A study of backward stochastic differential equation on a Riemannian manifold

Suppose $N$ is a compact Riemannian manifold, in this paper we will introduce the definition of $N$-valued BSDE and $L^2(\mathbb{T}^m;N)$-valued BSDE for which the solution are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to $L^2(\mathbb{T}^m;N)$-valued BSDE will be proved without any convexity condition on $N$.