A Short Course in Differential Geometry and Topology by A. T. Fomenko

By A. T. Fomenko

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If r < min(m, n) at a point x, then the point x is called a singular point of the mapping f . The following relations can exist between the dimensions m and n: a) m < n. In this case, in a neighborhood Ua of a regular point a, a mapping f is injective. If a ∈ M is a regular point of the mapping f , then b = f (a) is a regular point of the submanifold V = f (M ) ⊂ N , and the map f carries a sufficiently small neighborhood of the point a into a spherical neighborhood of the point b = f (a). Moreover, the tangent subspace Tb (V ) at a regular point b is an m-dimensional subspace of the tangent subspace Tb (N ) whose dimension is equal to n.

Then we may choose the points of a projective frame {Au } in such a way that they form an autopolar simplex with respect to this hyperquadric, and we normalize the vertices of this simplex. 71) satisfy the equations ωuv + ωvu = 0. 92) The elliptic transformations are those projective transformations of the space Pn that preserve the hyperquadric Q. These transformations depend on 1 2 n(n + 1) parameters, and the latter number coincides with the number of independent forms among the forms ωvu . If the hyperquadric Q is of signature (1, n), then it defines the hyperbolic geometry in Pn , which is also called the Lobachevsky geometry.

N, we complete ω 1 , . . , ω p to a basis for T ∗ . Then θa = lab ω b + laξ ω ξ . 32), we obtain lab ω a ∧ ω b + ω a ∧ laξ ω ξ = 0, which implies laξ = 0 and lab = lba . Cartan’s lemma is of pure algebraic nature. But if the forms ω a and θa are given on a differentiable manifold M , then Cartan’s lemma is also valid, and the quantities lab are smooth functions on M . In the algebra of differential forms, another operation—the exterior differentiation—can be defined. , exterior forms of degree zero, this operation coincides with ordinary differentiation, and for exterior forms of type θ = adxi1 ∧ .

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