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# Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 ( real ) dimensions.
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# and Every
# Every type of state is a powerful institution of the ruling class ; the state is an instrument which one class uses to secure its rule and enforce its preferred production relations ( and its exploitation ) onto society.
# " Personality " Argument: this argument is based on a quote from Hegel: " Every man has the right to turn his will upon a thing or make the thing an object of his will, that is to say, to set aside the mere thing and recreate it as his own ".
# Moral law of karma: Every action ( by way of body, speech, and mind ) will have karmic results ( a. k. a. reaction ).
# Every finitely generated ideal of A is principal ( i. e., A is a Bézout domain ) and A satisfies the ascending chain condition on principal ideals.
Good Trouble ( 1982 ) and Wheels Are Turnin ' ( 1984 ) were follow-up albums which also did well commercially, the former containing the hit singles " Keep the Fire Burnin '" ( U. S. # 7 ), " Sweet Time " ( U. S. # 26 ) and the un-ranked " The Key " and the latter containing the # 1 hit single " Can't Fight This Feeling " plus three more hits: " I Do ' Wanna Know " ( U. S. # 29 ), " One Lonely Night " ( U. S. # 19 ), " Live Every Moment " ( U. S. # 34 ) and the un-ranked " Break His Spell ".
# Every simple path from a given node to any of its descendant leaves contains the same number of black nodes.
# and Riemannian
# Gauss – Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ ( M ) where χ ( M ) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.
# The Cartan – Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R ^ n with n = dim M via the exponential map at any point.
# The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.
# If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT ( k ) space.
# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).
# Myers ' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by, then the manifold has diameter, with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.
# The Bishop – Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space.
# The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a line, meaning a geodesic γ such that for all, then it is isometric to a product space.
# Pontryagin numbers of closed Riemannian manifold ( as well as Pontryagin classes ) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
# and metric
# Apparent lack of regular rhythm, especially in the siguiriyas: the melodic rhythm of the sung line is different from the metric rhythm of the accompaniment.
# If A is an open or closed subset of R < sup > n </ sup > ( or even Borel set, see metric space ), then A is Lebesgue measurable.
# The metric is translation-invariant ; i. e., d ( x + a, y + a ) = d ( x, y ) for all x, y and a in V
The chain in use on modern bicycles has a 1 / 2 " pitch, which is ANSI standard # 40, where the 4 indicates the pitch of the chain in eighths of an inch, and metric # 8, where the 8 indicates the pitch in sixteenths of an inch.
# Complex Euclidean space C < sup > n </ sup > with the standard Hermitian metric is a Kähler manifold.
# A torus C < sup > n </ sup >/ Λ ( Λ a full lattice ) inherits a flat metric from the Euclidean metric on C < sup > n </ sup >, and is therefore a compact Kähler manifold.
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