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A hypersphere S^n is parallelizable if there are n vector fields that are linearly independent at each point. There exist only three parallelizable spheres: S^1, S^3, and S^7 ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...

The notion of parallel transport on a manifold M makes precise the idea of translating a vector field V along a differentiable curve to attain a new vector field V^' which is ...

An orientation on an n-dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an ...

From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In ...

The plane spanned by the normal vector N and the binormal vector B.

The plane spanned by the tangent vector T and binormal vector B.

Let M^n be a compact n-dimensional oriented Riemannian manifold without boundary, let O be a group representation of pi_1(M) by orthogonal matrices, and let E(O) be the ...

A branch of mathematics which brings together ideas from algebraic geometry, linear algebra, and number theory. In general, there are two main types of K-theory: topological ...

The infinitesimal algebraic object associated with a Lie groupoid. A Lie algebroid over a manifold B is a vector bundle A over B with a Lie algebra structure [,] (Lie ...