aligned}$$

(5.13)

and the conclusion follows directly. Let us note that 1) corresponds to condition $(2.1_1)$ while 2) corresponds to condition $(2.1_2)$. $\square$

Let us remark that

$$\begin{aligned} \beta \eta ^2\circ \bar{A}\left( \frac{\delta }{\delta x^j}, \frac{\delta }{\delta x^k}\right) =\eta ^1\circ \bar{A}\left( \frac{\delta }{\delta x^j}, \frac{\partial }{\partial y^k}\right) -\eta ^1\circ \bar{A}\left( \frac{\delta }{\delta x^k}, \frac{\partial }{\partial y^j}\right) , \end{aligned}$$

(5.14)

and then the vanishing of $\eta ^1\circ \bar{A}\left( \frac{\delta }{\delta x^a}, \frac{\partial }{\partial y^b}\right)$ implies the vanishing of $\eta ^2\circ \bar{A}\left( \frac{\delta }{\delta x^u}, \frac{\delta }{\delta x^v}\right)$. The vanishing of the former expression means that $y_k$ is an eigenvector for $\frac{\delta }{\delta x^j}$

$$\begin{aligned} \frac{\delta y_k}{\delta x^j}=\left( -\frac{N^a_jy_a}{F^2}\right) y_k \end{aligned}$$

(5.15)

and then $y_k$ is an eigenvector for the geodesic spray

$$\begin{aligned} S_F(y_k)=\left( -\frac{N^a_jy^jy_a}{F^2}\right) y_k. \end{aligned}$$

(5.16)

Such condition holds in the Euclidian space $(\mathbb {R}^m, g_{ij}=\delta _{ij})$ but here the expression $\eta ^2\circ \bar{A}(\frac{\delta }{\delta x^j}, \frac{\partial }{\partial y^k})$ is non-vanishing since

$$\begin{aligned} y_v\frac{\partial }{\partial y^k}\left( \frac{y_jy^v}{F^2}\right) =\delta _{jk}-\frac{y_jy^k}{F^2}\ne 0 \end{aligned}$$

(5.17)

and then it remains an open problem to find Riemannian and/or Finsler manifolds satisfying the conditions of Theorem 5.1 with $\beta \ne 1$.