Universal geometric algebra
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A vector manifold is a set of "points" (abstract vectors endowed with an algebraic structure) such that at each point there "is" a pseudoscalar of definite dimension. This (unit) pseudoscalar is a function of the points "on" the vector manifold. The dimension of the pseudoscalar is the dimension of the manifold. The algebraic structure which contains these vectors and pseudoscalars is called Universal Geometric Algebra (UGA).[1]
Background
To better introduce the notion of vector manifold one must first understand what UGA is.
UGA is the Geometric Algebra generated by an infinite dimensional vector space with nondegenerate signature (p,q), Vn, with p+q=n→∞. The vectors in this space generate the algebra by the definition of a geometric product. This product makes the manipulation of vectors more similar to that of scalars. One of the main (and very important) differences between the multiplication of vectros and scalars is that it is noncommutative. From UGA one can generate all finite dimensional Geometric Algebras (GA).
Since a Geometric Algebra is a type of abstract algebra one can place many interpretations on it. Among these are two very important interpretations: the metrical and the projective. For example the algebra G(R3) can be interpreted metrically in such a way that a vector corresponds to an equivalence class of oriented line segments. It can also be interpreted projectively as a point in the real projective plane. Metrically a bivector(see below) is an equivalence class of oriented plane segments of R3 while projectively it is a line in P_2(R). All elements of GA can be interpreted geometrically in one way or another, so can operations in GA. The duality operation in GA is the multiplication by the (unit) pseudoscalar (or it's inverse). Metrically duality can be interpreted as is done in differential forms by the Hodge mapping. Alternatively one can interpret it as the duality operation of projective geometry.
The elements of UGA are called multivectors. Every multivector can be written as the sum of various r-vectors. Some special r-vectors are scalars (r=0), vectors (r=1) and bivectors (r=2). Scalars are identical to the real numbers. Complex number are not used as scalars because there exists structures in UGA which are isomorphic to the complex numbers so they need not be invoked. In this article we refer to various Geometric Algebras but always use upper case letters, when lower case letters are used they are referring to something else. Given a geometry, a geometric algebra (if it exists) is an abstarct algebra in which every expression has a clear geometric meaning. Meaning that every expresion of the algebra represents a geometrical object, or a relaion between geometric objects. So GA is a specific type of geometric algebra, it is the algebra of orthogonal geometry. The geometric algebra of Euclidean geometry is called Conformal Geometric Algebra.[2]
The way one generates a finite dimensional GA is by the election of a unit pseudoscalar (I). The set of all vectors for which satisfy
is a vector space. The dimension of the pseudoscalar is the dimension of the vector space. The geometric product of the vectors in this vector space then defines the GA -of which I is a member. Since every finite dimensional GA has a unique I one can define or characterize the GA by it.
Definition
A vector manifold is defined similar to how a GA can b e defined, by its unit pseudoscalar. Consider a set of vectors in UGA, addition and multiplication are not closed within the set. This set is NOT a vector space. Call the vectors "points on the vector manifold". At every point "place" a unit pseudoscalar, all of the same dimension. The dimension of the vector manifold is the dimension chosen for these unit pseudoscalars. The unit pseudoscalar of the vector manifold is a (pseudoscalar valued-)function of the points on the vector manifold. This function along with its domain and range define the vector manifold. If i.e. this function is smooth then so is the vector manifold[3]. A manifold can be defined as a set isomorphic to a vector manifold. The points of a manifold do not have any algebraic structure. The differential geometry of a manifold can be carried out in a vector manifold which is algebraically rich. All quantities relevant to differential geometry can be calculated from the pseudoscalar of the vector manifold. This is the original motivation behind it's definition. Vector manifolds allow the differential geometry of manifolds without the introduction of other structures such as metrics, connections, fiber bundles, etc. The relevant structure of a vector manifold is it's tangent algebra.
References
- David Hestenes, Garrett Sobczyk (Authors), Clifford Algebra to Geometric Calculus: a Unified Language for mathematics and Physics Publisher: Springer, ISBN-10: 9027725616 ISBN-13: 978-9027725615
- Chris Doran and Anthony Lasenby (Authors), Geometric Algebra for Physicists, section 6.5 Embedded Surfaces and Vector Manifolds Publisher: Cambridge University Press, ISBN-10: 0521715954, ISBN-13: 978-0521715959
- Leo Dorst and Joan Lasenby (Editors), Guide to Geometric Algebra in Practice, Chapter 19, Publisher: Springer; 2011 edition (August 31, 2011), ISBN-10: 0857298100 ISBN-13: 978-0857298102
- Hongbo Li (Author), Invariant Algebras And Geometric Reasoning Publisher: World Scientific, ISBN-10: 9812708081 ISBN-13: 978-9812708083