In this paper we obtain some Cheeger-Gromov-Taylor type compactness theorems for a forward complete and connected Finsler manifold of dimensional n 2 via weighted Ricci curvatures. The proofs are based on the index form of a minimal unit speed geodesic segment, Bochner-Weitzenböck formula and Hessian comparison theorem.
Abstract:Product experience encompasses all five sensory modalities through which information is received (Shifferstein and Desmet, 2008) therefore designing taking into account all the five senses is important. Besides, advances in technology make it easier to accomplish designing for five senses in many industries. On the other hand, particularly in the early years of undergraduate level industrial design education, sense of vision is often emphasized more compared to the other four senses. In order to underline the importance of the remaining senses, a unique studio course has been designed at Istanbul Medipol University, aiming second year industrial design students. In this particular studio course, in addition to their regular sketchbooks, students are expected to keep a dairy of sound, tactile, smell and taste. The course is designed in four modules and this paper elaborates on the details of Module I that focuses on the sense of hearing. Module I is interdisciplinary in the sense that a psychologist and a music composer/virtuoso have been invited to contribute to the classes so that designs can be treated at the visceral, behavioural and reflective levels (Norman, 2004) more profoundly. This paper aims to share the outcomes related to the 15 projects carried out during the course such that the advantages and the disadvantages of a multi-sensory design studio set up are revealed.
In this paper, we shall give some structure equations for RB h-almost solitons which generalize previous results for Ricci almost solitons. As a consequence of these equations we also derive an integral formula for the compact gradient RB h-almost solitons which enables us to show that a compact nontrivial almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal.
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