Tangent space of a Grassmannian - Imported Knowlpedia Content

Statement

Let $V$ be a finite-dimensional real or complex vector space, and let $W$ be a point of the Grassmannian $\operatorname{Gr}_k(V)$ . Its tangent space has a canonical linear identification

T_W\operatorname{Gr}_k(V)\cong\operatorname{Hom}(W,V/W).

In the complex case this is an isomorphism of complex vector spaces. Explicitly, represent a tangent vector by a smooth curve $W(t)$ of $k$ -planes with $W(0)=W$ . For $w\in W$ , choose a smooth curve $w(t)\in W(t)$ with $w(0)=w$ ; the tangent vector sends $w$ to the class of $w'(0)$ in $V/W$ . This class is independent of all choices Harris, Lecture 6.

Derivation from graph charts

Choose a complementary subspace $U$ with $V=W\oplus U$ . Every plane sufficiently near $W$ is the graph of a unique linear map $A:W\to U$ . This chart identifies the tangent space at $W$ with $\operatorname{Hom}(W,U)$ . The quotient projection restricts to an isomorphism $U\to V/W$ , producing the displayed identification. Although the chart used $U$ , the quotient-space description does not, which proves canonicity.

Naturality

If $F:V\to V'$ is a linear isomorphism, its action on Grassmannians sends $W$ to $F(W)$ . Under the canonical tangent-space identifications, its differential sends $\phi:W\to V/W$ to

F_{\mathrm{quot}}\circ\phi\circ(F|_W)^{-1},

where $F_{\mathrm{quot}}:V/W\to V'/F(W)$ is the induced quotient map. Thus the identification respects changes of coordinates and group actions.

Consequences and examples

The dimension follows immediately:

\dim_{\mathbb F}T_W\operatorname{Gr}_k(V)=k(\dim_{\mathbb F}V-k).

Using the dual space, the same result can be written $T_W\operatorname{Gr}_k(V)\cong W^*\otimes(V/W)$ . For $k=1$ , it gives $T_L\mathbb P(V)\cong\operatorname{Hom}(L,V/L)$ . If an inner product identifies $V/W$ with the orthogonal complement $W^\perp$ , one obtains the convenient but noncanonical model $\operatorname{Hom}(W,W^\perp)$ . The quotient model remains valid without an inner product.

References

Joe Harris, Algebraic Geometry: A First Course, Springer, 1992. Springer DOI record. Relevant: Lecture 6, Grassmannians and their tangent spaces.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. Publisher DOI record. Relevant: §§5–6, Grassmann manifolds and their local structure.