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If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples ( such as the complement of a trefoil knot ) where a manifold may have a finite volume geometric structure of either type.
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If and manifold
If g is the metric on this manifold, one defines the action S ( g ) as
If the base manifold is four-dimensional, the Kaluza – Klein manifold P is five-dimensional.
If the underlying manifold is supersymmetric, the resulting theory is a supersymmetric Yang – Mills theory.
If the underlying manifold is allowed to be infinite dimensional ( for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group.
# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are ( more or less ) Kac – Moody algebras.
* If M is a smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f ( x )
If M is an open subset of R < sup > n </ sup >, then M is a C < sup >∞</ sup > manifold in a natural manner ( take the charts to be the identity maps ), and the tangent spaces are all naturally identified with R < sup > n </ sup >.
If the manifold is the surface of the Earth,
If the range of each chart is the n-dimensional Euclidean space, then M is said to be an n-dimensional manifold.
If the manifold M is smooth ( respectively analytic )--- that is, the change of coordinates are smooth ( respectively analytic )--- then one can make sense of the notion of smooth ( respectively analytic ) vector fields.
If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold.
If γ: b → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L ( γ ) in analogy with the example above by
If the field is a real vector field, then the target manifold is isomorphic to R < sup > 3 </ sup >.
If is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not null.
If M is a simply connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1 / 4 and 1 then M is diffeomorphic to a sphere.
If M is a non-compact complete non-negatively curved n-dimensional Riemannian manifold, then M contains a compact, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S ( S is called the soul of M .) In particular, if M has strictly positive curvature everywhere, then it is diffeomorphic to R < sup > n </ sup >.
# 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 a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line ( i. e. a geodesic which minimizes distance on each interval ) then it is isometric to a direct product of the real line and a complete ( n-1 )- dimensional Riemannian manifold which has nonnegative Ricci curvature.
If and is
If the circumstances are faced frankly it is not reasonable to expect this to be true.
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.
If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;
If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
If it is an honest feeling, then why should she not yield to it??
If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If he is good, he may not be legal ; ;
If the man on the sidewalk is surprised at this question, it has served as an exclamation.
If the existent form is to be retained new factors that reinforce it must be introduced into the situation.
If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??
If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.
If he is a traditionalist, he is an eclectic traditionalist.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.
If and non-compact
* If π < sub > 1 </ sub >( M ) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.
If and then
If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.
I showed her the shower and tub, and she said, smiling, `` If you really don't mind, I think I'll get clean in the shower, then soak for a few minutes in your tub.
If internal responsibility suggests acceptance of the socialist ideal of equality, then external responsibility implies adherence to principles of ideological supranationalism.
If Franklin was an authentic genius, then Alexander Hamilton, with his exceptional precocity, consuming energy, and high ambition, was a political prodigy.
If a child watched its progress he whispered, `` Hay, hay, load of hay -- make a wish and turn away '', and then stared rigidly in the opposite direction until the sound of the horses' feet returned no more.
If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.
Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.
If the historian was convinced of his own correctness, then he should not allow his vision to become fogged by disturbing facts.
If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.
If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.
If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??
If the President of the United States really feels he won himself a `` lousy job '', then heaven help us all.
If we make it established custom that whenever butchery on the highways grows excessive, say beyond 25,000 per annum, then somebody is going to hang, it follows that the more eminent the victim, the more impressive the lesson.
If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.
If the UN troops are not mercenaries then the Hessians were not mercenaries either.
`` If they are here, then surely I have the right to be here '', Rousseau said.
If he failed to reach the riverbank in five minutes, say, then the skiffs would pull away and leave him groping in the mud.
If we had, then one of us would have had to go instead.
If our national interest lies in being able to fight and win a war rather than committing national suicide, then we must take a much more penetrating look at ballistic missiles.
If, for some reason such as economy, we are not going to develop aircraft nuclear propulsion with a sense of national urgency, then we should turn our effort to developing jet engines with a thrust-to-weight ratio of 12 or 15 to one.
If your pool is located on or near sloping ground, it may have natural drainage which is certainly more desirable than to be faced with the annual expense and labor of first pumping out the water and then scooping out all the debris.
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