Journal of Indian Acad. Math.  
Vol. 47, No. 1 (2025) pp. 81-87  
ISSN: 0970-5120  
Jeyanthi  
AXION FIXED POINT THEOREM: A NEW  
FRAMEWORK BRIDGING HILBERT  
MANIFOLDS AND HILBERT SPACES  
Venkatapathy1  
and  
Madhan Velayuthan2  
Abstract: This paper presents a novel fixed-point framework on Hilbert  
manifolds, called Axion. The local and global structure of manifolds can be  
better understood by using contraction mappings to define axion points. By  
using an Axion structure(a, , ), where  
is a diffeomorphism and  
is  
its inverse meeting a contraction condition, the Axion Fixed Point Theorem  
extends conventional fixed-point findings to infinite-dimensional spaces. By  
establishing the existence and uniqueness of axion points, this approach  
advances our knowledge of fixed points in functional spaces.  
Key Words and Phrases: Axion Set, Embedding, Hilbert Manifold,  
Hilbert Space.  
Mathematics Subject Classification (2020) No.: Primary: 46T10, 57R40;  
Secondary: 46C05.  
1. Introduction and Preliminaries  
A Hilbert space is an infinite-dimensional generalization of Euclidean space,  
equipped with an inner product that induces a norm and a complete metric topology.  
Fixed point theorems are essential in analysis, topology, and geometry, providing  
fundamental results in nonlinear functional analysis, differential equations, and  
dynamical systems. The Banach Fixed Point Theorem, one of the most well-known  
results, guarantees the existence and uniqueness of fixed points under contraction  
82  
JEYANTHI VENKATAPATHY AND MADHAN VELAYUTHAN  
mappings in complete metric spaces. In 1956, Nash established the fundamental  
theory for embedding abstract Riemannian manifolds into Euclidean spaces, which  
remains a cornerstone in differential geometry [10]. A few years later, Hamilton  
(1982) contributed critical insights into curvature evolution, significantly influencing  
modern perspectives in differential geometry and general relativity [6]. In 2006,  
Chavel provided an extensive treatment of modern Riemannian geometry, focusing  
on embedding theorems and geometric flows [5]. Lee (2013) presented a  
contemporary perspective on smooth manifolds and Lie groups, which has been  
instrumental in advancing research in differential structures [8]. Between 2020 and  
2024, significant progress was made in isometric embeddings and Hilbert manifold  
structures. Chattopadhyay et al. (2020) investigated the isometric embeddability of  
Sqm into Spn , contributing to a deeper understanding of embeddings between finite-  
dimensional spaces [4]. In 2024, Capdeville examined the isometric embeddings of  
n-point spaces for n 4 , laying the groundwork for further studies in discrete metric  
spaces [3]. Looking ahead to 2025, Madhan Velayuthan and Jeyanthi Venkatapathy  
have extended embedding theories by addressing diffeomorphic embeddings of  
higher-dimensional Hilbert manifolds into Hilbert spaces. Their work introduces  
innovative techniques for handling infinite-dimensional structures and preserving  
geometric and topological properties [9].  
Definition 1.1 ([8]): A topological space  
manifold if:  
is called an n-dimensional  
1.  
Local Euclidean Property: p , a neighborhood U  
and a homeomorphism :U V n , such that  
and 1 are  
continuous.  
2.  
3.  
Hausdorff Property:  
is Hausdorff, i.e., p, q , p q,   
disjoint open sets Up, Uq such that p Up and q Uq  
.
Second-Countability: The topology of  
has a countable basis.  
If, in addition,  
is equipped with an atlas {(Uj,j )}jsuch that  
for any two overlapping charts (Uj,j ) and (Uk,k ), the transition maps  
AXION FIXED POINT THEOREM  
83  
k j1 : j(Uj Uk ) k(Uj Uk ) are infinitely differentiable (C), then  
is called a smooth manifold.  
Definition 1.2 ([5]): A Hilbert manifold is a smooth manifold  
modeled on an Hilbert space  
. Specifically, satisfies:  
1. ∃  
an  
atlas  
{(U, )}   
such  
that  
each  
chart  
:U(U) is a bijective homeomorphism mapping onto  
an open subset of  
.
2. Transition  
maps  
between  
overlapping  
charts,  
1 : (UU) (UU), are infinitely differentiable  
(C)  
.
3. The topology of is induced by  
, i.e.,A is open if and  
only if (A) is open in for each chart  
  
.
Definition 1.3 [9]: Let  
and  
be Hilbert manifolds. A mapping  
:   
is called a diffeomorphism if it satisfies the following  
conditions:  
1.  
2.  
Bijectivity: The map  
and surjective.  
is a bijection, meaning it is both injective  
Smoothness: The map  
is infinitely differentiable, i.e.,  
C(, )  
.
3.  
Smooth Inverse: The inverse mapping 1 exists and is  
also smooth, ensuring that  
correspondence between  
establishes a smooth one-to-one  
and  
.
84  
JEYANTHI VENKATAPATHY AND MADHAN VELAYUTHAN  
If such a map exists, we say that  
denoted as   
and  
are diffeomorphic,  
.
Definition 1.4 [8]: Given a local chart :UV, the induced  
metric on Uis defined by d(x,y) (x) (y) , where ·  
denotes the norm in the Hilbert space  
.
Theorem 1.5 ([8]): Let d be a complete metric space and let  
c [0,1]  
C : d d be a contraction mapping, i.e., such that  
has a unique fixed point  
d((h), (t)) c·d(h,t),h, t d . Then,  
hd.  
2. Axion Fixed Point Theorem  
This section presents new definition Axion and Axion fixed point theorem.  
Definition 2.1: Let  
be a Hilbert manifold modeled on a Hilbert  
space  
, with an atlas{(U, )} . An Axion is an ordered triplet  
(a, , ) satisfying:  
1. Accumulation: a is an accumulation point of  
S M , i.e.,  
U'   
,
a U' U' S    
.
2. Smooth Chart: ∃ a chart (U,)  
diffeomorphism :U(U)   
with a Uand a smooth  
.
3. Inverse Mapping:   1 : (U) Uis smooth.  
The set of all Axions  
{(a, , ) | :U(U)  
is a  
diffeomorphism,   1 }.  
AXION FIXED POINT THEOREM  
85  
Theorem 2.2 (Axion Fixed Point Theorem): Let (a, , ) be an Axion in  
a Hilbert manifold with respect to a chart (U, ), where:  
1.  
:U(U) is a smooth diffeomorphism.  
2.  
3.  
: (U) Uis the inverse of  
, i.e., 1  
.
satisfies the contraction condition:  
a constant c [0,1) such  
that d((x), (y)) c·d(x,y), x, y U  
.
Then,  
a unique fixed point aUsuch that (a) a  
.
Proof: The local chart :UVinduces a metric on Udefined  
by: d(x,y) (x) (y) , where ·is the norm in . This metric  
provides a distance measure for elements of U. To apply the Banach Fixed  
Point Theorem, we must show that (U, d) is a complete metric space. Let (xn)  
be  
a
Cauchy sequence in U  
with respect to  
d  
.
By definition,  
d(xn, xm) (xn) (xm) .  
Since, (xn) is Cauchy in U, the sequence (xn) is Cauchy in  
. Since,  
is a Hilbert space, it is complete, and thus, a limit point y such that:  
(xn) y as n   . Since, 1 is a diffeomorphism. By the continuity of  
, xn xn (y) as n   . Since y  (U) , we have (y) U  
,
proving that Uis complete. By assumption, such that:  
c [0,1)  
d((x), (y)) c·d(x,y), x, y U. This confirms that Γ is a strict  
contraction mapping.  
Since (U, d) is a complete metric space and Γ is a contraction, the  
Banach Contraction Theorem guarantees the existence of a unique fixed point  
86  
JEYANTHI VENKATAPATHY AND MADHAN VELAYUTHAN  
aUsuch that: (a) a. Suppose there exist two fixed points a1, a2 U  
such that (a1) a1 and (a2) a2  
.
Then, d(a1, a2) d((a1), (a2)) c·d(a1, a2)  
.
Since, c [0,1), it follows that d(a1, a2) 0 , implying a1 a2  
.
Thus, the fixed point is unique.  
3. Conclusion  
The Axion triplet (a, , ) is introduced in the Axion fixed point Theorem,  
which extends the standard Banach fixed point theorem to Hilbert manifolds. This  
framework preserves the underlying geometric structure of the manifold while  
enabling the analysis of contraction mappings inside local charts. In order to provide  
stability under smooth transformations, the theorem ensures that fixed points for such  
mappings exist and are unique. The Banach contraction theorem is used in the proof  
to demonstrate convergence, taking use of the contraction quality of Γ and the  
completeness of the induced metric space.  
REFERENCES  
[1] M. Belkin and P. Niyogi (2003): Laplacian Eigenmaps for Dimensionality Reduction and  
Data Representation, Neural Comput., Vol. 15, pp. 1373-1396.  
[2] M. Bonk and O. Schramm (2000): Embeddings of Gromov hyperbolic spaces, Geom.  
Funct. Anal. (GAFA), Vol. 10, pp. 266-306.  
[3] B. Capdeville (2024): Isometric embedding of the n-point spaces into the space of spaces  
for n 4 , arXiv preprint arXiv:2402.18156.  
[4] A. Chattopadhyay, G. Hong, A. Pal, C. Pradhan, and S. K. Ray (2020): Isometric  
Embeddability of Sqm into Spn , arXiv preprint arXiv:2008.13164.  
[5] I. Chavel (2006): Riemannian Geometry A Modern Introduction, Second edition,  
Cambridge Univ. Press, Cambridge.  
[6] R. S. Hamilton (1982): Three-manifolds with positive Ricci curvature, J. Differential  
Geom., Vol. 17(2), pp. 255-306.  
 
AXION FIXED POINT THEOREM  
87  
[7] S. Lang (1995): Differential and Riemannian Manifolds, Third edition, Springer-Verlag,  
New York.  
[8] J. M. Lee: Introduction to Smooth Manifolds, Second edition, Springer, New York, 2013.  
[9] V. Madhan and V. Jeyanthi (2025): Diffeomorphic embedding of higher-dimensional  
Hilbert manifolds into Hilbert spaces, Creat. Math. Inform., Vol. 34(1), pp. 133-141.  
[10] J. Nash (1956): The embedding problem for Riemannian manifolds, Ann. Math., Vol.  
63(1), pp. 20-63.  
1, 2 Department of Mathematics,  
Sri Krishna Arts and Science  
(Received, January 5, 2025)  
(Revised-1, January 12, 2025)  
(Revised-2, January 22, 2025)  
College, Coimbatore, 641008 - India  
1. E-mail: jeyanthiv@skasc.ac.in  
2. E-mail: madhanvmaths@gmail.com