Peter Gilkey’s research while affiliated with University of Oregon and other places

... It was proven in [20] that any space covered by R n or isometric to a rank one symmetric space is harmonic. In the compact case, these spaces include the sphere S n , the real projective space RP n , the complex projective space CP n , the quaternionic projective space HP n , and the Cayley projective plane OP 2 ; for more details see [21]. The negative curvature or pseudo-Riemannian analogs of the listed spaces are also harmonic, as they can be obtained through analytical continuation. ...

Reference:

Banana diagrams as functions of geodesic distance

... It is thus an example of a homogeneous Riemannian manifold which has this property, but which is not a globally symmetric space, see e.g. [GPVL15,p. 105]. We will use the explicit fibered decomposition to derive an upper bound for the geodesic complexity of three-dimensional lens spaces of type L(p; 1). ...

Reference:

Geodesic complexity via fibered decompositions of cut loci

... The analog of Theorem 3.2 fails for Witten's type perturbations of the Dolbeault complex on Kähler manifolds[2]. ...

Reference:

ZETA INVARIANTS OF MORSE FORMS

... There is a vast literature on this subject; we refer to [1,2,5,8,11,13,14] and the references cited therein for further details. Note that if M is a harmonic space, then we can rescale the metric to replace g by c 2 g for any c > 0 to obtain another harmonic space M c := (M, c 2 g). ...

Reference:

Maximal Domains of Radial Harmonic Functions

... The elements of T 1 are called instantons. 1 X can be C ∞ -approximated by gradient-like Smale vector fields that agree with X around X [20, Proposition 2.4] (this follows from [69, Theorem A]). A well-known consequence is that, for any Morse function h, there is a C ∞ -dense set of Riemannian metrics g on M such that − grad g h is Smale; this density is also true in the subspace of metrics that are Euclidean with respect to Morse coordinates on given neighborhoods of the critical points. ...

Reference:

ZETA INVARIANTS OF MORSE FORMS

... Previous work on integrability properties in the presence of torsion in black hole spacetimes has focused on purely axial (or "skew") torsion [33][34][35] or has been more formal in nature [36][37][38]. In the present work, we want to close this gap by discussing-to the best of our knowledge-for the first time the complete integrability of the autoparallel equation of motion in a wide range of off-shell geometries with non-vanishing torsion, before specializing to the case of an exact Schwarzschild black hole solution of quadratic Poincaré gravity endowed with an GM/r 2 torsion profile. ...

Reference:

Translation invariant' black hole: autoparallels and complete integrability

... Further, both 2-dimensional and 3-dimensional harmonic Riemannian manifolds have constant sectional curvature. Known examples of harmonic Riemannian manifolds are the Euclidean space, rank one symmetric spaces and Damek-Ricci spaces [1,Chapter 5] and [2,4]. ...

Reference:

Harmonic Finsler manifolds of (α, β)-type

... Opozda [18] classified the locally homogeneous affine surfaces without torsion. Subsequently, Arias-Marco and Kowalski [1] extended this classification to the more general setting; a different proof of this result has been given recently by Brozos-Vázquez et al. [2]. Previous studies of locally homogeneous surfaces in the torsion free setting include [13,14]. ...

Reference:

Symmetric affine surfaces with torsion

... To that end, we will consider here a simplified scenario. First, we limit our considerations to two dimensions (and, later, suitably generalized to spherical symmetry in the four-dimensional sense). 1 And sec- * jens.boos@kit.edu 1 Poincaré gauge gravities in two dimensions have been explored in [10], whereas the mathematical classification of two-dimensional spaces with torsion is fairly recent [11]; see also [12]. ...

Reference:

Torsion in two dimensions: autoparallels, symmetries, and applications to black holes

... The heat content of polygonal subdomains in, possibly unbounded, domains Ω ⊂ R 2 can be defined analogously to (1.1) by considering the heat equation on Ω with some boundary condition imposed on ∂Ω (when the latter is non-empty). The small-time asymptotics for such cases have been obtained in [20,21,23] and we summarise these below. We denote the length of a segment A ⊂ ∂ D by L(A) so that L(∂ D) is the length of the boundary of D. ...

Reference:

Heat Flow in Polygons with Reflecting Edges