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I'm trying to find conditions for an endomorphism $L:TM\rightarrow TM$ verfying the condition

$$ \oint_{j,k,l} \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m = \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m + \left(\mathcal{T}_L\right)_{km}^i \left(\mathcal{T}_L\right)_{lj}^m + \left(\mathcal{T}_L\right)_{lm}^i \left(\mathcal{T}_L\right)_{jk}^m= 0, $$

where $\mathcal{T}_L$ denotes the Nijenhuis torsion of $L$ and a sum over $m$ is understood.

Assuming dim$(M)=3$,the system should have $n^3(n-1)/2 =_{n=3} 27 $ (independent) equations. So, there is a compact way (maybe the Table command mixed with DSolve) to write the 27 equations quickly in Mathematica? And does the programm help me and will it solve symbolicaly the system giving the endomorphism $L$ in function of some real parameters?

Thanks :)

Pd: The Nijenhuis torsion of $L$ is written in terms of $L_j^i$ and $\frac{\partial L_j^i}{\partial x^k}$, such as show eq. 5 https://arxiv.org/pdf/1405.5118.pdf

I'm trying to find conditions for an endomorphism $L:TM\rightarrow TM$ verfying the condition

$$ \oint_{j,k,l} \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m = \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m + \left(\mathcal{T}_L\right)_{km}^i \left(\mathcal{T}_L\right)_{lj}^m + \left(\mathcal{T}_L\right)_{lm}^i \left(\mathcal{T}_L\right)_{jk}^m= 0, $$

where $\mathcal{T}_L$ denotes the Nijenhuis torsion of $L$ and a sum over $m$ is understood.

Assuming $\mbox{dim}\,(M)=3$,the system should have $n^3(n-1)/2 =_{n=3} 27 $ (independent) equations. So, there is a compact way (maybe the Table command mixed with DSolve) to write the 27 equations quickly in Mathematica? And does the program help me and will it solve symbolically the system giving the endomorphism $L$ in function of some real parameters?

Thanks :)

Pd: The Nijenhuis torsion of $L$ is written in terms of $L_j^i$ and $\frac{\partial L_j^i}{\partial x^k}$, such as show eq. 5 https://arxiv.org/pdf/1405.5118.pdf

EDIT:

Following the @bills' advice, I have decided try with an easier example. Let $f(x,y)=\sin(x+y)$. Then, is very easy check the identity

$$ f(x,y) + \frac{\partial^2f}{\partial x\partial y} =0 . $$

I have introduced this partial differential equation in Mathematica using DSolve:

DSolve[f[x, y] + D[D[f[x, y], y], x] == 0, f, {x, y}]

and hoping for a condition about the function $f$, but the program returns the same code.

At this point, what (or where) is the problem? Is a syntax problem? Or maybe is because the function DSolve can not be used to get conditions about a function from a differential equation? Then, it is there some command in Mathematica which could do it?

I'm trying to find conditions for an endomorphism $L:TM\rightarrow TM$ verfying the condition

$$ \oint_{j,k,l} \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m = \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m + \left(\mathcal{T}_L\right)_{km}^i \left(\mathcal{T}_L\right)_{lj}^m + \left(\mathcal{T}_L\right)_{lm}^i \left(\mathcal{T}_L\right)_{jk}^m= 0, $$

where $\mathcal{T}_L$ denotes the Nijenhuis torsion of $L$ and a sum over $m$ is understood.

Assuming dim$(M)=3$,the system should have $n^3(n-1)/2 =_{n=3} 27 $ equations. So, there is a compact way (maybe the Table command mixed with DSolve) to write the 27 equations quickly in Mathematica? And does the programm help me and will it solve symbolicaly the system giving the endomorphism $L$ in function of some real parameters?

Thanks :)

Pd: The Nijenhuis torsion of $L$ is written in terms of $L_j^i$ and $\frac{\partial L_j^i}{\partial x^k}$, such as show eq. 5 https://arxiv.org/pdf/1405.5118.pdf

I'm trying to find conditions for an endomorphism $L:TM\rightarrow TM$ verfying the condition

$$ \oint_{j,k,l} \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m = \left(\mathcal{T}_L\right)_{jm}^i \left(\mathcal{T}_L\right)_{kl}^m + \left(\mathcal{T}_L\right)_{km}^i \left(\mathcal{T}_L\right)_{lj}^m + \left(\mathcal{T}_L\right)_{lm}^i \left(\mathcal{T}_L\right)_{jk}^m= 0, $$

where $\mathcal{T}_L$ denotes the Nijenhuis torsion of $L$ and a sum over $m$ is understood.

Assuming dim$(M)=3$,the system should have $n^3(n-1)/2 =_{n=3} 27 $ (independent) equations. So, there is a compact way (maybe the Table command mixed with DSolve) to write the 27 equations quickly in Mathematica? And does the programm help me and will it solve symbolicaly the system giving the endomorphism $L$ in function of some real parameters?

Thanks :)

Pd: The Nijenhuis torsion of $L$ is written in terms of $L_j^i$ and $\frac{\partial L_j^i}{\partial x^k}$, such as show eq. 5 https://arxiv.org/pdf/1405.5118.pdf