<p>In a previous work, the authors introduced the notion of 'coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss–Bonnet formulas on coherent tangent bundles on 2-dimensional manifolds were proven, and several applications to surface theory were given.</p><p>Let Mn (n ≥ 2) be an oriented compact n-manifold without boundary and TMn its tangent bundle. Let ℰ be a vector bundle of rank n over Mn, and φ: TMn → ℰ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss–Bonnet formulas can be generalized to an index formula for the bundle homomorphism φ under the assumption that φ admits only certain kinds of generic singularities.</p><p>We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.</p>
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