Search results for "Mathematics::Differential Geometry"
showing 10 items of 209 documents
Nonlocal Isoperimetric Inequality
2019
For the nonlocal perimeter, there is also an isoperimetric inequality, and here the main hypothesis used on J is that it is radially nonincreasing.
On the history of Levi-Civita's parallel transport
2016
In this historical note, we wish to highlight the crucial conceptual role played by the principle of virtual work of analytical mechanics, in working out the fundamental notion of parallel transport on a Riemannian manifold, which opened the way to the theory of connections and gauge theories. Moreover, after a detailed historical-technical reconstruction of the original Levi-Civita's argument, a further historiographical deepening and a related critical discussion of the question, are pursued.
Umbilicity of space-like submanifolds of Minkowski space
2004
We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3…
Generalized curvature and the equations of D=11 supergravity
2005
It is known that, for zero fermionic sector, the bosonic equations of Cremmer-Julia-Scherk eleven-dimensional supergravity can be collected in a compact expression which is a condition on the curvature of the generalized connection. Here we peresent the equation which collects all the bosonic equations of D=11 supergravity when the gravitino is nonvanishing.
Higher Order Integrability in Generalized Holonomy
2004
Supersymmetric backgrounds in M-theory often involve four-form flux in addition to pure geometry. In such cases, the classification of supersymmetric vacua involves the notion of generalized holonomy taking values in SL(32,R), the Clifford group for eleven-dimensional spinors. Although previous investigations of generalized holonomy have focused on the curvature \Rm_{MN}(\Omega) of the generalized SL(32,R) connection \Omega_M, we demonstrate that this local information is incomplete, and that satisfying the higher order integrability conditions is an essential feature of generalized holonomy. We also show that, while this result differs from the case of ordinary Riemannian holonomy, it is n…
A Remark on an Overdetermined Problem in Riemannian Geometry
2016
Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let \(\varOmega \subset M\) be a bounded domain, with \(O \in \varOmega \), and consider the problem \(\varDelta _p u = -1\ \mathrm{in}\ \varOmega \) with \(u=0\ \mathrm{on}\ \partial \varOmega \), where \(\varDelta _p\) is the p-Laplacian of g. We prove that if the normal derivative \(\partial _{\nu }u\) of u along the boundary of \(\varOmega \) is a function of d satisfying suitable conditions, then \(\varOmega \) must be a geodesic ball. In particular, our result applies to open balls of \(\mathbb {R}^n\) equipped with a rotationally symmetric metr…
The c-map on groups
2019
We study the projective special Kaehler condition on groups, providing an intrinsic definition of homogeneous projective special Kaehler that includes the previously known examples. We give intrinsic defining equations that may be used without resorting to computations in the special cone, and emphasise certain associated integrability equations. The definition is shown to have the property that the image of such structures under the c-map is necessarily a left-invariant quaternionic Kaehler structure on a Lie group.
A Rainich-like approach to the Killing-Yano tensors
2002
The Rainich problem for the Killing-Yano tensors posed by Collinson \cite{col} is solved. In intermediate steps, we first obtain the necessary and sufficient conditions for a 2+2 almost-product structure to determine the principal 2--planes of a skew-symmetric Killing-Yano tensor and then we give the additional conditions on a symmetric Killing tensor for it to be the square of a Killing-Yano tensor.We also analyze a similar problem for the conformal Killing-Yano and the conformal Killing tensors. Our results show that, in both cases, the principal 2--planes define a maxwellian structure. The associated Maxwell fields are obtained and we outline how this approach is of interest in studying …
Homogeneous three-dimensional Riemannian spaces
2020
The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a transitive group of isometries are obtained. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic, and they offer an IDEAL labeling of these geometries. It is shown that the transitive action of the group naturally falls into an unfolding of some of the ten types in the Bianchi-Behr classification. Explicit conditions, depending on the Ricci tensor, are obtained that characterize all these types.
Noncompact Topological Quantum Groups
1995
A star-product construction of quantum semisimple real Lie groups is performed for the noncompact case.