Is there Geometric Interpretation of Spinors? Usually in Physics we define a spinor to be an element of the $\left(\frac{1}{2},0\right)$ representation space of the Lorentz group. Essentially this boils down to the 'n-tuple of numbers that transforms like a spinor' definition that physicists tend to use for vectors, covectors, and tensors.
However, vectors, covectors, and tensors also have geometric definitions that are much nicer than, and also equivalent to, the 'n-tuple of numbers' definition. For example, a vector can be thought of as an equivalence class of curves tangent at a point, or the directional derivative at a point. A covector can be thought of as a differential 1-form or as an equivalence class of functions with equal gradient at a point. Tensors are then tensor products of these spaces.
I was wondering if there is a similar definition of spinors based in differential geometry rather than just the representation theory of the Lorentz group. If so, are these specific to certain manifolds (complex, Lorentzian, etc), or are they general to all manifolds?
 A: Spinors can be represented in geometric algebra as even multivectors, that is, even-graded elements of a geometric algebra.
A geometric algebra is like an exterior algebra (indeed, in a geometric algebra you can and usually do define an exterior product as well) but with a different (still associative) product between its elements: the geometric product. On vectors specifically, the geometric product is defined so that $v^2=\langle v,v\rangle$, where $\langle\cdot ,\cdot\rangle$ is the scalar product between vectors. So a geometric algebra is a Clifford algebra from another perspective.
Even-graded elements of a geometric algebra are those that are formed from products of an even number of vectors and the sums of such products. These even multivectors can act on vectors by two-sided multiplication to rotate and scale the vector (in the same way that quaternions are used to rotate vectors):
$$v'=\overline\psi v \psi$$
($\overline\psi$ is the reverse of the multivector $\psi$: just reversing the order of all factors in the geometric products that make up $\psi$.)
This is why spinors are sometimes called the "square roots" of vectors: because by two-sided multiplication, a spinor can transform a given reference vector into another vector.
So a spinor can essentially be thought of as a transformation that rotates and scales a vector. (In spacetime contexts this is a Lorentz rotation, so a spatial rotation plus a boost.) This is the most intuitive way of thinking about spinors that I have been able to find.
In differential geometry, this means that spinors are even multivectors in the geometric algebra formed from the tangent space of the manifold, and the bilinear covariants of the spinors are the results of the spinors acting by two-sided multiplication on vectors of a reference orthonormal frame on the manifold.
A little more info can be found here and here.
