Левоинвариантная параконтактная метрическая структура на группе Sol

Among the eight three-dimensional Thurston geometries, there is the Heisenberg group, the nilpotent Lie group of real 3x3 matrices of a special form. It is known that this group has a left-invariant Sasakian structure. This article proves that there is also a paracontact metric structure on the Heisenberg group, which is also Sasakian. This group has a unique contact metric connection with skew-symmetric torsion, which is invariant under the group of automorphisms of the para-Sasakian structure. The discovered connection is proved to be a contact metric connection for any para-Sasakian structure. The concept of a connection compatible with the distribution is introduced. It is found that the Levi-Civita connection and the contact metric connection on the Heisenberg group endowed with a para-Sasakian structure are compatible with the contact distribution. Their orthogonal projections on this distribution determine the same truncated connection. It is proved that Levi-Civita contact geodesics and truncated geodesics coincide. It is found that contact geodesics are either straight lines lying in the contact planes or parabolas the orthogonal projections of which on the contact planes are straight lines. The results obtained in this article are also valid for the multidimensional Heisenberg group. AMS Mathematical Subject Classification: 53D10, 53C50

Descent methods for optimization on homogeneous manifolds

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\Mc$ is the Maurer-Cartan moduli space of the graded differential Lie algebra $\Omega^\ast (M, \adp)$, the differential forms with values in the associated adjoint bundle of $P$. For a 2-dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman in \cite{G2}. We treat different Lie algebra structures on $\H_{2\ast}(LM)$ depending on the choice of the linear reductive Lie group $G$ in our discussion.

Flat connections on a punctured sphere and geodesic polygons in a Lie group

We invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H*dR(G) itself. Then, we show that when a flat connection A exists on a principal G-bundle P, we may construct a homomorphism EA: H*dR(G)→H*dR(P), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we show that that both the flat principal bundle and its adjoint bundle have the real coefficient cohomology isomorphic to that of the trivial bundle.

We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive definite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups, and describe the equivalence classes in terms of some basic isometric invariants. The classification naturally splits into two cases. In the first case, it reduces to a classification of left-invariant sub-Riemannian structures. In the second case, we find a canonical frame with which to directly compare equivalence classes.

Let $E$ be a flat complex vector bundle over a closed oriented odd dimensional manifold $M$ endowed with a flat connection $\nabla$. The refined analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex with an odd degree closed differential form $H$, other than one form, as a flux and with coefficients in $E$. In this paper we generalize the construction of the refined analytic torsion to the twisted de Rham complex. We show that the refined analytic torsion of the twisted de Rham complex is independent of the choice of the Riemannian metric on $M$ and the Hermitian metric on $E$. We also show that the twisted refined analytic torsion is invariant (under a natural identification) if $H$ is deformed within its cohomology class. We prove a duality theorem, establishing a relationship between the twisted refined analytic torsion corresponding to a flat connection and its dual. We also define the twisted analogue of the Ray-Singer metric and calculate the twisted Ray-Singer metric of the twisted refined analytic torsion. In particular we show that in case that the Hermtitian connection is flat, the twisted refined analytic torsion is an element with the twisted Ray-Singer norm one.

The classical relation between two-dimensional spaces of constant curvature and certain nonlinear partial differential equations is formulated in group-theoretic terms by means of the underlying semisimple isometry group. Rather than working with the metric and curvature in the given constant curvature space, it is then possible to consider the equivalent system consisting of a pair of first-order partial differential equations in flat space for two so(2,1) vectors satisfying a pair of SO(2,1)-invariant algebraic constraints. Such equations determine a SL(2,R) principal bundle with flat connection. The construction is carried out in detail for the case of the sine-Gordon equation. The connection is shown to give rise to a spectral problem of the inverse scattering type by means of a gauge transformation. Bäcklund transformations are shown to be automorphisms of the connection characterized by a certain so(2,1) null vector. The generation of solutions of the nonlinear equation from known ones is seen to be determined by gauge transformations leaving invariant a fixed set of gauge conditions.

This is qualitative research about the didactical design of distance concept in solid geometry, aiming to develop mathematical representation ability. This research employs a qualitative method with didactical design research (DDR) consisting of three stages of the didactical situation: didactical and pedagogical anticipation, metapedadidactical analysis, and retrospective analysis. Hypothetical didactical design was arranged by learning obstacle and hypothetical learning trajectory with the theory of didactical situations. The research subjects were three students of State Vocational High School in Garut as representatives of 30 students who had experience learning Distance Concepts in Solid Geometry and had been given an identification test for learning obstacles. The study identified that learning obstacles are the didactic and epistemological obstacle, which covered five inaccuracies in (1) Students’ ability in presenting concepts in various mathematical representation; (2) Understanding of the concept orthogonal projection between points on a line in the geometry; (3) construct geometric shapes of space to clarify the problem; (4) Understanding students’ visual representations related to the ability to determine the position of a point or a line perpendicular to a line; (5) Understanding of calculation procedures using the Pythagorean theorem and algebraic concepts.

Geometrical approach to the gauge field mass problem. Possible reasons for which the Higgs bosons are unobservable.

The incompressible Navier-Stokes equations are solved in a channel, using a Discontinuous Galerkin method over staggered grids. The resulting linear systems are studied both in terms of the structure and in terms of the spectral features of the related coefficient matrices. In fact, the resulting matrices are of block type, each block showing Toeplitz-like, band, and tensor structure at the same time. Using this rich matrix-theoretic information and the Toeplitz, Generalized Locally Toeplitz technology, a quite complete spectral analysis is presented, with the target of designing and analyzing fast iterative solvers for the associated large linear systems. Quite promising numerical results are presented, commented, and critically discussed for elongated two- and three-dimensional geometries.

We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed Hodge structures which is equipped with a flat Gauss-Manin connection. We construct graded differential rings of special functions on these moduli spaces and show that they contain rings of quasi-modular forms. We show that the algebra of derivations of quasi-modular forms can be obtained from the Gauss--Manin connection contracted with vector fields on the enhanced moduli spaces. We provide examples for this construction given by the mirrors of the canonical bundles of $\mathbb{P}^2$ and $\mathbb{F}_2$.

In the last years, three-dimensional (3D) medical imaging techniques have taken an increasing importance in patient care and medical research. Volume images provide medical specialists with a direct access to the interior of a patient's body and reduce the need for invasive exploration. The use of volume imaging modalities such as X-ray CT, PET or MRI has therefore become essential for medical diagnosis and surgical planning. Computer visualization techniques such as extraction of planar slices of arbitrary orientation (Multiplanar Reprojection), surface rendering of anatomic structures, and volume rendering provide medical users with the tools for exploiting 3D volume images. Surface or respectively volume rendering provides information about the 3D geometry and 3D context of the structures of interest but does not allow to directly visualize original intensities, respectively colors located within the 3D structures. In addition, surface rendering requires the segmentation of the volume data and volume rendering often requires a classification of the volume image pixels. In contrast, the extraction of planar sections provides interactivity, requires no pre-processing and the original intensity, respectively color of each slice element may be directly inspected. However, it does not allow the visualization of curved anatomic structures within a single slice. In this thesis, we propose to overcome this limitation by generalizing the concept of planar section to the extraction of curved cross-sections. In the first part, we focus on the interactive extraction of curved surfaces from volume images. Unlike planar slices, curved cross-sections may follow the trajectory of tubular structures such as the Aorta or follow structures with an irregular shape such as the Pelvis. In the second part of this work, we focus on the visualization of curved surfaces. We would like to offer the possibility of carrying out distance measurements along a structure of interest both for medical applications and for anatomical studies. Orthogonal or perspective projection of curved surfaces induces angular and metric distortions as well as surface overlapping. In order to enable measurements, we propose to use surface flattening methods, which preserve distances along specific orientations and minimize distortions around a focus point. Flattening of curved cross-sections enables inspecting spatially complex relationship between anatomic structures and their neighbourhood. They also allow the visualization of a curved anatomic structure within a single planar view and therefore to precisely inspect the original intensity, respectively color at each surface point. Thanks to a multi-resolution approach, surfaces are flattened at interactive rates, allowing users to displace the focus point during the visualization of the flattened surface. We also propose a new efficient method for computing a flattened surface minimizing global distortions and still preserving distances along one orientation. Surface extraction and flattening techniques are integrated into an interactive visualization Java applet. This Java applet enables anyone to precisely and interactively inspect the Visible Human anatomy. Besides medical visualization, the presented methods may also be useful for creating new interesting views of anatomic structures for didactic purposes.

From a global viewpoint, a symmetric space is a Riemannian manifold which possesses a symmetry about each point, that is, an involutive isometry leaving the point fixed. This generalizes the notion of reflection in a point in ordinary Euclidean geometry. The theory of symmetric spaces implies that such spaces have a transitive group of isometries and can be represented as coset spaces G/K, where G is a connected Lie group with an involutive automorphism G whose fixed point set is (essentially) K.KeywordsSymmetric SpaceTransitive GroupIwasawa DecompositionBibliographical NoteInvolutive AutomorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Assessing thermophysical properties of parameterized woven composite models using image-based simulations