Conversely, in a negatively curved space, the hot spots would look smaller than they would in flat space. If a spacetime is not curved, it is said to be flat, which just means that it has the geometry of Minkowski space. This has the effect of making the hot spots look larger than they would in flat space. In positively curved space, though, light travels along converging lines. Numerical results can be used to explain recent experiments. In flat space, light from the hot spots travels along straight lines. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. The physical, mathematical, and biological meanings of the equilibrium differential equations and the geometrical constraint equations are discussed. Additionally, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are presented. By minimizing the potential functional, the equilibrium differential equations and the boundary conditions of two-component lipid vesicles are derived. The FriedmannLematreRobertsonWalker metric is a curved metric which forms the current foundation for the description of the. Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. We introduce two kinds of virtual displacement modes: the normal one and the tangential one. Curved space often refers to a spatial geometry which is not 'flat', where a flat space is described by Euclidean geometry. To take into account the influences of the local compositions together with the mean curvature and Gaussian curvature of the membrane surface, a general potential functional is constructed.
This paper aims at the general mathematical framework for the equilibrium theory of two-component lipid bilayer vesicles. All these symmetrical patterns in soft matters should be just the reason-able and natural results of the symmetrical analytical system. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of net-works of biomembrane nanotubes. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived.
Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor. To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry.
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