Methods operating on Vision Transformer (ViT) feature spaces typically rely on Euclidean distance or cosine similarity. This assumes that every direction is equally meaningful, but there is no reason to believe the true task geometry has this property. The task-sensitive geometry of the feature space is given by the pullback metric
g(F)=J(F)⊤J(F), where
J is the Jacobian of the decoder's output fed to a task-specific distance, with respect to the features. Storing the full
g is infeasible at modern scales, and for dense outputs such as depth maps even forming
J is impractical. We show that whether a low-rank approximation of this metric can be learned depends on the model-decoder pair, and we characterize this with a matrix-free diagnostic
κcap(r) computable with a low number of Jacobian-vector products. For tractable pairs, we develop the Spectral Pullback Network (SPN), which learns a low-rank version of the metric from randomized power iteration, and we distill it into a
310K-parameter importance head that predicts token importance directly from the features. When the Jacobian spectrum is too spread out for a low-rank approximation, passing the decoder's input features through a VAE bottleneck can restore tractability. Across DPT, DINOv2, CLIP, and VGGT backbones,
κcap(r) predicts which learned-metric architectures are viable. The importance head reaches Spearman
ρ=0.998 on DINOv2 CLS, and our geometric token pruning reduces the additional depth error of ToMe-based token selection by
25% on DPT depth at prune ratio
0.5, without fine-tuning the ViT. Project page: https://cyberiada.github.io/TaskInducedViTs/