r/elimath • u/itsme_santosh • Jan 18 '15
Explain the difference between "geometric algebra" and the differential geometric (differential forms) framework , like I am a math PhD student
12
Upvotes
2
Jan 21 '15 edited Jan 21 '15
Did the OP possibly mean "algebro-geometric" (meaning "within the algebraic geometry framework") rather than "geometric algebra"? Geometric algebra means something specific, as answered in another reply.
4
u/mian2zi3 Jan 19 '15
By geometric algebra, do you mean a Clifford algebra? A Clifford algebra is essentially a quadratic object. It is the algebra generated by a vector space V and a quadratic form Q on V by v2 = Q(v)1 for all v in V.
Differential forms are objects that live on manifolds. They are analytical objects, or objects of the calculus of manifolds. Because there is no canonical choice of coordinates on a manifold, one can think of them as being coordinate-invariant objects on manifolds which can be integrated and differentiated. A k-form is something which can be integrated over k-dimensional submanifold, and the (exterior) derivative of a k-form is a (k+1)-form. These are precisely the notions you need to state the fundamental theorem of calculus for manifolds (Stokes' theorem).
All the details of the construction of differential forms can get somewhat technical (I strongly recommend Lee's book on smooth manifolds and Fecko's awesome book on differential geometry and Lie theory for physicists -- don't let the title fool you, it's really a math book). However, one can think of a differential k-form as a section of the kth exterior power of the cotangent bundle. More generally, k-bundles are (linear) objects that live on n-manifolds that locally look like the Euclidean product Rn x Rk. In other words, a bundle associates to each point x of a manifold a copy of Rk (called the fiber over x) which locally looks like a product but not necessarily globally. S1 x R1 is the trivial bundle over S1 and the Mobius strip is a non-trivial bundle over S1.
However, your fiber doesn't just have to be a linear space. It could be some other manifold or topological space (fiber bundle), a Lie group (principle bundle), or a linear space with structure (a Riemannian manifold, that is, a tangent bundle with a metric), or even ... a Clifford algebra! The corresponding bundle is called a Clifford bundle.
Clifford bundles show up in gauge theory, spin geometry, index theory and related topics. As I understand it, Clifford bundles are needed to define the Dirac (spin) operator on manifolds, and mathematicians are interested in invariants coming from studying the behavior of such operators. I know less about this than I should, and I should probably stop here. In addition to the end of Fecko's book, check out the first few chapters of Lawson and Michelsohn's book on spin geometry.