A Wisdom Archive on Manifolds

A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ... Including:

• Manifold - Introduction
• Manifold - Motivational example: the circle
• Manifold - Charts atlases and transition maps
• Manifold - Construction
• Manifold - Charts
• Manifold - Patchwork
• Manifold - Zeros of a function
• Manifold - Identifying points of a manifold
• Manifold - Cartesian products
• Manifold - Gluing along boundaries
• Manifold - Topological manifolds
• Manifold - Differentiable manifolds
• Manifold - Orientability
• Manifold - Möbius strip
• Manifold - Klein bottle
• Manifold - Real projective plane
• Manifold - Other types and generalizations of manifolds
• Manifold - History

Read more here: » Manifold: Encyclopedia - Manifold

manifold: Varied. Having many forms, aspects, parts.

(See also: Manifold, Hinduism, Body Mind and Soul )

Ratha Saptami
This falls on the 7th day of the bright fortnight of the month of Margaseersha (December-January). People worship the sun in the early morning and recite the Surya Sahasranama. Good actions done on this day give manifold results.
 
From Hindu Fasts & Festivals by Sri Swami Sivananda.
 

Read more here: » Ratha Saptami: Fasts in Hinduism - Ratha Saptami

Body And Soul: The individual souls are eternal, manifold, eternally separate from one another, and distinct from the body, senses and mind; and yet capable of apprehension, volition, desire, aversion, pleasure, pain, merit and demerit. They are infinite, ubiquitous or omnipresent and diffused everywhere throughout space. A mans soul is as much in New York as in Bombay, although it can only apprehend and feel and act where the body is. The soul and the mind are not objects of perception.
 
Excerpt from All About Hinduism by Sri Swami Sivananda
 

Read more here: » Body And Soul: Vedic Philosophy - Body And Soul

Baha'u'llah And the New Faith
Echoing the suffering of all the messengers of God before him and the privations they suffered to fulfil their sacred task, Baha'u'l-lah, the founder of the Baha'i religion, wrote: "We have accepted to be abased, O believers in the Unity of God, that ye may be exalted, and have suffered manifold afflictions, that ye might prosper and flourish".
 
Born in Persia on November 12, 1817, Baha'u'llah's unusual undertaking at age 27 has gradually brought into its fold several million people from diverse backgrounds. For Baha'u'llah claimed to be the Messenger of God - the Bearer of a Divine Revelation that fulfils the promises made in earlier religions and which will generate the spiritual forces for the unification of the world.
 

Read more here: » Bahai - Bah‡'’: Baha'u'llah And the New Faith

Elohim
(Genesis 1.26) "Gods" (plural), "God" Eloha. Manifold expression of The Source in order to create diversity of Creation

 
(See also: Elohim, Body Mind and Soul )

Visvarupa (Sanskrit) [from visva all + rupa form]
 
Having all forms, manifold, omnipresent; often applied to Vishnu and at times to Krishna in the Bhagavad-Gita; likewise to Siva.

 
(See also: Visvarupa, Mysticism, Mysticism Dictionary, Body mind and Soul )

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field theory, and gauge theory. 3-manifold theory is a part o ... Including:

• 3-manifold - Some famous examples of 3-manifolds
• 3-manifold - Some important classes of 3-manifolds
• 3-manifold - Some important structures on 3-manifolds
• 3-manifold - Foundational results
• 3-manifold - Some famous conjectures

Read more here: » 3-manifold: Encyclopedia - 3-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, the topological and smooth categories are not equivalent. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifol ... Including:

• 4-manifold - Handle decomposition

Read more here: » 4-manifold: Encyclopedia - 4-manifold

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cn, such that the change of coordinates between charts are holomorphic. Complex manifolds can be regarded as a special case of differentiable manifolds. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface. The requirement that the transition functions b ... Including:

• Complex manifold - Examples of complex manifolds
• Complex manifold - Integrable almost-complex structures
• Complex manifold - Kähler and Calabi-Yau manifolds

Read more here: » Complex manifold: Encyclopedia - Complex manifold