scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

scholarly journals A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres

2013 ◽  
Vol 9 (17) ◽  
pp. 11-20
Author(s):  
Carlos Cadavid ◽  
Juan Diego Vélez
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Critical Points ◽  
Morse Function ◽  
Initial Condition ◽  
Morse Functions ◽  
Flat Tori

Let (M, g)be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of pointsp, q∈M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to∂f /∂t= ∆gf, f (·,0) =f0is such that for sufficiently larget, f(·, t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

scholarly journals Umbilical Submanifolds and Morse Functions

1972 ◽  
Vol 48 ◽  
pp. 197-201 ◽  
Author(s):  
Katsumi Nomizu ◽  
Lucio Rodríguez
Keyword(s):  
Euclidean Space ◽  
Differentiable Function ◽  
Critical Points ◽  
Morse Function ◽  
Differentiable Manifold ◽  
Morse Functions

Let Mn be a differentiable manifold (of class C∞). By a Morse function on Mn we mean a differentiable function whose critical points are all non-degenerate. If f is an immersion of Mn into a Euclidean space Rm, we may obtain Morse functions on Mn in the following way.

scholarly journals Distance functions and umbilic submanifolds of hyperbolic space

1979 ◽  
Vol 74 ◽  
pp. 67-75 ◽  
Author(s):  
Thomas E. Cecil ◽  
Patrick J. Ryan
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Hyperbolic Space ◽  
Critical Points ◽  
Morse Function ◽  
Complete Riemannian Manifold ◽  
Distance Functions

In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em.

scholarly journals Discrete Morse Inequalities on Infinite Graphs

10.37236/127 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rafael Ayala ◽  
Luis M. Fernández ◽  
José A. Vilches
Keyword(s):  
Morse Function ◽  
Special Kind ◽  
Infinite Graphs ◽  
Morse Inequalities ◽  
Morse Functions ◽  
Critical Elements ◽  
Finite Case ◽  
Discrete Morse Function

The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous, namely, decreasing rays. In addition, we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them.

scholarly journals Cusp cobordism group of Morse functions

2021 ◽  
pp. 1-35
Author(s):  
Dominik J. Wrazidlo
Keyword(s):  
Critical Points ◽  
Compact Manifold ◽  
Infinite Order ◽  
Morse Function ◽  
Explicit Description ◽  
Morse Functions ◽  
Cobordism Group ◽  
Manifold With Boundary ◽  
Singular Fibers ◽  
Odd Dimensions

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all nondegenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine’s cusp elimination technique as well as the complementary process of creating pairs of cusps along fold lines. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. For Morse functions on surfaces our result yields an explicit description of Saeki–Yamamoto’s cobordism invariant which they constructed by means of the cohomology of the universal complex of singular fibers.

Harmonic G-structures

2009 ◽  
Vol 146 (2) ◽  
pp. 435-459 ◽  
Author(s):  
J. C. GONZÁLEZ–DÁVILA ◽  
F. MARTÍN CABRERA
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Compact Riemannian Manifold ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Second Variation ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Intrinsic Torsion ◽  
First And Second Variation

AbstractFor closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold (M, 〈⋅, ⋅〉), where G-structures are considered as sections of the quotient bundle (M)/G. We deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related to the study of G-structures. In this direction, we show the rôle in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for 2n-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into (M)/U(n).

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

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