The spectral geometry of the canonical Riemannian submersion of a compact Lie Group

Let G be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy be the multiplication operator. We show the associated fibration m mapping GxG to G is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show the pull back of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions that the pull back of a form on the base is harmonic on the total space

Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series

Let G be a compact Lie group acting transitively on Riemannian manifolds M and N. Let p be a G equivariant Riemannian submersion from M to N. We show that a smooth differential form on N has finite Fourier series if and only if the pull back has finite Fourier series on

Complete curvature homogeneous pseudo-Riemannian manifolds

We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.Comment: Update paper to fix misprints in original versio

Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds

Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher order Jordan Osserman for $2\le k\le s$; they are k-timelike higher order Jordan Osserman for $s+2\le k\le 2s$, and they are not k timelike higher order Jordan Osserman for $2\le s\le s+1$.Comment: Update bibliography, fix minor misprint

Thermal stress promotes host mitochondrial degradation in symbiotic cnidarians: are the batteries of the reef going to run out?

The symbiotic relationship between cnidarians and their dinoflagellate symbionts, Symbiodinium spp, which underpins the formation of tropical coral reefs, can be destabilized by rapid changes to environmental conditions. Although some studies have concluded that a breakdown in the symbiosis begins with increased reactive oxygen species (ROS) generation within the symbiont due to a decoupling of photosynthesis, others have reported the release of viable symbionts via a variety of host cell derived mechanisms. We explored an alternative model focused upon changes in host cnidarian mitochondrial integrity in response to thermal stress. Mitochondria are often likened to being batteries of the cell, providing energy in the form of ATP, and controlling cellular pathway activation and ROS generation. The overall morphology of host mitochondria was compared to that of associated symbionts under an experimental thermal stress using confocal and electron microscopy. The results demonstrate that hyperthermic stress induces the degradation of cnidarian host mitochondria that is independent of symbiont cellular deterioration. The potential sites of host mitochondrial disruption were also assessed by measuring changes in the expression of genes associated with electron transport and ATP synthesis using quantitative RT-PCR. The primary site of degradation appeared to be downstream of complex III of the electron transport chain with a significant reduction in host cytochrome c and ATP synthase expression. The consequences of reduced expression could limit the capacity of the host to mitigate ROS generation and maintain both organelle integrity and cellular energy supplies. The disruption of host mitochondria, cellular homeostasis, and subsequent cell death irrespective of symbiont integrity highlights the importance of the host response to thermal stress and in symbiosis dysfunction that has substantial implications for understanding how coral reefs will survive in the face of climate change