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| In [[differential geometry]], a '''pseudo-Riemannian manifold''' <ref>{{citation | last1=Benn|first1=I.M.|last2=Tucker|first2=R.W. | title = An introduction to Spinors and Geometry with Applications in Physics| publisher=Adam Hilger | year=1987| page=172}}
| | Whether youre struggling to lose 30 lbs or those last 5 lbs your diet will greatly influence the success. No matter how various crunches we do, laps around the track field like Rocky or time spent on the elliptical while checking out the girl on the stair master. It can all be for nothing if your diet is not inside check. Dieting may be difficult plus depressing, but should you plan it out and consistently create changes youll find it may be fun and simple. These steps will enable you to shape your own diet to help we in the fat reduction goals.<br><br>Calculating your BMR with a [http://safedietplans.com/bmr-calculator bmr calculator] is an important step. This tells us how countless calories you burn a day by really existing. Breathing, hearts beating, kidneys working plus everything our bodies do takes calories. Knowing how numerous calories these functions use is important knowledge inside you weight reduction program.<br><br>Surely this might be the upcoming question that comes to mind today. The speed of calorie shedding may differ from individual to individual. Every person will have their own Body Mass Index plus basal metabolic rate, plus the individual habits of the individual usually also determine, how much is burned whenever a individual is sleeping. The healthy consumption of calories for an average human body is mentioned to be about 2.000 per day. And for each 3,500 which are added to the body, an individual usually put 1 1 pound of fat. Given below are several tips that will show we how to burn more during sleep.<br><br>A low calorie diet may do you more harm than superior. If you are craving more calories, this can therefore affect the metabolism. Where can the body get the power it requirements? It will really get power from your muscle, since there are no food reserves available. Whenever this happens, it may be very detrimental to a health, because the right way to lose fat is to gain muscle, not take it away.<br><br>So my friend's bmr is 1969 calories a day. With this information plus her doctor's recommendation, we've choose to go for 1400-1500 calories consumed a day plus 50-60 minutes of rather little exercise a day. According to the objective calculator a reduction of 400-500 calories a day within the BMR might cause regarding 1 pound of weight lose per week, which is ideal.<br><br>Then you have to incorporate the general physical escapades which are done on a daily bases. Based on how active we are a would add the following to your BMR.<br><br>If you like to lose weight, we want to program on eating a number of calories which is someplace between the BMR and the total amount of calories we burn in a day. If you want to gain weight, we have to eat more calories than we burn in a day. |
| </ref><ref>{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds|publisher=The Macmillan Company | year=1968|page=208}}</ref> (also called a '''semi-Riemannian manifold''') is a generalization of a [[Riemannian manifold]]. It is one of many mathematical objects named after [[Bernhard Riemann]]. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the [[metric tensor]] need not be [[Definite bilinear form|positive-definite]]. Instead a weaker condition of [[nondegenerate|nondegeneracy]] is imposed. | |
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| == Introduction ==
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| === Manifolds ===
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| {{main|Manifold|Differentiable manifold}}
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| In [[differential geometry]], a [[differentiable manifold]] is a space which is locally similar to a [[Euclidean space]]. In an <var>n</var>-dimensional Euclidean space any point can be specified by <var>n</var> real numbers. These are called the [[coordinate]]s of the point.
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| An <var>n</var>-dimensional differentiable manifold is a generalisation of <var>n</var>-dimensional Euclidean space. In a manifold it may only be possible to define coordinates ''locally''. This is achieved by defining [[coordinate patch]]es: subsets of the manifold which can be mapped into <var>n</var>-dimensional Euclidean space.
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| See [[Manifold]], [[differentiable manifold]], [[coordinate patch]] for more details.
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| === Tangent spaces and metric tensors ===
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| {{main|Tangent space|metric tensor}}
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| Associated with each point <math>\scriptstyle p</math> in an <math>\scriptstyle n</math>-dimensional differentiable manifold <math>\scriptstyle M</math> is a [[tangent space]] (denoted <math>\scriptstyle T_pM</math>). This is an <math>\scriptstyle n</math>-dimensional [[vector space]] whose elements can be thought of as [[equivalence class]]es of curves passing through the point <math>\scriptstyle p</math>.
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| A [[metric tensor]] is a [[non-degenerate]], smooth, symmetric, [[bilinear map]] which assigns a [[real number]] to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by <math>\scriptstyle g</math> we can express this as
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| :<math>g : T_pM \times T_pM \to \mathbb{R}</math>.
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| The map is symmetric and bilinear so if <math>\scriptstyle X,Y,Z \in T_pM</math> are tangent vectors at a point <math>\scriptstyle p</math> to the manifold <math>\scriptstyle M</math> then we have
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| * <math>\,g(X,Y) = g(Y,X)</math>
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| * <math>\,g(aX + Y, Z) = a g(X,Z) + g(Y,Z)</math>
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| for any real number <math>\scriptstyle a\in\mathbb{R}</math>.
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| That <math>\scriptstyle g</math> is [[non-degenerate]] means there are no non-zero <math>X \in T_pM</math> such that <math>\,g(X,Y) = 0</math> for all <math>Y \in T_pM</math>.
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| === Metric signatures ===
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| {{main|Metric signature}}
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| Given a metric tensor ''g'' on an ''n''-dimensional real manifold, the [[quadratic form]] {{nowrap|1=''q''(''x'') = ''g''(''x'',''x'')}} associated with the metric tensor applied to each vector of any [[orthogonal basis]] produces ''n'' real values. By [[Sylvester's law of inertia#Law of inertia for quadratic forms|Sylvester's rigidity theorem]], the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor, independent of the choice of orthogonal basis. The '''[[Metric signature|signature]]''' (''p'',''q'',''r'') of the metric tensor gives these numbers, shown in the same order. For a non-degenerate metric tensor {{nowrap|1=''r'' = 0}} and the signature may be denoted (''p'',''q''), where {{nowrap|1=''p'' + ''q'' = ''n''}}.
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| == Definition ==
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| A '''pseudo-Riemannian manifold''' <math>\,(M,g)</math> is a [[differentiable manifold]] <math>\,M</math> equipped with a non-degenerate, smooth, symmetric [[metric tensor]] <math>\,g</math> which, unlike a [[Riemannian metric]], need not be [[Definite bilinear form|positive-definite]], but must be non-degenerate. Such a metric is called a '''pseudo-Riemannian metric''' and its values can be positive, negative or zero.
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| The signature of a pseudo-Riemannian metric is (<var>p</var>, <var>q</var>) where both <var>p</var> and <var>q</var> are non-negative.
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| ==Lorentzian manifold==
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| A '''Lorentzian manifold''' is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (1, <var>n</var>−1) (or sometimes (<var>n</var>−1, 1), see [[sign convention]]). Such metrics are called '''Lorentzian metrics'''. They are named after the physicist [[Hendrik Lorentz]].
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| === Applications in physics ===
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| After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of [[general relativity]].
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| A principal basis of [[general relativity]] is that [[spacetime]] can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). Unlike Riemannian manifolds with positive-definite metrics, a signature of (<var>p</var>, 1) or (1, <var>q</var>) allows tangent vectors to be classified into ''timelike'', ''null'' or ''spacelike'' (see [[Causal structure]]).
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| ==Properties of pseudo-Riemannian manifolds==
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| Just as [[Euclidean space]] <math>\mathbb{R}^n</math> can be thought of as the model [[Riemannian manifold]], [[Minkowski space]] <math>\mathbb{R}^{n-1,1}</math> with the flat [[Minkowski metric]] is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (<var>p</var>, <var>q</var>) is <math>\mathbb{R}^{p,q}</math></sup> with the metric
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| :<math>g = dx_1^2 + \cdots + dx_p^2 - dx_{p+1}^2 - \cdots - dx_{p+q}^2</math>
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| Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the [[fundamental theorem of Riemannian geometry]] is true of pseudo-Riemannian manifolds as well. This allows one to speak of the [[Levi-Civita connection]] on a pseudo-Riemannian manifold along with the associated [[Riemann curvature tensor|curvature tensor]]. On the other hand, there are many theorems in Riemannian geometry which do not hold in the generalized case. For example, it is ''not'' true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain [[topology|topological]] obstructions. Furthermore, a [[submanifold]] does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any [[Minkowski space#Causal structure|light-like]] [[curve]]. The [[Clifton–Pohl torus]] provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that the [[Hopf–Rinow theorem]] disallows for Riemannian manifolds.<ref>{{citation|title=Semi-Riemannian Geometry With Applications to Relativity|volume=103|series=Pure and Applied Mathematics|first=Barrett|last=O'Neill|publisher=Academic Press|year=1983|isbn=9780080570570|page=193|url=http://books.google.com/books?id=CGk1eRSjFIIC&pg=PA193}}.</ref>
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| == See also ==
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| *[[Spacetime]]
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| *[[Hyperbolic partial differential equation]]
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| *[[Causality conditions]]
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| *[[Globally hyperbolic manifold]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{citation | last1=Benn|first1=I.M.|last2=Tucker|first2=R.W. | title = An introduction to Spinors and Geometry with Applications in Physics| publisher=Adam Hilger | year=1987|edition=First published 1987|isbn=0-85274-169-3}}
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| * {{citation | last1=Bishop|first1=Richard L.|last2=Goldberg|first2=Samuel I. | title = Tensor Analysis on Manifolds|publisher=The Macmillan Company | year=1968|edition=First Dover 1980|isbn=0-486-64039-6}}
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| * {{citation | last1=Chen|first1=Bang-Yen| | title = Pseudo-Riemannian Geometry, [delta]-invariants and Applications| publisher=World Scientific Publisher | year=2011|isbn=978-981-4329-63-7}}
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| * G. Vrănceanu & R. Roşca (1976) ''Introduction to Relativity and Pseudo-Riemannian Geometry'', Bucarest: Editura Academiei Republicii Socialiste România.
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| [[Category:Lorentzian manifolds]]
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| [[Category:Smooth manifolds]]
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Whether youre struggling to lose 30 lbs or those last 5 lbs your diet will greatly influence the success. No matter how various crunches we do, laps around the track field like Rocky or time spent on the elliptical while checking out the girl on the stair master. It can all be for nothing if your diet is not inside check. Dieting may be difficult plus depressing, but should you plan it out and consistently create changes youll find it may be fun and simple. These steps will enable you to shape your own diet to help we in the fat reduction goals.
Calculating your BMR with a bmr calculator is an important step. This tells us how countless calories you burn a day by really existing. Breathing, hearts beating, kidneys working plus everything our bodies do takes calories. Knowing how numerous calories these functions use is important knowledge inside you weight reduction program.
Surely this might be the upcoming question that comes to mind today. The speed of calorie shedding may differ from individual to individual. Every person will have their own Body Mass Index plus basal metabolic rate, plus the individual habits of the individual usually also determine, how much is burned whenever a individual is sleeping. The healthy consumption of calories for an average human body is mentioned to be about 2.000 per day. And for each 3,500 which are added to the body, an individual usually put 1 1 pound of fat. Given below are several tips that will show we how to burn more during sleep.
A low calorie diet may do you more harm than superior. If you are craving more calories, this can therefore affect the metabolism. Where can the body get the power it requirements? It will really get power from your muscle, since there are no food reserves available. Whenever this happens, it may be very detrimental to a health, because the right way to lose fat is to gain muscle, not take it away.
So my friend's bmr is 1969 calories a day. With this information plus her doctor's recommendation, we've choose to go for 1400-1500 calories consumed a day plus 50-60 minutes of rather little exercise a day. According to the objective calculator a reduction of 400-500 calories a day within the BMR might cause regarding 1 pound of weight lose per week, which is ideal.
Then you have to incorporate the general physical escapades which are done on a daily bases. Based on how active we are a would add the following to your BMR.
If you like to lose weight, we want to program on eating a number of calories which is someplace between the BMR and the total amount of calories we burn in a day. If you want to gain weight, we have to eat more calories than we burn in a day.