The Variation of the Manifold Under the Time

Received: November 30, 2010 Accepted: December 24, 2010 doi:10.5539/jmr.v3n2p119 Abstract In this papre, we introduce the variation of the manifold under the time, the variation in the volume, the variation in the dimension and the variation in the position are discussed. AMS Subject Classification 2000: 51H10, 57N10. Keyword: Dynamical manifold 1. Definitions and Background 1Dynamical system: The notation of a dynamical system is the mathematical formalization of the general scientific concept of a deterministic process. The evolution of a dynamical system means change in the state of the system with time t∈T where T is an ordered set.We will consider two types of dynamical systems. Some with continuous time T=R, and another with discrete(integer) time T=Z. System of the first type are called continuous-time dynamical systems,the second are termed discrete-time dynamical system.[M.M.A.Ramadan, 2008]. 2n-manifolds: The n-dimensional manifold is a Housdorff space (Space stisfies the T2 separation axiom) such that each point has an open neighborhood homeomorphic to the open ndimensional disc. Un = {x ∈ Rn : ||x|| < 1}. Where n is positive integer.[Massey W.S, 1973]. 3Housdorff space: A topolgical space X which satisfy the following axiom ”Every two distinct point of X have disjoint neighborhoods in X”..[Massey W.S, 1973]. 2. The main results Aiming to our study, we will introduce the variation of manifold under the time. 2.1 The variation in the volume 2.1.1 The variaton in the volume and the shape not variation See Fig.(1) form the 1-manifold L after some time it will be L\ ,L\ is 1-manifold (The same shape of L). The limit in this case(L\) goes to R1. See Fig.(2) form the 2-manifold c after some time it will be c\ ,c\ is 2-manifold (The same shape of c). The limit in this case(c\) goes to R2. See Fig.(3) form the 3-manifold b after some time it will be b\ ,b\ is 3-manifold (The same shape of b). The limit in this case(b\) goes to R3. 2.1.2 The variation in the volume and the shape variation See Fig.(4) form the 1-manifold L after some trime it will be L\ ,L\ is 1-manifold (The shape variant). The limit in this case(L\) goes to R1.

1-Dynamical system: The notation of a dynamical system is the mathematical formalization of the general scientific concept of a deterministic process.
The evolution of a dynamical system means change in the state of the system with time t∈T where T is an ordered set.We will consider two types of dynamical systems.Some with continuous time T=R, and another with discrete(integer) time T=Z.System of the first type are called continuous-time dynamical systems,the second are termed discrete-time dynamical system.[M.M.A.Ramadan, 2008].

2-n-manifolds:
The n-dimensional manifold is a Housdorff space (Space stisfies the T 2 separation axiom) such that each point has an open neighborhood homeomorphic to the open n-dimensional disc.
3-Housdorff space: A topolgical space X which satisfy the following axiom "Every two distinct point of X have disjoint neighborhoods in X".. [Massey W.S, 1973].

The main results
Aiming to our study, we will introduce the variation of manifold under the time.

The variation in the volume
2.1.1The variaton in the volume and the shape not variation See Fig.
(1) form the 1-manifold L after some time it will be L \ ,L \ is 1-manifold (The same shape of L).
The limit in this case(L \ ) goes to R 1 .See Fig. (2) form the 2-manifold c after some time it will be c \ ,c \ is 2-manifold (The same shape of c).
The limit in this case(c \ ) goes to R 2 .See Fig. (3) form the 3-manifold b after some time it will be b \ ,b \ is 3-manifold (The same shape of b).
The limit in this case(b \ ) goes to R 3 .

2.1.2
The variation in the volume and the shape variation See Fig.( 4) form the 1-manifold L after some trime it will be L \ ,L \ is 1-manifold (The shape variant).
The limit in this case(L \ ) goes to R 1 .
Published by Canadian Center of Science and Education Vol. 3, No. 2;May 2011 See Fig.(5) form the 2-manifold c after some time it will be c \ ,c \ is 2-manifold (The shape variant).
The limit in this case(c \ ) goes to R 2 .See Fig.( 6) form the 3-manifold b after some time it will be b \ , b \ is 3-manifold (The shape variant).
The limit in this case(b \ ) goes to R 3 .

The variation in the dimension
2.2.1 From 0-dimension to 1-dimension See Fig.( 7) form the 0-manifold a after some time it will be L ,L is 1-dimensiona.
The limit in this case(L) gose to R 1 .

From 1 -dimension to 2 -dimension
See Fig.( 8) form 1-manifold L after some time it will be c ,c is 2-dimension.
The limit in this case(c) goes to R 2 .

From 0 -dimension to 2 -dimension
See Fig.( 9) form the 0-manifold a after some time it will be c ,c is 2-dimension.
The limit in the case(c) goes to R 2 .This the variation is ungradually.13) form the 1-manifold L after some time it will be L \\\ ,L \\\ is 1-manifold (The outside manifold and the inside manifold are identical).

The variation in the position
The limit in this case(L \\\ ) goes to R 1 .

There are many 2-manifold inside the manifold
See Fig.( 14) refer to the inside manifolds increasing, and the limit of the inside manifold goes to R 2 .See Fig.( 15) refer to the inside manifolds decreasing, and the limit of the inside manifold goes to R 2 .
2.6 There are many 3-manifold inside the manifold See Fig.( 16) refer to the inside manifolds increasing, and the limit of the inside manifold goes to R 3 .See Fig.( 17) refer to the inside manifolds decreasing, and the limit of the inside manifold goes to R 3 .

The variation of the manifold with boundary under the time
See Fig. (18) The limit in this case goes to R 3 .See Fig. (19) The limit in this case goes to R 2 .

Theorem 1:
The variation of the manifold of dimension n by the time into another n-manifold is in a volume, curvature and torsion.
Let M be a manifold of dimension n, L arc length, K is the curvature, τ is a torsion, then: L= |I|dt, dx.dx = I Where I is the first fundamental form.
Lemma 1: The variation in a manifold of dimension n without boundary is a manifold with boundaries. Proof: Where M is a manifold without boundary, and M \ is a manifold with boundary.
Lemma 2: The converse of the above lemma is true. Proof: See Fig.( 22).

Let G: M→M \
Where M is a manifold with boundary, and M \ is a manifold with out boundary.

Applications:
1-Growth of seeds of plants.
2-A leaf is sick and the sickness is spreaded in every side of it then it will form a net.