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The Gaussian measure on algebraic varieties

Ilka Agricola, Thomas Friedrich (1999)

Fundamenta Mathematicae

We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety M n is dense in the Hilbert space L 2 ( M , e - | x | 2 d μ ) , where dμ denotes the volume form of M and d ν = e - | x | 2 d μ the Gaussian measure on M.

The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant

Xavier Gómez-Mont, Pavao Mardešić (1997)

Annales de l'institut Fourier

Given a real analytic vector field tangent to a hypersurface V with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra B Ann B ( h ) associated with the singularity of the vector field on V . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...

The Laplace-Beltrami operator in almost-Riemannian Geometry

Ugo Boscain, Camille Laurent (2013)

Annales de l’institut Fourier

We study the Laplace-Beltrami operator of generalized Riemannian structures on orientable surfaces for which a local orthonormal frame is given by a pair of vector fields that can become collinear.Under the assumption that the structure is 2-step Lie bracket generating, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence, a quantum particle cannot cross the singular set (i.e., the set where the vector fields become collinear) and the...

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