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Xiaobo Liu - One of the best experts on this subject based on the ideXlab platform.
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Homogeneity of infinite dimensional isoparametric submanifolds
arXiv: Differential Geometry, 1999Co-Authors: Ernst Heintze, Xiaobo LiuAbstract:A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of Palais-Terng and Dadok, finally classified irreducible isoparametric submanifolds of a finite dimensional Euclidean space of rank at least 3. While Thorbergsson's proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos. The main purpose of this paper is to extend Thorbergsson's result to the infinite dimensional case.
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Homogeneity of infinite dimensional isoparametric submanifolds
Annals of Mathematics, 1999Co-Authors: Ernst Heintze, Xiaobo LiuAbstract:A subsetS of a Riemannian manifoldN is called extrinsically homogeneous ifS is an orbit of a subgroup of the isometry group ofN. In [Th], Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a flnite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of [PT1] and [Da], flnally classifled irreducible isoparametric submanifolds of a flnite dimensional Euclidean space of rank at least 3. While Thorbergsson’s proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos (cf. [O2]). The main purpose of this paper is to extend Thorbergsson’s result to the inflnite dimensional case. The study of inflnite dimensional isoparametric submanifolds of a Hilbert space (always assumed to be separable) was initiated by Terng [T2]. Besides its intrinsic interest, the theory of inflnite dimensional isoparametric submanifolds is a very useful tool in studying the submanifold geometry of compact symmetric spaces (cf. [TT] as well as [HL] and [Ew]). Although much progress has been made (especially, many basic properties of flnite dimensional isoparametric submanifolds having been successfully extended to the inflnite dimensional case (cf. [T2], [T5], and [HL])), the classiflcation of inflnite dimensional isoparametric submanifolds is far from being solved. To this end, the understanding of the Homogeneity of inflnite dimensional isoparametric submanifolds will certainly play a very important role, as it does in the flnite dimensional case. Examples of non-homogeneous inflnite dimensional isoparametric hypersurfaces have been found by Terng and Thorbergsson (cf. [TT]). In this paper, we will prove the following theorem which solves a long standing open problem (cf. [T3], [T4], [TT]). Theorem A. Let M be a complete, connected, irreducible isoparametric submanifold in a Hilbert space V . Assume that the set of all the curvature normals of M at some point is not contained in any a‐ne line. Then M is extrinsically homogeneous in the Hilbert space V . Remark. In this theorem, the dimension of M and thus V could be either flnite or inflnite. Without loss of generality, we may assume that M is full, i.e.
Bernhard Fink - One of the best experts on this subject based on the ideXlab platform.
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colour Homogeneity and visual perception of age health and attractiveness of male facial skin
Journal of The European Academy of Dermatology and Venereology, 2011Co-Authors: Bernhard Fink, P J Matts, D Demiliano, L Bunse, Bettina Weege, Susanne RoderAbstract:Background: Evolutionary psychology suggests that skin signals aspects of mate value, yet only limited empirical evidence exists for this assertion. Objectives: We sought to study the relationship between perception of skin condition and Homogeneity of color/chromophore distribution. Methods: Cropped skin cheek images from 170 girls and women (11-76 years) were blind-rated for attractiveness, healthiness, youthfulness, and biological age by 353 participants. These skin images and corresponding melanin/hemoglobin concentration maps were analyzed objectively for Homogeneity. Results: Homogeneity of unprocessed images correlated positively with perceived attractiveness, healthiness, and youthfulness (all r [ 0.40; P \.001), but negatively with estimated age (r = � 0.45; P \ .001). Homogeneity of hemoglobin and melanin maps was positively correlated with that of unprocessed images (r = 0.92, 0.68; P\.001) and negatively correlated with estimated age (r = � 0.32, � 0.38; P\.001). Limitations: Female skin only was studied. Conclusions: Skin color Homogeneity, driven by melanin and hemoglobin distribution, influences perception of age, attractiveness, health, and youth. ( J Am Acad Dermatol 2007;57:977-84.)
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color Homogeneity and visual perception of age health and attractiveness of female facial skin
Journal of The American Academy of Dermatology, 2007Co-Authors: P J Matts, Bernhard Fink, Karl Grammer, Maria BurquestAbstract:Background Evolutionary psychology suggests that skin signals aspects of mate value, yet only limited empirical evidence exists for this assertion. Objectives We sought to study the relationship between perception of skin condition and Homogeneity of color/chromophore distribution. Methods Cropped skin cheek images from 170 girls and women (11-76 years) were blind-rated for attractiveness, healthiness, youthfulness, and biological age by 353 participants. These skin images and corresponding melanin/hemoglobin concentration maps were analyzed objectively for Homogeneity. Results Homogeneity of unprocessed images correlated positively with perceived attractiveness, healthiness, and youthfulness (all r > 0.40; P r = −0.45; P r = 0.92, 0.68; P r = −0.32, −0.38; P Limitations Female skin only was studied. Conclusions Skin color Homogeneity, driven by melanin and hemoglobin distribution, influences perception of age, attractiveness, health, and youth.
Ernst Heintze - One of the best experts on this subject based on the ideXlab platform.
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Homogeneity of infinite dimensional isoparametric submanifolds
arXiv: Differential Geometry, 1999Co-Authors: Ernst Heintze, Xiaobo LiuAbstract:A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of Palais-Terng and Dadok, finally classified irreducible isoparametric submanifolds of a finite dimensional Euclidean space of rank at least 3. While Thorbergsson's proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos. The main purpose of this paper is to extend Thorbergsson's result to the infinite dimensional case.
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Homogeneity of infinite dimensional isoparametric submanifolds
Annals of Mathematics, 1999Co-Authors: Ernst Heintze, Xiaobo LiuAbstract:A subsetS of a Riemannian manifoldN is called extrinsically homogeneous ifS is an orbit of a subgroup of the isometry group ofN. In [Th], Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a flnite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of [PT1] and [Da], flnally classifled irreducible isoparametric submanifolds of a flnite dimensional Euclidean space of rank at least 3. While Thorbergsson’s proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos (cf. [O2]). The main purpose of this paper is to extend Thorbergsson’s result to the inflnite dimensional case. The study of inflnite dimensional isoparametric submanifolds of a Hilbert space (always assumed to be separable) was initiated by Terng [T2]. Besides its intrinsic interest, the theory of inflnite dimensional isoparametric submanifolds is a very useful tool in studying the submanifold geometry of compact symmetric spaces (cf. [TT] as well as [HL] and [Ew]). Although much progress has been made (especially, many basic properties of flnite dimensional isoparametric submanifolds having been successfully extended to the inflnite dimensional case (cf. [T2], [T5], and [HL])), the classiflcation of inflnite dimensional isoparametric submanifolds is far from being solved. To this end, the understanding of the Homogeneity of inflnite dimensional isoparametric submanifolds will certainly play a very important role, as it does in the flnite dimensional case. Examples of non-homogeneous inflnite dimensional isoparametric hypersurfaces have been found by Terng and Thorbergsson (cf. [TT]). In this paper, we will prove the following theorem which solves a long standing open problem (cf. [T3], [T4], [TT]). Theorem A. Let M be a complete, connected, irreducible isoparametric submanifold in a Hilbert space V . Assume that the set of all the curvature normals of M at some point is not contained in any a‐ne line. Then M is extrinsically homogeneous in the Hilbert space V . Remark. In this theorem, the dimension of M and thus V could be either flnite or inflnite. Without loss of generality, we may assume that M is full, i.e.
P J Matts - One of the best experts on this subject based on the ideXlab platform.
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colour Homogeneity and visual perception of age health and attractiveness of male facial skin
Journal of The European Academy of Dermatology and Venereology, 2011Co-Authors: Bernhard Fink, P J Matts, D Demiliano, L Bunse, Bettina Weege, Susanne RoderAbstract:Background: Evolutionary psychology suggests that skin signals aspects of mate value, yet only limited empirical evidence exists for this assertion. Objectives: We sought to study the relationship between perception of skin condition and Homogeneity of color/chromophore distribution. Methods: Cropped skin cheek images from 170 girls and women (11-76 years) were blind-rated for attractiveness, healthiness, youthfulness, and biological age by 353 participants. These skin images and corresponding melanin/hemoglobin concentration maps were analyzed objectively for Homogeneity. Results: Homogeneity of unprocessed images correlated positively with perceived attractiveness, healthiness, and youthfulness (all r [ 0.40; P \.001), but negatively with estimated age (r = � 0.45; P \ .001). Homogeneity of hemoglobin and melanin maps was positively correlated with that of unprocessed images (r = 0.92, 0.68; P\.001) and negatively correlated with estimated age (r = � 0.32, � 0.38; P\.001). Limitations: Female skin only was studied. Conclusions: Skin color Homogeneity, driven by melanin and hemoglobin distribution, influences perception of age, attractiveness, health, and youth. ( J Am Acad Dermatol 2007;57:977-84.)
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color Homogeneity and visual perception of age health and attractiveness of female facial skin
Journal of The American Academy of Dermatology, 2007Co-Authors: P J Matts, Bernhard Fink, Karl Grammer, Maria BurquestAbstract:Background Evolutionary psychology suggests that skin signals aspects of mate value, yet only limited empirical evidence exists for this assertion. Objectives We sought to study the relationship between perception of skin condition and Homogeneity of color/chromophore distribution. Methods Cropped skin cheek images from 170 girls and women (11-76 years) were blind-rated for attractiveness, healthiness, youthfulness, and biological age by 353 participants. These skin images and corresponding melanin/hemoglobin concentration maps were analyzed objectively for Homogeneity. Results Homogeneity of unprocessed images correlated positively with perceived attractiveness, healthiness, and youthfulness (all r > 0.40; P r = −0.45; P r = 0.92, 0.68; P r = −0.32, −0.38; P Limitations Female skin only was studied. Conclusions Skin color Homogeneity, driven by melanin and hemoglobin distribution, influences perception of age, attractiveness, health, and youth.
Susanne Roder - One of the best experts on this subject based on the ideXlab platform.
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colour Homogeneity and visual perception of age health and attractiveness of male facial skin
Journal of The European Academy of Dermatology and Venereology, 2011Co-Authors: Bernhard Fink, P J Matts, D Demiliano, L Bunse, Bettina Weege, Susanne RoderAbstract:Background: Evolutionary psychology suggests that skin signals aspects of mate value, yet only limited empirical evidence exists for this assertion. Objectives: We sought to study the relationship between perception of skin condition and Homogeneity of color/chromophore distribution. Methods: Cropped skin cheek images from 170 girls and women (11-76 years) were blind-rated for attractiveness, healthiness, youthfulness, and biological age by 353 participants. These skin images and corresponding melanin/hemoglobin concentration maps were analyzed objectively for Homogeneity. Results: Homogeneity of unprocessed images correlated positively with perceived attractiveness, healthiness, and youthfulness (all r [ 0.40; P \.001), but negatively with estimated age (r = � 0.45; P \ .001). Homogeneity of hemoglobin and melanin maps was positively correlated with that of unprocessed images (r = 0.92, 0.68; P\.001) and negatively correlated with estimated age (r = � 0.32, � 0.38; P\.001). Limitations: Female skin only was studied. Conclusions: Skin color Homogeneity, driven by melanin and hemoglobin distribution, influences perception of age, attractiveness, health, and youth. ( J Am Acad Dermatol 2007;57:977-84.)
