Particle filtering is currently a popular numerical method for solving stochastic filtering problems. This paper outlines its application to continuous time filtering problems with observations in a manifold. Such problems include a variety of important engineering situations and have received independent interest in the mathematical literature. The paper begins by stating a general stochastic filtering problem where the observation process, conditionally on the unknown signal, is an elliptic diffusion in a differentiable manifold. Using a geometric structure (a Riemannian metric and a connection) which is adapted to the observation model, it expresses the solution of this problem in the form of a Kallianpur-Striebel formula. The paper proposes a new particle filtering algorithm which implements this formula using sequential Monte Carlo strategy. This algorithm is based on an original use of the concept of connector map, which is here applied for the first time in the context of filtering problems. The paper proves the convergence of this algorithm, under the assumption that the underlying manifold is compact, and illustrates it with two worked examples. In the first example, the observations lie in the special orthogonal group SO(3). The second example is concerned with the case of observations in the unit sphere S².