Let $f: X \to Y$ be a separated morphism between $k$-varieties or more general schemes of finite type. The most common way in standard literature on algebraic geometry to define the sheaf of relative Kähler differentials $\Omega_{X/Y}$ is to observe that the diagonal map $\Delta: X \to X \times_Y X$ is a closed embedding (we assume $f$ separated) and let $I \subset O_{X \times_Y X}$ ideal sheaf define image $\Omega(X)$. The sheaf of relative Kähler differentials is defined as

$$ \Omega_{X/Y}:= \Delta^* (I/I^2) $$

and I'm interested in the geometric motivation behind this definition. As is so often the case, the origins are in differential geometry.

Let $f: X \to Y$ a equidimensional surjective map between connected $k$-manifolds $Y, X$ for $k= \mathbb{R}, \mathbb{C}$ and we moreover assume that every fiber $F:= f^{-1}(y) \subset X$ for $y \in Y$ is also a connected submanifold of same dimension. Most natural examples: a vector bundle $X$ over a manifold $Y$ or a fibration with 'nice' behavior on fibers. We obtain an exact sequence of tangent spaces

$$ 0 \to T_{X/Y} \to T_X \to f^*T_Y \to 0 $$

where $T_{X/Y}$ is the kernel of induced map of tangent bundles. intuitively, for every $x \in f^{-1}(y)$, $(T_{X/Y})_x$ is the tangent space of the fiber at $x$. The relative space of Kähler differentials $\Omega_{X/Y}$ is defined as the dual of $T_{X/Y}$ and sits in the sequence which we obtain if we dualize the previous sequence above of tangent spaces:

$$ 0 \to f^*\Omega_Y \to \Omega_X \to \Omega_{X/Y} \to 0 $$

Now let us translate the first definition from modern algebraic geometry also to framework of differential geometry:

Let $f: X \to Y$ as above surjective map between connected manifolds with equidim submanifold fibers and now let us embed $X$ via diagonal map $\Delta: X \to X \times_Y X$ into the fiber product (a problem: does the fiber product exist as manifold of $X,Y$ nice enough?).

Since $\Delta(X) \subset X \times_Y X$ is a smooth submanifold, we can pick local coordinates $( x_1 , ... , x_n )$ around a $x \in \Delta(X)$ such that $ \Delta(X) $ is locally defined by $x_{k + 1} = ... = x_n = 0$; then with this choice of coordinates

$$T_x X \times_Y X = k{\frac{\partial}{\partial x_1} \vert _x,... \frac{\partial}{\partial x_n} \vert _x }$$

$$T_x \Delta(X) = k{\frac{\partial}{\partial x_1} \vert _x,... \frac{\partial}{\partial x_k} \vert _x }$$

$$(N_{X \times_Y X/\Delta(X)})_x = k{\frac{\partial}{\partial x_{k+1}} \vert _x,... \frac{\partial}{\partial x_n} \vert _x }$$

and the ideal sheaf is locally generated by $x_{k+1}, ..., x_n $. The bundle $N_{X \times_Y X/\Delta(X)}$ is the normal bundle of embedding $\Delta$ and fits in following canonical sequence:

$$0 \to T_{\Delta(X)} \to T_{X \times_Y X} \to N_{X \times_Y X/X} \to 0$$

Also, it is well known that the pairing

$$(I/I^2)_x \times (N_{X \times_Y X/\Delta(X)})_x \to k$$

is perfect and therefore $I/I^2 \cong (N_{X \times_Y X/\Delta(X)})^{\vee}$. As pointed out above in algebraic geometry we define sheaf (or bundle in more old fashioned language) of relative Kähler differentials $\Omega_{X/Y}$ as $ \Omega_{X/Y}:= \Delta^* (I/I^2) $.

Therefore it is reasonable to conjecture although I nowhere found a proof that if there is any justice n this world then these two definitions of relative differential bundles should coinside in the setting of differential geometry. That is for $f: X \to Y$ surjective map between connected manifolds with equidim submanifold fibers, we should have

$$T_{X/Y} \cong \Delta^* N_{X \times_Y X/\Delta(X)}$$

Can we write down an explicit isomorphism and understand what is geometrically going on there? I would like also remark that I asked almost identical question in MathStackExchange

and suppose that possibly MO might be a better place to ask about it.