اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA curvature form for pseudoconnections
162
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We obtain the curvature form $F^ abla=Pcirc d^ ablacirc abla-d^ ablacirc Pcirc abla+d^ ablacirc ablacirc P$ for a vector bundle pseudoconnection $ abla$, where $d^ abla$ is the exterior derivative associated to $ abla$. We use $F^ abla$ to obtain the curvature of $ abla$. We also prove that $F^ abla=0$ is a necessary (but not sufficient) condition for $d^ abla$ to be a chain complex. Instead we prove that $F^ abla=0$ and $d^ ablacirc d^ ablacirc abla=0$ are necessary and sufficient conditions for $d^ abla$ to be a {em chain $2$-complex}, i.e., $d^ ablacirc d^ ablacirc d^ abla=0$.
قيم البحث
اقرأ أيضاً
In this note we explain how a flow in the space of Riemmanian metrics (including Riccis cite{mt}) induces one in the space of pseudoconnections.
Let $M$ be a compact $n$-manifold of $operatorname{Ric}_Mge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (
i) There is $rho>0$ such that for any $xin M$, the open $rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_rho,x^*)to (B_rho(x),x)$, has the maximal volume i.e., the volume of a $rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].
We establish a pointwise estimate of A along the mean curvature flow in terms of the initial geometry and the jHAj bound. As corollaries we obtain the extension theorem of HA and the blowup rate estimate of HA.
In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalised) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initia
l data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalises the theorem that spaces of the same curvature, dimension and metric tensor sign must be locally isometric to a class of spaces that have non-constant curvature. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields.
In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean mathbb{R}^{n+1} with speed u^alpha f^beta (alpha, betainmathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric
, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If alpha leq 0<betaleq 1-alpha, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
