# Definition:Affine Subspace

## Definition

Let $\mathcal E$ be an affine space with tangent space $E$.

Let $\mathcal F \subseteq \mathcal E$ be a subset of $\mathcal E$.

Then $\mathcal F$ is an affine subspace of $\mathcal E$ if and only if there exists a point $p \in \mathcal E$ such that:

$F_p := \set {q - p: q \in \mathcal F}$

is a vector subspace of the vector space $E$.

## Also known as

Some sources give this as affine manifold.