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The classical Cohn-Vossen inequality states that for any Riemannian2-manifold the difference between 27πX(M) and the total curvature ∫M KdA isalways nonnegative. For complete open surfaces in E3 this curvature defect can beinterpreted in terms of the length of the curve "at infinity" . The goal of this paperis to investigate higher dimensional analogues for open submanifolds of euclideanspace with cone-like ends. This is based on the extrinsic Gauss-Bonnet formulafor compact submanifolds with boundary and its extension "to infinity". It turnsout that the curvature defect can be positive, zero, or negative, depending on theshape of the ends "at infinity". Furthermore we study the variational problem forthe total curvature of hypersurfaces where the ends are not fixed. It turns outthat for open hypersurfaces with cone-like ends the total curvature is stationaryif and only if each end has vanishing Gauss-Kronecker curvature in the sphere"at infinity". For this case of stationary total curvature we prove a result on thequantization of the total curvature., Differential Geometry : Proceedings of the First Intenational Symposiumu on Differential Geometry, February 22-24, 2001 Josai University, edited by Qing-Ming Cheng.

Through the human evolution, process of hominisation and bipedal locomotion the lower limbs undertake essentially changed biomechanical conditions resulting by an evolutionary adaptation of the human body to erect position. Lower limbs undertake entire loading force from the body and different biomechanic and kinematic role as well a different process of functional adaptation than the upper limbs. Logically is that the different evolutionary adaptation of the lower limbs may result also by a different patology for instance by the diabetic foot. Interrelationship between arterial cross-sectional area and tissue volume for the empties (cadaveric) arteries is five times better for upper limbs in relation to lower limbs. Međutim, in living subjects this difference, in diastolic and sistolic case both, was something manja. This finding confirm the hypothesis that another possibilities of haemodynamic and structural prilagodbe exist too.

Ishaemic lesions of tissues like the diabetic foot syndrom, are more common in foots than in hands. We tried to explaine if there are significant differences in crossectional area and blood flow of the main arteries supplying upper and lower limbs in relation to volume unit of tissue between the upper and the lower limb. In spite of faster blood flow in a. femoralis prokrvljenost of the legs is slabija due to significantly larger volume. Consequently every pathological factor e.g. in diabetes may disturbe the balance in blood supply in legs.