Figure 3 - uploaded by Harish Seshadri
Content may be subject to copyright.
The holonomy on a sphere
Source publication
Context in source publication
Similar publications
Citations
... The Ricci flow theory [1] [2] became attractive for research in mathematics and physics after G. Perelman successfully carried out his program [3] [4] [5] which resulted in proofs of Thurston and Poincaré conjectures, see reviews of results in Refs. [6] [7] [8]. The profound impact of such results on understanding the topology and geometric structure of curved spacetime and fundamental properties of classical and quantum interactions was used as a motivation to study the geometric evolution of regular Lagrange systems [9] [10] on tangent bundles and nonholonomic (pseudo) Riemannian and Einstein manifolds. ...
The approach to nonholonomic Ricci flows and geometric evolution of regular
Lagrange systems [S. Vacaru: J. Math. Phys. 49 (2008) 043504 & Rep. Math. Phys.
63 (2009) 95] is extended to include geometric mechanics and gravity models on
Lie algebroids. We prove that such evolution scenarios of geometric mechanics
and analogous gravity can be modeled as gradient flows characterized by
generalized Perelman functionals if an equivalent geometrization of Lagrange
mechanics [J. Kern, Arch. Math. (Basel) 25 (1974) 438] is considered. The R.
Hamilton equations on Lie algebroids describing Lagrange-Ricci flows are
derived. Finally, we show that geometric evolution models on Lie algebroids is
described by effective thermodynamical values derived from statistical
functionals on prolongation Lie algebroids.
... Most of the above is proved in section 11 (especially 11.8) of [14], the last part concerning the topology of X C can be found in section 1 of [15]; the proof uses a compactness argument (go to ancient time) involving the soul theorem, see for example chapter 9 of [13], [12] and [3]. ...
... Therefore the limit contains a line and splits; it must be the cylinder and that is a contradiction. See also [12], [13], [3] for more details. ...
... Moreover, on each time slice, the curvature is bounded, positive; see [17] Theorem 4.14. See also 12.1 of [13] for an alternative argument. ...
We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.
... it was shown that for a closed 3-manifold M 3 we have cat S 1 M 3 = 2 if and only if π 1 (M 3 ) is cyclic. By results of Olum [10] and Perelman [9] this implies that cat S 1 M 3 = 2 if and only if M 3 is a lens space or M 3 is the non-orientable S 2 -bundle over S 1 . For the case n > 3 we showed [6] that cat S 1 M n = 2 implies that π 1 (M n ) is cyclic or a nontrivial product with amalgamation A * C B of cyclic groups. ...
A closed topological n-manifold M n is of S1-category 2 if it can be covered by two open subsets W1,W2 such that the inclusions Wi ! M n factor homotopically through maps Wi ! S1. We show that for n > 3 the fundamental group of such an n-manifold is either trivial or infinite cyclic. 1 2
A closed topological n-manifold M
n
is of S
1-category 2 if it can be covered by two open subsets W
1, W
2 such that the inclusions W
i
→ M
n
factor homotopically through maps W
i
→ S
1. We show that for n > 3, if $${{\rm cat}_{S^1}(M^n )=2}$$ then M
n
is homeomorphic to S
n
or S
n–1 × S
1 or the non-orientable S
n–1-bundle over S
1. We also obtain an unknotting theorem for locally flat knots of S
n–2 in S
n
and a characterization of S
1 × S
n–1.
In [2], Roger Brockett derived a necessary condition for the existence of a feedback control law asymptotically stabilizing an equilibrium for a given nonlinear control system. The intuitive appeal and the ease with which it can be applied have made this criterion one of the standard tools in the study of the feedback stabilizability of nonlinear control systems. Brockett's original proof used an impressive combination of Liapunov theory and algebraic topology, in part to cope with a lacuna in our understanding of the topology of the sublevel sets of Liapunov functions. In [33], F. W. Wilson, Jr. extended the converse theorems of Liapunov theory to compact attractors and proved some fundamental results about the topology of their domain of attraction and the level sets of their Liapunov functions. In particular, Wilson showed that the level sets Mc = V −1 (c) are diffeomorphic to S n−1 for n = 4, 5 using the proof of the generalized Poincaré Conjecture of Smale. He observed that the excluded cases would from the validity of the Poincaré Conjecture in dimension 3 and 4 and showed that, for n = 5, the assertion ∂Mc ≃ S 4 would imply the Poincaré Conjecture for 4-manifolds. Of course, the topological Poincaré Conjecture for S 4 was subsequently proved by Freedman in 1980 and with the remarkable recent solution by Perelman of the classical Poincaré Conjecture, Wilson's Theorem now holds for all n. In this paper we describe the sublevel, and therefore as a corollary the level, sets of proper smooth functions V : R n → R having a compact set C(V) of critical points. Among the main results in this paper is the assertion that an arbitrary sublevel set Mc = V −1 [0, c] of such a function is homeomorphic to D n , the unit disk. For n = 2, this assertion is a consequence of the Schönflies Theorem, a classical enhancement of the Jordan Curve Theorem. For arbitrary n it follows from the generalized Schönflies Theorem of Mazur and Brown, from [33] and from the verification of the Poincaré Conjecture in all dimensions by Perelman, Freedman and Smale. We also describe the smooth structure of Mc and its boundary, generalizing the results of [33]. This result has several corollaries. In particular, using the Brouwer Fixed Point Theorem this gives a straightforward proof of Brockett's criterion and some of its enhancements to global attractors. These results in turn imply a new necessary condition for Input-to State Stability with respect to a compact set and an extension of Brockett's Theorem to the practical stabilizability of equilibria. Our main results can be further enhanced using the Poincaré-Hopf Theorem and, in this way, also lead to a streamlined version of Coron's proof [7] that Brockett's Theorem holds for continuous feedback laws, using a classical topological argument on the unit disc D n .
We characterize those smooth 1-connected open 4-manifolds with certain finite type properties which admit proper special generic maps into 3-manifolds. As a corollary, we show that a smooth 4-manifold homeomorphic to R 4 admits a proper special generic map into R n for some n = 1, 2 or 3 if and only if it is diffeomorphic to R 4 . We also characterize those smooth 4-manifolds homeomorphic to L × R for some closed orientable 3-manifold L which admit proper special generic maps into R 3 .
A closed topological n-manifold M
n
is of S
1-category 2 if it can be covered by two open subsets W
1, W
2 such that the inclusions W
i
→ M
n
factor homotopically through maps W
i
→ S
1. We show that for n>3 the fundamental group of such an n-manifold is either trivial or infinite cyclic.
KeywordsLusternik–Schnirelmann category-Coverings of n-manifolds with open S
1-contractible subsets
Mathematics Subject Classification (2000)57N10-57N13-57N15-57M30
We determine which three-manifolds are dominated by products. The result is
that a closed, oriented, connected three-manifold is dominated by a product if
and only if it is finitely covered either by a product or by a connected sum of
copies of the product of the two-sphere and the circle. This characterization
can also be formulated in terms of Thurston geometries, or in terms of purely
algebraic properties of the fundamental group. We also determine which
three-manifolds are dominated by non-trivial circle bundles, and which
three-manifold groups are presentable by products.
There were elaborated different models of Finsler geometry using the Cartan
(metric compatible), or Berwald and Chern (metric non-compatible) connections,
the Ricci flag curvature etc. In a series of works, we studied (non)commutative
metric compatible Finsler and nonholonomic generalizations of the Ricci flow
theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and
references therein]. The goal of this work is to prove that there are some
models of Finsler gravity and geometric evolution theories with generalized
Perelman's functionals, and correspondingly derived nonholonomic Hamilton
evolution equations, when metric noncompatible Finsler connections are
involved. Following such an approach, we have to consider distortion tensors,
uniquely defined by the Finsler metric, from the Cartan and/or the canonical
metric compatible connections. We conclude that, in general, it is not possible
to elaborate self-consistent models of geometric evolution with arbitrary
Finsler metric noncompatible connections.
We give a simple proof of a result originally due to Dimca and Suciu: a group
that is both Kaehler and the fundamental group of a closed three-manifold is
finite. We also prove that a group that is both the fundamental group of a
closed three-manifold and of a non-Kaehler compact complex surface is infinite
cyclic or the direct product of an infinite cyclic group and a group of order
two.
