Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and $$\mathcal {I}(X)$$
be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from $$\mathcal {L}(X)$$
, the subset $$\mathcal {I}(X)$$
is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $$\mathscr {G}(X)$$
, via the map $$\kappa $$
that sends $$Q\in \mathcal {I}(X)$$
to Q(X), such that the affine-Banach space structure on each fiber is the one induced from $$\mathcal {L}(X)$$
. Using this, we show that if H is a real or complex Hilbert space, then the assignment $$\begin{aligned} (E,T)\mapsto T^*\circ P_{E^\bot } + P_{E}, \quad \text {where}\quad E\in \mathscr {G}(H)\quad \text {and}\quad T\in \mathcal {L}(E,E^\bot ), \end{aligned}$$
induces a real bi-analytic bijection from the total space of the tangent bundle, $$\mathbf {T}(\mathscr {G}(H))$$
, of $$\mathscr {G}(H)$$
onto $$\mathcal {I}(H)$$
(here, $$E^\bot $$
is the orthogonal complement of E, $$P_E\in \mathcal {L}(H)$$
is the orthogonal projection onto E, and $$T^*$$
is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every $$E\in \mathscr {G}(H)$$
, we identify $$\mathcal {L}(E,E^\bot )$$
with a subspace of $$\mathcal {L}(H)$$
via the embedding $$S\mapsto S\circ P_E$$
, then the inclusion map from $$\mathbf {T}(\mathscr {G}(H))$$
to the trivial Banach bundle $$\mathscr {G}(H)\times \mathcal {L}(H)$$
is a real analytic immersion. Through this, we give a concrete idempotent in $$M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )$$
that represents the K-theory class of the tangent bundle $$\mathbf {T}(\mathscr {G}(\mathbb {K}^n))$$
, when $$\mathbb {K}$$
is either the real field or the complex field.