In real-world applications, it is commonplace that the computational models have field responses, i.e., the temporal or spatial fields. It has become a critical task to develop global sensitivity analysis (GSA) methods to measure the effect of each input variable on the full-field. In this paper, a new sensitivity analysis method based on the manifold of feature covariance matrix (FCM) is developed for quantifying the impact of input variables on the field response. The method firstly performs feature extraction on the field response to obtain a low-dimensional FCM. An adaptive feature selection method is proposed to avoid the FCM from singularity. Thereby, the field response is represented by a FCM, which lies on a symmetric positive-definite matrix manifold. Then, the GSA technique based on the Cramér-von Mises distance for output valued on the Riemannian manifold is introduced for estimating the sensitivity indices for field response. An example of a temporal field and an example of a 2-D displacement field are introduced to demonstrate the applicability of the proposed method in estimating global sensitivity indices for field solution. Results show that the proposed method can distinguish the important input variables correctly and can yield robust index values. Besides, the proposed method can be implemented for GSA for field responses of different dimensionalities.