On Horizontal Recurrent Finsler Connections

TL;DR

This paper explores horizontally recurrent Finsler connections using a pullback approach, establishing existence and uniqueness results, and introduces a special class called HRF-connections with specific properties.

Contribution

It generalizes the Cartan connection theorem by proving existence and uniqueness of horizontally recurrent Finsler connections for any scalar 1-form.

Findings

Existence and uniqueness of horizontally recurrent Finsler connections for any scalar 1-form

Introduction and analysis of special HRF-connections with unique properties

Extension of classical Finsler connection theory to recurrent cases

Abstract

In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($\pi$)1-form $A$, there exists a unique horizontally recurrent Finsler connection whose $h$-recurrence form is $A$. This result generalizes the existence and uniqueness theorem of Cartan connection. We then study some properties of a special kind of horizontally recurrent Finsler connection, which we call special HRF-connection.

Equations

$$0⟶{\pi }^{-1}(TM)\stackrel{\gamma }{⟶}T(\mathcal{T}M)\stackrel{\rho }{⟶}{\pi }^{-1}(TM)⟶0,\text{vspace}-0.1cm$$

$$T_u(\mathcal{T}M)=V_u(\mathcal{T}M)\oplus H_u(\mathcal{T}M)\forall u\in \mathcal{T}M.$$

$$Q(\overline{X},\overline{Y})=\text{T}(\beta \overline{X}\beta \overline{Y}),T(\overline{X},\overline{Y})=\text{T}(\gamma \overline{X},\beta \overline{Y})\forall \overline{X},\overline{Y}\in \mathfrak{X}(\pi (M)),\text{vspace}-0.2cm$$

$$R(\overline{X},\overline{Y})\overline{Z}=\text{K}(\beta \overline{X}\beta \overline{Y})\overline{Z},P(\overline{X},\overline{Y})\overline{Z}=\text{K}(\beta \overline{X},\gamma \overline{Y})\overline{Z},S(\overline{X},\overline{Y})\overline{Z}=\text{K}(\gamma \overline{X},\gamma \overline{Y})\overline{Z},$$

$$\hat{R}(\overline{X},\overline{Y})=R(\overline{X},\overline{Y})\overline{\eta },\hat{P}(\overline{X},\overline{Y})=P(\overline{X},\overline{Y})\overline{\eta },\hat{S}(\overline{X},\overline{Y})=S(\overline{X},\overline{Y})\overline{\eta }.$$

$$\Gamma =[J,G],\text{vspace}-0.3cm$$

$${\overline{D}}_{\bar{\beta }\overline{X}}g=A(\overline{X})g\forall \overline{X}\in \mathfrak{X}(\pi (M)),$$

$$2g({\overline{D}}_{\gamma \overline{X}}\overline{Y},\overline{Z})$$

$$\bar{\beta }\overline{X}=\beta \overline{X}+\gamma {\overline{X}}_t,$$

$$\rho [\gamma \overline{X},\bar{\beta }\overline{Y}]=\rho [\gamma \overline{X},\beta \overline{Y}].$$

$${\overline{D}}_{\gamma \overline{X}}\overline{Y}={\nabla }_{\gamma \overline{X}}\overline{Y}.\text{vspace}-0.1cm$$

$$2g({\overline{D}}_{\bar{\beta }\overline{X}}\overline{Y},\overline{Z})$$

$$\rho [\bar{\beta }\overline{X},\bar{\beta }\overline{Y}]=\rho [\beta \overline{X},\beta \overline{Y}]+\rho [\beta \overline{X},\gamma {\overline{Y}}_t]+\rho [\gamma {\overline{X}}_t,\beta \overline{Y}],$$

$$2g({\overline{D}}_{\bar{\beta }\overline{X}}\overline{Y},\overline{Z})$$

$$2{\overline{\eta }}_t=2A(\overline{\eta })\overline{\eta }-L^2\overline{a},$$

$$\bar{\beta }\overline{X}-\beta \overline{X}=\gamma {\overline{X}}_t=\frac{1}{2}\{A(\overline{X})\gamma \overline{\eta }+A(\overline{\eta })\gamma \overline{X}-g(\overline{X},\overline{\eta })\gamma \overline{a}+L^2\gamma T(\overline{a},\overline{X})\}.$$

$$2T({\overline{X}}_t,\overline{Y},\overline{Z})=A(\overline{\eta })T(\overline{X},\overline{Y},\overline{Z})-g(\overline{X},\overline{\eta })T(\overline{a},\overline{Y},\overline{Z})+L^{2}T(T(\overline{a},\overline{X}),\overline{Y},\overline{Z}),$$

$$2{D^{\circ }}_{\gamma {\overline{X}}_t}\overline{Y}=A(\overline{X}){D^{\circ }}_{\gamma \overline{\eta }}\overline{Y}+A(\overline{\eta }){D^{\circ }}_{\gamma \overline{X}}\overline{Y}-g(\overline{X},\overline{\eta }){D^{\circ }}_{\gamma \overline{a}}\overline{Y}+L^2{D^{\circ }}_{\gamma T(\overline{a},\overline{X})}\overline{Y}.$$

$${\overline{D}}_{\bar{\beta }\overline{X}}\overline{Y}$$

$${\overline{D}}_X\overline{Y}$$

$${\overline{D}}_X\overline{Y}={\nabla }_X\overline{Y}+N(\rho X,\overline{Y});$$

$$N(\rho X,\overline{Y})$$

$$\overline{G}=G+A(\overline{\eta })\mathcal{C}-\frac{1}{2}{L}^2\gamma \overline{a}.$$

$$\overline{\Gamma }(X)=\Gamma (X)+\{A(\rho X)\mathcal{C}+A(\overline{\eta })JX-g(\rho X,\overline{\eta })\gamma \overline{a}+L^2\gamma T(\overline{a},\rho X)\}.$$

$$\overline{P}(\overline{X},\overline{Y})\overline{Z}$$

$$\hat{\overline{P}}(\overline{X},\overline{Y})=\hat{P}(\overline{X},\overline{Y})-\frac{1}{2}\{LT(\overline{X},\overline{Y})-\ell (\overline{X})\overline{Y}+L^{-1}g(\overline{X},\overline{Y})\overline{\eta }\}.$$

$$\frac{1}{2}L\varphi (\overline{Y})=0\forall \overline{Y}\in \mathfrak{X}(\pi (M)).$$

$$\overline{R}(\overline{X},\overline{Y})\overline{Z}$$

$$P(\overline{X},\overline{Y})\overline{Z}-P(\overline{Y},\overline{X})\overline{Z}=-({\nabla }_{\beta \overline{\eta }}S)(\overline{X},\overline{Y},\overline{Z}).$$

$$\hat{\overline{R}}(\overline{X},\overline{Y})=\hat{R}(\overline{X},\overline{Y})+\frac{L}{4}\{\ell (\overline{X})\overline{Y}-\ell (\overline{Y})\overline{X}\}.$$

Full text

(a)

${{D}}^{\circ}_{\gamma\overline{X}}\overline{Y}=\nabla_{\gamma\overline{X}}\overline{Y}-T(\overline{X},\overline{Y})=\rho[\gamma\overline{X},\beta\overline{Y}]$,

(b)

${{D}}^{\circ}_{\beta\overline{X}}\overline{Y}=\nabla_{\beta\overline{X}}\overline{Y}+\widehat{P}(\overline{X},\overline{Y})=K[\beta\overline{X},\gamma\overline{Y}].$**

(a)

$({{D}}^{\circ}_{\gamma\overline{X}}\,g)(\overline{Y},\overline{Z})=2T(\overline{X},\overline{Y},\overline{Z})$,

(b)

$({{D}}^{\circ}_{\beta\overline{X}}\,g)(\overline{Y},\overline{Z})=-2g(\widehat{P}(\overline{X},\overline{Y}),\overline{Z})$,

where $T(\overline{X},\overline{Y},\overline{Z}):=g(T(\overline{X},\overline{Y}),\overline{Z})$.

We terminate this section by some concepts and results concerning the Klein-Grifone approach to intrinsic Finsler geometry. For more details, we refer to [3, 4, 7, 13].

A semi-spray is a vector field $X$ on $TM$, $C^{\infty}$ on ${\cal T}M$, $C^{1}$ on $TM$, such that $JX=\mathcal{C}$, where $J:=\gamma\circ\rho$ and $\mathcal{C}:=\gamma\overline{\eta}$. A semispray $X$ which is homogeneous of degree $2$ in the directional argument ($[\mathcal{C},X]=X$ ) is called a spray.

A nonlinear connection on $M$ is a vector $1$-form $\Gamma$ on $TM$, $C^{\infty}$ on ${\cal T}M$, $C^{0}$ on $TM$, such that $J\Gamma=J$ and $\Gamma J=-J.$ The horizontal and vertical projectors associated with $\Gamma$ are defined by $h:=\frac{1}{2}(I+\Gamma)$ and $v:=\frac{1}{2}(I-\Gamma)$, respectively. The torsion and curvature of $\Gamma$ are defined by $t:=\frac{1}{2}[J,\Gamma]$ and $\mathfrak{R}:=-\frac{1}{2}[h,h]$, respectively. A nonlinear connection $\Gamma$ is homogenous if $[\mathcal{C},\Gamma]=0$. It is conservative if $d_{h}E=0$.

[TABLE]

*where $G$ is the canonical spray.

This nonlinear connection is called the canonical or the Barthel connection associated with $(M,L)$.*

3.  Horizontal recurrent Finsler connection

In this section, we prove that for each $\pi$-form $A$ there exists a unique horizontally recurrent Finsler connection whose $h$-recurrence form is $A$. Throughout, we use the notions and results of [14].

[TABLE]

where $\bar{\beta}$ is the horizontal map of $\overline{D}$. The scalar form $A$ is called the h-recurrence form of $\overline{D}$.

Let $(M,L)$ be a Finsler manifold. Let $\nabla$ be the Cartan connection associated with $(M,L)$. We denote by $K,\beta,T$ the connection map, the horizontal map and the (h)hv-torsion of $\nabla$, respectively. We also denote by $R,P,S$ the h-, hv- and v-curvature tensors of $\nabla$, respectively. Now, we announce the main result of the present section.

(C1)

$\overline{D}$* is horizontally recurrent with h-recurrence form $A$ : $(\overline{D}g)\circ\bar{\beta}=A\otimes g$,*

(C2)

the metric $g$ is $\overline{D}$-vertically parallel : $\overline{D}_{\gamma\overline{X}}g=0$,

(C3)

the (h)h-torsion $\overline{Q}$ of $\overline{D}$ vanishes : $\overline{Q}=0$,

(C4)

the (h)hv-torsion $\overline{T}$ of $\overline{D}$ satisfies $g(\overline{T}(\overline{X},\overline{Y}),\overline{Z})=g(\overline{T}(\overline{X},\overline{Z}),\overline{Y})$.

Such a connection is called the horizontally recurrent Finsler (HRF-) connection with h-recurrence form $A$.

In view of axioms (C2) and (C4), one can show that, for all $\overline{X},\overline{Y},\overline{Z}\in\mathfrak{X}(\pi(M))$,

[TABLE]

As the difference between $\bar{\beta}\overline{X}$ and ${\beta}\overline{X}$ is vertical, one can set

[TABLE]

for some $\overline{X}_{t}\in\mathfrak{X}(\pi(M))$. Since $J^{2}=0,\,[J,J]=0$ and $\rho\circ J=0$, we get

[TABLE]

Making use of (3.3) and Theorem 4**(a)** of [14], Equation (3.1) implies that

[TABLE]

Similarly, using axioms (C1) and (C3), we obtain, for all $\overline{X},\overline{Y},\overline{Z}\in\mathfrak{X}(\pi(M))$,

[TABLE]

From (3.2), it is easy to show that

[TABLE]

Using the above relation, Proposition 2.1, Proposition 2.2 and Theorem 4**(b)** of [14], (3.5) becomes

[TABLE]

Setting $\overline{X}=\overline{Y}=\overline{\eta}$ in (3.6), noting that $\bar{K}\circ\bar{\beta}=K\circ\beta=0$ and $i_{\overline{\eta}}T=0$, we get

[TABLE]

where $g(\overline{a},\overline{X}):=A(\overline{X})$. From which, by setting again $\overline{Y}=\overline{\eta}$ into Equation (3.6), we obtain

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From which, one can show that

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[TABLE]

Now, using (3.7), (3.8) and (3.9), Equation (3.6) reduces to

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Consequently, from (3.4), (Proof.) and taking into account (3.7), the full expression of $\overline{D}_{X}\overline{Y}$ is given by:

[TABLE]

Hence ${\overline{D}}_{X}\overline{Y}$ is uniquely determined by the right-hand side of (3.11).

Now, we prove the existence of $\overline{D}$. For a given 1-scalar form $A$ on $\pi^{-1}(TM)$, we define $\overline{D}$ by the requirement that (3.11) holds, or equivalently, (3.4), (Proof.) and (3.7) hold. Then, using the properties of Cartan connection $\nabla$ and the results of [14], it is not difficult to show that the connection $\overline{D}$ satisfies the conditions (C1)-(C4). ∎

[TABLE]

[TABLE]

In view of Theorem 3.4, we have

The spray $\overline{G}$ associated with $\overline{D}$ is related to the canonical spray $G$ by :

[TABLE]

(b)

The nonlinear connection $\overline{\Gamma}$ associated with $\overline{D}$ is related to the Barthel connection $\Gamma$ by :

[TABLE]

(a)

${{\overline{S}}}(\overline{X},\overline{Y})\overline{Z}=S(\overline{X},\overline{Y})\overline{Z}$.

(b)

${{\overline{P}}}(\overline{X},\overline{Y})\overline{Z}=P(\overline{X},\overline{Y})\overline{Z}+(\nabla_{\gamma\overline{Y}}N)(\overline{X},\overline{Z})+N({T}(\overline{Y},\overline{X}),\overline{Z})$

${\qquad\qquad\ \ \ }+\frac{1}{2}\{A(\overline{\eta})S(\overline{X},\overline{Y})\overline{Z}-L\ell(\overline{X})S(\overline{a},\overline{Y})\overline{Z}+L^{2}S(T(\overline{a},\overline{X}),\overline{Y})\overline{Z}\}$.

(c)

${{\overline{R}}}(\overline{X},\overline{Y})\overline{Z}=R(\overline{X},\overline{Y})\overline{Z}+P(\overline{X},\overline{Y}_{t})\overline{Z}-P(\overline{Y},\overline{X}_{t})\overline{Z}+S(\overline{X}_{t},\overline{Y}_{t})\overline{Z}$

${\qquad\qquad\ \ \ }+\mathfrak{U}_{\overline{X},\overline{Y}}\{(\nabla_{\bar{\beta}\overline{Y}}N)(\overline{X},\overline{Z})+N(\overline{Y},N(\overline{X},\overline{Z}))+N(T(\overline{Y}_{t},\overline{X}),\overline{Z})\}$,

where $\overline{Y}_{t}$ is given by (3.7).

4 Special HRF-connection

In this section, we investigate a special horizontally recurrent Finsler connection for which the h-recurrence form $A$ is taken to be $\ell:=L^{-1}i_{\overline{\eta}}\,g$. In what follows $\overline{D}$ will denote the HRF-connection whose h-recurrence form is $\ell$, and will be called the special HRF-connection. In this case $\overline{a}=L^{-1}\,\overline{\eta}$.

The following two lemmas are useful for subsequence use.

Lemma 4.1**.**

The nonlinear connection $\overline{\Gamma}$ associated with the special HRF-connection $\overline{D}$ is given by $\overline{\Gamma}=\Gamma+LJ.$ Consequently, ${\bar{\beta}}=\beta+\frac{1}{2}L\gamma$ and ${\overline{K}}=K-\frac{1}{2}L\rho$.

Proof.

The proof follows from Proposition 3.7 and the identities $\overline{a}=L^{-1}\overline{\eta}$, $\ell(\overline{\eta})=L$ and $i_{\overline{\eta}}T=0$. ∎

Lemma 4.2**.**

For the special HRF-connection $\overline{D}$, we have

(a)

$N(\overline{X},\overline{Y})=\frac{1}{2}\{L^{-1}g(\overline{X},\overline{Y})\overline{\eta}-\ell(\overline{X})\overline{Y}-\ell(\overline{Y})\overline{X}\}$.

(b)

$N(\overline{X},\overline{\eta})=-\frac{1}{2}L\overline{X}$.

(c)

$(\nabla_{\beta\overline{X}}N)(\overline{Y},\overline{Z})=0$.

(d)

$(\nabla_{\gamma\overline{X}}N)(\overline{Y},\overline{Z})=\frac{1}{2L}\{g(\overline{Y},\overline{Z})\phi(\overline{X})-\hbar(\overline{X},\overline{Y})\overline{Z}-\hbar(\overline{X},\overline{Z})\overline{Y}\}$.

(e)

$(\nabla_{\bar{\beta}\overline{X}}N)(\overline{Y},\overline{Z})=\frac{1}{4}\{g(\overline{Y},\overline{Z})\phi(\overline{X})-\hbar(\overline{X},\overline{Y})\overline{Z}-\hbar(\overline{X},\overline{Z})\overline{Y}\}$,

where $\phi(\overline{X}):=\overline{X}-L^{-1}\ell(\overline{X})\overline{\eta}$ and  $\hbar(\overline{X},\overline{Y})=g(\phi(\overline{X}),\overline{Y})$ is the angular metric.

Proof.

The proof follows from Corollary 3.6 and fact that $T$ is symmetric and indicatory, taking the identities $\nabla g=0$, $\nabla_{\beta\overline{X}}L=0$, $\nabla_{\beta\overline{X}}\ell=0$, $\nabla_{\gamma\overline{X}}L=\ell(\overline{X})$ and $(\nabla_{\gamma\overline{X}}\ell)(\overline{Y})=L^{-1}\hbar(\overline{X},\overline{Y})$ into account. ∎

Theorem 4.3**.**

The (v)hv-torsion tensor $\widehat{\overline{P}}$ of the special HRF-connection $\overline{D}$ never vanishes.

Proof.

From Proposition 3.8, taking into account Lemma 4.1, Lemma 4.2 and the fact that $T$ and $S$ are indicatory, one can show that

[TABLE]

From wihch

[TABLE]

Now, assume the contrary, that is $\widehat{\overline{P}}=0$. Hence, setting $\overline{X}=\overline{\eta}$ in (4.1), using the fact that $\widehat{P}$ and $T$ are indicatory, we obtain

[TABLE]

Consequently, $(n-1)L=0$ (since $Tr\,\phi=n-1$), which is a contradiction. ∎

Corollary 4.4**.**

In view of Equation (4.1), we have

(a)

$\widehat{\overline{P}}(\overline{X},\overline{Y})-\widehat{\overline{P}}(\overline{Y},\overline{X})=\frac{1}{2}\{\ell(\overline{X})\overline{Y}-\ell(\overline{Y})\overline{X}\}.$**

(b)

$g(\widehat{\overline{P}}(\overline{X},\overline{Y}),\overline{\eta})=-\frac{1}{2}\hbar(\overline{X},\overline{Y}).$**

Proposition 4.5**.**

The h-curvature tensor ${\overline{R}}$ of the special HRF-connection $\overline{D}$ has the form

[TABLE]

Proof.

The proof follows from Proposition 3.8, Lemma 4.1 and Lemma 4.2. ∎

Definition 4.6**.**

[15]* A Finsler manifold $(M,L)$, with $dim\,M\geq 3$, is said to be:*

(a)

$h$-isotropic with a scalar $k_{o}$ if the h-curvature tensor $R$ of Cartan connection has the form $R(\overline{X},\overline{Y})\overline{Z}=k_{o}\{g(\overline{X},\overline{Z})\overline{Y}-g(\overline{Y},\overline{Z})\overline{X}\}.$

(b)

of constant curvature $k$ if the (v)h-torsion tensor $\widehat{R}$ of Cartan connection satisfies the relation $\widehat{R}(\overline{\eta},\overline{X})=kL^{2}\phi(\overline{X}).$

Definition 4.7**.**

[15]* A Finsler manifold $(M,L)$ is said to be P-symmetric if the mixed curvature tensor $P$ of Cartan connection satisfies $P(\overline{X},\overline{Y})\overline{Z}=P(\overline{Y},\overline{X})\overline{Z}$, for all $\overline{X},\overline{Y},\overline{Z}\in\mathfrak{X}(\pi(M))$.*

Theorem 4.8**.**

Let $(M,L)$ be an h-isotropic Finsler manifold with scalar $(-\frac{1}{4})$. The h-curvature tensor ${\overline{R}}$ of the special HRF-connection $\overline{D}$ vanishes if and only if the v-curvature tensor $S$ of Cartan connection has property that $\nabla_{\beta\overline{\eta}}\,S=\frac{L}{2}S$.

Proof.

The proof follows from proposition 4.5, Lemma 4.1 and the fact that [17]

[TABLE]

∎

Theorem 4.9**.**

Let $(M,L)$ be a $P$-symmetric h-isotropic Finsler manifold with scalar $(-\frac{1}{4})$. The h-curvature tensor ${\overline{R}}$ of the special HRF-connection $\overline{D}$ vanishes if and only if the v-curvature tensor $S$ vanishes.

Proof.

The proof follows from Proposition 4.5 and Relation (4.2). ∎

Theorem 4.10**.**

If the (v)h-torsion tensor ${\widehat{\overline{R}}}$ of $\overline{D}$ vanishes, then $(M,L)$ is of constant curvature.

Proof.

From proposition 4.5, we obtain

[TABLE]

Now, if the (v)h-torsion tensor ${\widehat{\overline{R}}}$ of $\overline{D}$ vanishes, the above relation implies that

[TABLE]

Hence, by Definition 4.1(b), the result follows. ∎