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Affine Killing complete and geodesically complete homogeneous affine surfaces
Gilkey, P.B.,Park, J.H.,Valle-Regueiro, X. Elsevier 2019 Journal of mathematical analysis and applications Vol.474 No.1
<P><B>Abstract</B></P> <P>An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting of homogeneous affine surfaces.</P>
A NOTE ON THE EIGENFUNCTIONS OF THE LAPLACIAN FOR A TWISTED HOLOMORPHIC PRODUCT
Gilkey, Peter B.,Park, Jeong-Hyeong 대한수학회 1997 대한수학회논문집 Vol.12 No.2
Let $Z = X \times Y$ where X and Y are complex manifolds. We suppose that projection $\pi$ on the second factor is a Riemannian submersion, that TX is perpendicular to TY, and that the metrics on Z and on Y are Hermetian; we do not assume Z is a Riemannian product. We study when the pull-back of an eigenfunction of the complex Laplacian on Y is an eigenfunction of the complex Laplacian on Z.
Virtual displays and virtual environments
Gilkey, R.H.,Isabelle, S.K.,Simpson, B.B. Ergonomics Society of Korea 1997 大韓人間工學會誌 Vol.16 No.2
Our recent work on virtual environments and virtual displays is reviewed, including our efforts to establish the Virtual Environment Research, Interactive Technology, And Simulation (VERITAS) facility and our research on spatial hearing. VERITAS is a state-of -the-art multisensory facility, built around the ${CAVE}^{TM}$ technology. High-quality 3D audio is included and haptic interfaces are planned. The facility will support technical and non-technical users working in a wide variety of application areas. Our own research emphasizes the importance of auditory stimulation in virtual environments and complex display systems. Experiments on auditory-aided visual target acquistion, sensory conflict, sound localization in noise, and loxalization of speech stimuli are discussed.
Gilkey, Peter V.,Kim, Chan Yong,Park, JeongHyeong MATHEMATICAL INSTITUTE OF TOHOKU UNIVERSITY 2017 Tohoku mathematical journal Vol.69 No.1
<P>We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs are second order. We examine when the surfaces are asymptotically minimal.</P>
MODULI SPACES OF ORIENTED TYPE ${\mathcal{A}}$ MANIFOLDS OF DIMENSION AT LEAST 3
Gilkey, Peter,Park, JeongHyeong Korean Mathematical Society 2017 대한수학회지 Vol.54 No.6
We examine the moduli space of oriented locally homogeneous manifolds of Type ${\mathcal{A}}$ which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.
MODULI SPACES OF ORIENTED TYPE A MANIFOLDS OF DIMENSION AT LEAST 3
Peter Gilkey,박정형 대한수학회 2017 대한수학회지 Vol.54 No.6
We examine the moduli space of oriented locally homogeneous manifolds of Type~$\mathcal{A}$ which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.
VOLUME DENSITY ASYMPTOTICS OF CENTRAL HARMONIC SPACES
Peter B. Gilkey,박정형 대한수학회 2023 대한수학회보 Vol.60 No.6
We show the asymptotics of the volume density function in the class of central harmonic manifolds can be specified arbitrarily and do not determine the geometry.
Spectral geometry, homogeneous spaces and differential forms with finite Fourier series
Dunn, C,Gilkey, P,Park, J H The Institute of Physics 2008 JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL Vol.41 No.13
<P>Let <I>G</I> be a compact Lie group acting transitively on Riemannian manifolds <I>M<SUB>i</SUB></I> and let π:<I>M</I><SUB>1</SUB> → <I>M</I><SUB>2</SUB> be a <I>G</I>-equivariant Riemannian submersion. We show that a smooth differential form &phis; on <I>M</I><SUB>2</SUB> has finite Fourier series on <I>M</I><SUB>2</SUB> if and only if the pull back π*&phis; has finite Fourier series on <I>M</I><SUB>1</SUB>.</P>