ترغب بنشر مسار تعليمي؟ اضغط هنا

Convolution of valuations on manifolds

228   0   0.0 ( 0 )
 نشر من قبل Andreas Bernig
 تاريخ النشر 2015
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J.~Fu and the second named author.



قيم البحث

اقرأ أيضاً

54 - Andreas Bernig 2016
We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formul as on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. $mathbb C^n$, it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature $4lambda$, and the dependence of the formulas on the parameter $lambda$. We will also describe Wannerers recent proof of local additive kinematic formulas for unitarily invariant area measures.
The dimensions of the spaces of $k$-homogeneous $mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation the ory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of $mathrm{Val}_2^{mathrm{Spin}(9)}$.
A compact complex manifold $V$ is called Vaisman if it admits an Hermitian metric which is conformal to a Kahler one, and a non-isometric conformal action by $mathbb C$. It is called quasi-regular if the $mathbb C$-action has closed orbits. In this c ase the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of $V$. It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kahler orbifold obtained as a quasi-regular quotient of $V$. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold $M$ is either infinite or equal to the sum of all Betti numbers of a Kahler orbifold obtained as an $S^1$-quotient of $M$.
115 - Qingsong Wang , Bo Yang , 2016
In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi -invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-Kahler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.
Given smooth manifolds $M_1,ldots, M_n$ (which may have a boundary or corners), a smooth manifold $N$ modeled on locally convex spaces and $alphain({mathbb N}_0cup{infty})^n$, we consider the set $C^alpha(M_1timescdotstimes M_n,N)$ of all mappings $f colon M_1timescdotstimes M_nto N$ which are $C^alpha$ in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $leq alpha_j$ in the $j$th variable for $jin{1,ldots, n}$, in local charts. We show that $C^alpha(M_1timescdotstimes M_n,N)$ admits a canonical smooth manifold structure whenever each $M_j$ is compact and $N$ admits a local addition. The case of non-compact domains is also considered.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا