In mathematics, the Grassmannian ${\displaystyle \mathbf {Gr} _{k}(V)}$ (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all ${\displaystyle k}$-dimensional linear subspaces of an ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$ over a field ${\displaystyle K}$ that has a differentiable structure. For example, the Grassmannian ${\displaystyle \mathbf {Gr} _{1}(V)}$ is the space of lines through the origin in ${\displaystyle V}$, so it is the same as the projective space ${\displaystyle \mathbf {P} (V)}$ of one dimension lower than ${\displaystyle V}$. When ${\displaystyle V}$ is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension ${\displaystyle k(n-k)}$. In general they have the structure of a nonsingular projective algebraic variety.

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to ${\displaystyle \mathbf {Gr} _{2}(\mathbf {R} ^{4})}$, parameterizing them by what are now called Plücker coordinates. (See § Plücker coordinates and Plücker relations below.) Hermann Grassmann later introduced the concept in general.

Notations for Grassmannians vary between authors; they include ${\displaystyle \mathbf {Gr} _{k}(V)}$, ${\displaystyle \mathbf {Gr} (k,V)}$,${\displaystyle \mathbf {Gr} _{k}(n)}$, ${\displaystyle \mathbf {Gr} (k,n)}$ to denote the Grassmannian of ${\displaystyle k}$-dimensional subspaces of an ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$.

By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differentiable manifold, one can talk about smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space. Suppose we have a manifold ${\displaystyle M}$ of dimension ${\displaystyle k}$ embedded in ${\displaystyle \mathbf {R} ^{n}}$. At each point ${\displaystyle x\in M}$, the tangent space to ${\displaystyle M}$ can be considered as a subspace of the tangent space of ${\displaystyle \mathbf {R} ^{n}}$, which is also just ${\displaystyle \mathbf {R} ^{n}}$. The map assigning to ${\displaystyle x}$ its tangent space defines a map from M to ${\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}$. (In order to do this, we have to translate the tangent space at each ${\displaystyle x\in M}$ so that it passes through the origin rather than ${\displaystyle x}$, and hence defines a ${\displaystyle k}$-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This can with some effort be extended to all vector bundles over a manifold ${\displaystyle M}$, so that every vector bundle generates a continuous map from ${\displaystyle M}$ to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.

For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-space, so it is the same as the projective space ${\displaystyle \mathbf {P} ^{n-1}}$of n − 1 dimensions.

For k = 2, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces Gr(2, 3), Gr(1, 3), and P² (the projective plane) may all be identified with each other.