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EUCLIDEAN MANIFOLDS
This chapter is the first where the algebraic concepts developed thus far are combined with
ideas from analysis. The main concept to be introduced is that of a manifold. We will discuss here
only a special case cal1ed a Euclidean manifold. The reader is assumed to be familiar with certain
elementary concepts in analysis, but, for the sake of completeness, many of these shall be inserted
when needed.
Section 43 Euclidean Point Spaces
Consider an inner produce space V and a set E . The set E is a Euclidean point space if
there exists a function f :E×E →V such that:
(a) f(x,y)=f(x,z)+f(z,y), x,y,z∈E
and
(b) For every x ∈E and v ∈V there exists a unique element y ∈E such that
f (x,y) = v .
The elements of E are called points, and the inner product space V is called the translation space.
We say that f (x,y) is the vector determined by the end point x and the initial point y . Condition
b) above is equivalent to requiring the function f : → x E V defined by f( ) =f( , ) x y x y to be one to
one for each x . The dimension of E , written dimE , is defined to be the dimension of V . If V
does not have an inner product, the set E defined above is called an affine space.
A Euclidean point space is not a vector space but a vector space with inner product is made
a Euclidean point space by defining 1 2 1 2 f(v ,v )≡v −v for all v ∈V . For an arbitrary point space
the function f is called the point difference, and it is customary to use the suggestive notation
f (x,y) = x − y
This chapter is the first where the algebraic concepts developed thus far are combined with
ideas from analysis. The main concept to be introduced is that of a manifold. We will discuss here
only a special case cal1ed a Euclidean manifold. The reader is assumed to be familiar with certain
elementary concepts in analysis, but, for the sake of completeness, many of these shall be inserted
when needed.
Section 43 Euclidean Point Spaces
Consider an inner produce space V and a set E . The set E is a Euclidean point space if
there exists a function f :E×E →V such that:
(a) f(x,y)=f(x,z)+f(z,y), x,y,z∈E
and
(b) For every x ∈E and v ∈V there exists a unique element y ∈E such that
f (x,y) = v .
The elements of E are called points, and the inner product space V is called the translation space.
We say that f (x,y) is the vector determined by the end point x and the initial point y . Condition
b) above is equivalent to requiring the function f : → x E V defined by f( ) =f( , ) x y x y to be one to
one for each x . The dimension of E , written dimE , is defined to be the dimension of V . If V
does not have an inner product, the set E defined above is called an affine space.
A Euclidean point space is not a vector space but a vector space with inner product is made
a Euclidean point space by defining 1 2 1 2 f(v ,v )≡v −v for all v ∈V . For an arbitrary point space
the function f is called the point difference, and it is customary to use the suggestive notation
f (x,y) = x − y