1,675 research outputs found
Intersection theory and the Alesker product
Alesker has introduced the space of {\it smooth
valuations} on a smooth manifold , and shown that it admits a natural
commutative multiplication. Although Alesker's original construction is highly
technical, from a moral perspective this product is simply an artifact of the
operation of intersection of two sets. Subsequently Alesker and Bernig gave an
expression for the product in terms of differential forms. We show how the
Alesker-Bernig formula arises naturally from the intersection interpretation,
and apply this insight to give a new formula for the product of a general
valuation with a valuation that is expressed in terms of intersections with a
sufficiently rich family of smooth polyhedra.Comment: further revisons, now 23 page
Convolution of convex valuations
We show that the natural "convolution" on the space of smooth, even,
translation-invariant convex valuations on a euclidean space , obtained by
intertwining the product and the duality transform of S. Alesker, may be
expressed in terms of Minkowski sum. Furthermore the resulting product extends
naturally to odd valuations as well. Based on this technical result we give an
application to integral geometry, generalizing Hadwiger's additive kinematic
formula for to general compact groups acting
transitively on the sphere: it turns out that these formulas are in a natural
sense dual to the usual (intersection) kinematic formulas.Comment: 18 pages; Thm. 1.4. added; references updated; other minor changes;
to appear in Geom. Dedicat
Riemannian curvature measures
A famous theorem of Weyl states that if is a compact submanifold of
euclidean space, then the volumes of small tubes about are given by a
polynomial in the radius , with coefficients that are expressible as
integrals of certain scalar invariants of the curvature tensor of with
respect to the induced metric. It is natural to interpret this phenomenon in
terms of curvature measures and smooth valuations, in the sense of Alesker,
canonically associated to the Riemannian structure of . This perspective
yields a fundamental new structure in Riemannian geometry, in the form of a
certain abstract module over the polynomial algebra that
reflects the behavior of Alesker multiplication. This module encodes a key
piece of the array of kinematic formulas of any Riemannian manifold on which a
group of isometries acts transitively on the sphere bundle. We illustrate this
principle in precise terms in the case where is a complex space form.Comment: Corrected version, to appear in GAF
Symmetric Criticality for Tight Knots
We prove a version of symmetric criticality for ropelength-critical knots.
Our theorem implies that a knot or link with a symmetric representative has a
ropelength-critical configuration with the same symmetry. We use this to
construct new examples of ropelength critical configurations for knots and
links which are different from the ropelength minima for these knot and link
types.Comment: This version adds references, and most importantly an
acknowledgements section which should have been in the original postin
Circles Minimize most Knot Energies
We define a new class of knot energies (known as renormalization energies)
and prove that a broad class of these energies are uniquely minimized by the
round circle. Most of O'Hara's knot energies belong to this class. This proves
two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies
not minimized by a round circle. The proof is based on a theorem of G. Luko on
average chord lengths of closed curves.Comment: 15 pages with 3 figures. See also http://www.math.sc.edu/~howard
Integral geometry of complex space forms
We show how Alesker's theory of valuations on manifolds gives rise to an
algebraic picture of the integral geometry of any Riemannian isotropic space.
We then apply this method to give a thorough account of the integral geometry
of the complex space forms, i.e. complex projective space, complex hyperbolic
space and complex euclidean space. In particular, we compute the family of
kinematic formulas for invariant valuations and invariant curvature measures in
these spaces. In addition to new and more efficient framings of the tube
formulas of Gray and the kinematic formulas of Shifrin, this approach yields a
new formula expressing the volumes of the tubes about a totally real
submanifold in terms of its intrinsic Riemannian structure. We also show by
direct calculation that the Lipschitz-Killing valuations stabilize the subspace
of invariant angular curvature measures, suggesting the possibility that a
similar phenomenon holds for all Riemannian manifolds. We conclude with a
number of open questions and conjectures.Comment: 68 pages; minor change
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