On the least positive eigenvalue of the Laplacian for compact group manifolds (original) (raw)
On the diagonalization of the Ricci flow on Lie groups
Proceedings of the American Mathematical Society, 2013
The main purpose of this note is to prove that any basis of a nilpotent Lie algebra for which all diagonal left-invariant metrics have diagonal Ricci tensor necessarily produce quite a simple set of structural constants; namely, the bracket of any pair of elements of the basis must be a multiple of one of them, and only the bracket of disjoint pairs can be a nonzero multiple of the same element. Some applications to the Ricci flow of left-invariant metrics on Lie groups concerning diagonalization are also given.
2004
, it is proved that all g-natural metrics on tangent bundles of m-dimensional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on m and on the assumption that the base manifold is oriented, or non-oriented, respectively. The result was originally stated in [8] for the oriented case, but the smoothness was assumed and not explicitly proved. In this note, we shall prove that, both in the oriented and non-oriented cases, the functions generating the g-natural metrics are, in fact, smooth on the set of all nonnegative real numbers.
Differential Geometry and Lie Groups A Computational Perspective
Differential Geometry and Lie Groups A Computational Perspective, 2020
To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard and Jane. discuss three results, one of which being the Hadamard and Cartan theorem about complete manifolds of non-positive curvature. The goal of Chapter 18 is to understand the behavior of isometries and local isometries, in particular their action on geodesics. We also intoduce Riemannian covering maps and Riemannian submersions. If π : M → B is a submersion between two Riemannian manifolds, then for every b ∈ B and every p ∈ π -1 (b), the tangent space T p M to M at p splits into two orthogonal components, its vertical component V p = Ker dπ p , and its horizontal component H p (the orthogonal complement of V p ). If the map dπ p is an isometry between H p and T b B, then most of the differential geometry of B can be studied by lifting B to M , and then projecting down to B again. We also introduce Killing vector fields, which play a technical role in the study of reductive homogeneous spaces. In Chapter 19, we return to Lie groups. Not every Lie group is a matrix group, so in order to study general Lie groups it is necessary to introduce left-invariant (and rightinvariant) vector fields on Lie groups. It turns out that the space of left-invariant vector fields is isomorphic to the tangent space g = T I G to G at the identity, which is a Lie algebra. By considering integral curves of left-invariant vector fields, we define the generalization of the exponential map exp : g → G to an arbitrary Lie group. The notion of immersed Lie subgroup is introduced, and the correspondence between Lie groups and Lie algebra is explored. We also consider the special classes of semidirect products of Lie algebras and Lie groups, the universal covering of a Lie group, and the Lie algebra of Killing vector fields on a Riemannian manifold. Chapter 20 deals with two topics: 1. A formula for the derivative of the exponential map for a general Lie group (not necessarily a matrix group). 2. A formula for the Taylor expansion of µ(X, Y ) = log(exp(X) exp(Y )) near the origin. The second problem is solved by a formula known as the Campbell-Baker-Hausdorff formula. An explicit formula was derived by Dynkin (1947), and we present this formula. Chapter 21 is devoted to the study of metrics, connections, geodesics, and curvature, on Lie groups. Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. As a consequence, it is possible to find explicit formulae for the Levi-Civita connection and the various curvatures, especially in the case of metrics which are both left and right-invariant. In Section 21.2 we give four characterizations of bi-invariant metrics. The first one refines the criterion of the existence of a left-invariant metric and states that every bi-invariant metric on a Lie group G arises from some Ad-invariant inner product on the Lie algebra g. In Section 21.3 we show that if G is a Lie group equipped with a left-invariant metric, then it is possible to express the Levi-Civita connection and the sectional curvature in terms Recall that a real symmetric matrix is called positive (or positive semidefinite) if its eigenvalues are all positive or null, and positive definite if its eigenvalues are all strictly positive. We denote the vector space of real symmetric n × n matrices by S(n), the set of symmetric positive matrices by SP(n), and the set of symmetric positive definite matrices by SPD(n). The next proposition shows that every symmetric positive definite matrix A is of the form e B for some unique symmetric matrix B. The set of symmetric matrices is a vector space, but it is not a Lie algebra because the Lie bracket [A, B] is not symmetric unless A and B commute, and the set of symmetric (positive) definite matrices is not a multiplicative group, so this result is of a different flavor as Theorem 2.6. Proposition 2.8. For every symmetric matrix B, the matrix e B is symmetric positive definite. For every symmetric positive definite matrix A, there is a unique symmetric matrix B such that A = e B . Proof. We showed earlier that e B = e B . If B is a symmetric matrix, then since B = B, we get e B = e B = e B , and e B is also symmetric. Since the eigenvalues λ 1 , . . . , λ n of the symmetric matrix B are real and the eigenvalues of e B are e λ 1 , . . . , e λn , and since e λ > 0 if λ ∈ R, e B is positive definite. To show the surjectivity of the exponential map, note that if A is symmetric positive definite, then by Theorem 12.3 from Chapter 12 of Gallier [50], there is an orthogonal matrix P such that A = P D P , where D is a diagonal matrix CHAPTER 2. THE MATRIX EXPONENTIAL; SOME MATRIX LIE GROUPS
Analytic and geometric aspects of Laplace operator on Riemannian manifold
Malaya Journal of Matematik
In the past decade there has been a flurry of work at intersection of spectral theory and Riemannian geometry. In this paper we present some of recent results on abstract spectral theory depending on Laplace-Beltrami operator on compact Riemannian manifold. Also, we will emphasize the interplay between spectrum of operator and geometry of manifolds by discussing two main problems (direct and inverse problems) with an eye towards recent developments.
The Lie derivative of currents on Lie groups
Lobachevskii Journal of Mathematics, 2012
The aim of this work is to study the properties of the Lie derivative of currents and generalized forms on Riemann manifolds. For an application, we give some results of the Lie derivative of currents and generalized forms on Lie groups.
First Eigenvalues of Geometric Operator under The Ricci-Bourguignon Flow
Journal of the Indonesian Mathematical Society, 2017
Let (M,g(t))(M,g(t))(M,g(t)) be a compact Riemannian manifold and the metric g(t)g(t)g(t) evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of geometric operator −Deltaphi+cR-\Delta_{\phi}+cR−Deltaphi+cR under the Ricci-Bourguignon flow, where Deltaphi\Delta_{\phi}Deltaphi is the Witten-Laplacian operator and RRR is the scalar curvature. In the final we show that some quantities dependent to the eigenvalues of the geometric operator are nondecreasing along the Ricci-Bourguignon flow on closed manifolds with nonnegative curvature.