The Anatomy of the Domestic Animals

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

THE dog has played by far the most important part in the elucidation of the difficult problems of physiology and pathology presented by the higher animal organism. It is by a firm reliance on the results of experimental researches, conducted largely upon this animal, that the modern physician is enabled to form some idea as to the causation of the symptoms of disease in man, and the mode of action of the remedies which he employs; while the modern surgeon, after a preliminary testing of an operation upon the dog, fearlessly proceeds to attack the most deeply-seated tumour, and to explore the most hidden recesses of the human organization. What, after all, are the services of friendship and companionship, or the more menial duties which are often laid upon the dog, compared with the alleviation of human suffering and the advancement of human knowledge for which he has served as the passive instrument, and this (pace the mendacious asseverations of fanatical essayists) at the expense of the least possible amount of suffering to himself?

Anatomy Of The Dog

and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

A short survey on biharmonic maps between Riemannian manifolds

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied by Jiang, Chen, Caddeo, Montaldo, and Oniciuc. We then apply the equation to show that the generalized Chen conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a 2-parameter family of conformally flat metrics and a 4-parameter family of multiply warped product metrics, each of which turns the foliation of an upper-half space of R m by parallel hyperplanes into a foliation with each leaf a proper biharmonic hypersurface. We also study the biharmonicity of Hopf cylinders of a Riemannian submersion.

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Biharmonic hypersurfaces in Riemannian manifolds

We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.

https://journals.tubitak.gov.tr/math/issues/mat-14-38-2/mat-38-2-16-1304-3.pdf

On generalized Robertson--Walker spacetimes satisfying some curvature condition

Recently, Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically im- mersed in Kenmotsu space forms. The equality case is considered. Some applications are derived.

Contact cr-warped product submanifolds in kenmotsu space forms

In the present article we study the rotational embedded surfaces in E4. The rotational embedded surface was first studied by G. Ganchev and V. Milousheva as a surface in E4. The Otsuki (non-round) sphere in E4 is one of the special examples of this surface. Finally, we give necessary and sufficient conditions for the flat Ganchev-Milousheva rotational surface to have pointwise 1-type Gauss map.

/pdf/rotational-embeddings-in-e-4-with-pointwise-1-type-gauss-map-440feshpmx.pdf

Rotational embeddings in E^4 with pointwise 1-type gauss map

Bu calismada, AW(k) ( k=1, 2 yada 3 ) tipindeki egri ve yuzeyler gozonune alindi. AW(k) sartini saglayan egri ve yuzeylere ornekler verildi

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CURVES AND SURFACES OF AW(k) TYPE

Recently, Chen established a general sharp inequality for warped products in real space forms. As applications, he obtained obstructions to minimal isometric immersions of warped products into real space forms. Afterwards, Matsumoto and one of the present authors proved the Sasakian version of this inequality. In the present paper, we obtain sharp estimates for the warping function in terms of the mean curvature for warped products isometrically immersed in Kenmotsu space forms. Some applications are derived.

Kadri Arslan

Papers

On generalized Robertson--Walker spacetimes satisfying some curvature condition

Contact cr-warped product submanifolds in kenmotsu space forms

Rotational embeddings in E^4 with pointwise 1-type gauss map

CURVES AND SURFACES OF AW(k) TYPE

Warped product submanifolds in kenmotsu space forms