Recall that a point in the Teichm�ller space is a pair ( X, f ) where X is an hyperbolic surface ( a Riemannian manifold with sectional curvatures all equal to-1 ) and f an homeomorphism, up to a natural equivalence relation.
42.
Moreover, there are relatively few examples of compact positively curved manifolds, leaving a lot of conjectures ( e . g ., the Hopf conjecture on whether there is a metric of positive sectional curvature on \ Bbb S ^ 2 \ times \ Bbb S ^ 2 ).
43.
Alexandrov spaces with curvature & ge; " k " are important as they form the limits ( in the Gromov-Hausdorff metric ) of sequences of Riemannian manifolds with sectional curvature & ge; " k ", as described by Gromov's compactness theorem.
44.
The scalar curvature of an " n "-manifold is defined as the trace of the Ricci tensor, and it can be defined as " n " ( " n " " 1 ) times the average of the sectional curvatures at a point.
45.
The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind of tidal force ( this is one way of thinking of the sectional curvature ).
46.
The specialness of varieties of low Kodaira dimension is analogous to the specialness of Riemannian manifolds of positive curvature ( and general type corresponds to the genericity of non-positive curvature ); see classical theorems, especially on " Pinched sectional curvature " and " Positive curvature ".
47.
In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of world-lines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.
48.
The maximum sectional curvature ( 4 ) is attained at a holomorphic 2-plane one for which " J " ( ? ) ?" ? while the minimum sectional curvature ( 1 ) is attained at a 2-plane for which " J " ( ? ) is orthogonal to ?.
49.
The maximum sectional curvature ( 4 ) is attained at a holomorphic 2-plane one for which " J " ( ? ) ?" ? while the minimum sectional curvature ( 1 ) is attained at a 2-plane for which " J " ( ? ) is orthogonal to ?.
50.
With Y . Li, Liu introduced a new geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature; proved the global existence of the solution; discussed the global convergence of its solution and possible applications to the Hopf conjecture, as well as its relation to the Navier-Stokes equations on manifolds.
