We define the notion of the extrinsic Itˆo projection of astochastic differential equation (SDE) on a submanifold. This allows oneto systematically develop low dimensional approximations to high dimensionalSDEs in a differential geometric setting. We consider the exampleof approximating the non-linear filtering problem with a Gaussian distributionand show how the Itˆo projection leads to improved approximationsin the Gaussian family. We briefly discuss the approximations formore general families of distribution. We perform a numerical comparisonof our projection filters with the classical Extended Kalman Filterto demonstrate the efficacy of the approach.