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Development (differential geometry) connection form; Cartan connection. affine connection; conformal connection; projective connection; method of moving frames; Cartan's equivalence method; Vierbein, tetrad; Cartan connection applications; Einstein–Cartan theory; connection (vector bundle) connection (principal bundle) Ehresmann connection ...
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), [1] states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).
An invariant depending on two forms. transversal A line meeting several other lines. For example, 4 generic lines in projective 3-space have 2 transversals meeting all of them. triad A set of 3 points tricircular A tricircular curve is one that passes through the circular points at infinity with order 3. tricuspidal Having three cusps trigonal
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
The negativity of the canonical line bundle makes projective spaces prime examples of Fano varieties, equivalently, their anticanonical line bundle is ample (in fact very ample). Their index ( cf. Fano varieties ) is given by I n d ( P n ) = n + 1 {\displaystyle \mathrm {Ind} (\mathbb {P} ^{n})=n+1} , and, by a theorem of Kobayashi-Ochiai ...
In the above identity, α, β, δ line up throughout and γ occurs twice in one term due to a contraction (once as an upper index and once as a lower index), and thus it is a valid expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) and once in another term, which is ...
As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs). + = where is constant and is a function of and . We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.