A talk at the 2009 Joint Mathematics Meeting in Washington, D.C., on relative critical sets and their properties. The talk ends with an open question whose answer will help extend our understanding of the local generic structure of relative critical sets.

1. Introduction Definition Structure Question References
Relative Critical Sets: Structure and application
Dr. Jason Miller
Truman State University
8 January 2009
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

2. Introduction Definition Structure Question References
About the Talk
Introduction
1
Definition
2
What’s known
3
Question
4
References
5
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

3. Introduction Definition Structure Question References
The concept of d-dimensional relative critical set generalizes the
concept of (zero dimensional) critical point of a differentiable
function.
Let U ⊂ Rn and f : U −→ R be a smooth function. Let x ∈ U.
Let H(f ) be the Hessian of f , λi ≤ λi +1 its eigenvalues and ei a
unit eigenvector for λi so that {ei }n=1 an orthonormal basis of Rn .
i
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

4. Introduction Definition Structure Question References
The concept of d-dimensional relative critical set generalizes the
concept of (zero dimensional) critical point of a differentiable
function.
Let U ⊂ Rn and f : U −→ R be a smooth function. Let x ∈ U.
Let H(f ) be the Hessian of f , λi ≤ λi +1 its eigenvalues and ei a
unit eigenvector for λi so that {ei }n=1 an orthonormal basis of Rn .
i
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

5. Introduction Definition Structure Question References
The concept of d-dimensional relative critical set generalizes the
concept of (zero dimensional) critical point of a differentiable
function.
Let U ⊂ Rn and f : U −→ R be a smooth function. Let x ∈ U.
Let H(f ) be the Hessian of f , λi ≤ λi +1 its eigenvalues and ei a
unit eigenvector for λi so that {ei }n=1 an orthonormal basis of Rn .
i
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

6. Introduction Definition Structure Question References
Critical Set, v.1
The x is a critical point iff ∇f = 0 at x.
Alternatively...
Critical Set, v.2
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
If we specify that λn < 0 at x, then x is a local maximum.
Structure
Generically, a function’s critical set is a set of isolated points.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

7. Introduction Definition Structure Question References
Critical Set, v.1
The x is a critical point iff ∇f = 0 at x.
Alternatively...
Critical Set, v.2
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
If we specify that λn < 0 at x, then x is a local maximum.
Structure
Generically, a function’s critical set is a set of isolated points.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

8. Introduction Definition Structure Question References
Critical Set, v.1
The x is a critical point iff ∇f = 0 at x.
Alternatively...
Critical Set, v.2
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
If we specify that λn < 0 at x, then x is a local maximum.
Structure
Generically, a function’s critical set is a set of isolated points.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

9. Introduction Definition Structure Question References
Critical Set, v.1
The x is a critical point iff ∇f = 0 at x.
Alternatively...
Critical Set, v.2
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
If we specify that λn < 0 at x, then x is a local maximum.
Structure
Generically, a function’s critical set is a set of isolated points.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

10. Introduction Definition Structure Question References
Critical Set, v.1
The x is a critical point iff ∇f = 0 at x.
Alternatively...
Critical Set, v.2
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
If we specify that λn < 0 at x, then x is a local maximum.
Structure
Generically, a function’s critical set is a set of isolated points.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

11. Introduction Definition Structure Question References
Let 0 < d < n.
0-dimensional Relative Critical Set
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

12. Introduction Definition Structure Question References
Let 0 < d < n.
0-dimensional Relative Critical Set
The x is a critical point iff, at x, ∇f · ei = 0 for all i .
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

13. Introduction Definition Structure Question References
Let 0 < d < n.
d-dimensional Relative Critical Set
The x is a critical point iff, at x, ∇f · ei = 0 for i ≤ n − d.
If we specify that λn−d < 0 at x, the x is a point in the function’s
d-dimensional height ridge.
Structure Question
What is the local generic structure of a function’s d-dimensional
ridge in Rn (esp. near partial umbilics)?
The d = 1 dimensional height ridge has applications in image
analysis, so knowing its generic structure is important.
[PE+ , PS, Ebe96]
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

14. Introduction Definition Structure Question References
Let 0 < d < n.
d-dimensional Relative Critical Set
The x is a critical point iff, at x, ∇f · ei = 0 for i ≤ n − d.
If we specify that λn−d < 0 at x, the x is a point in the function’s
d-dimensional height ridge.
Structure Question
What is the local generic structure of a function’s d-dimensional
ridge in Rn (esp. near partial umbilics)?
The d = 1 dimensional height ridge has applications in image
analysis, so knowing its generic structure is important.
[PE+ , PS, Ebe96]
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

15. Introduction Definition Structure Question References
Let 0 < d < n.
d-dimensional Relative Critical Set
The x is a critical point iff, at x, ∇f · ei = 0 for i ≤ n − d.
If we specify that λn−d < 0 at x, the x is a point in the function’s
d-dimensional height ridge.
Structure Question
What is the local generic structure of a function’s d-dimensional
ridge in Rn (esp. near partial umbilics)?
The d = 1 dimensional height ridge has applications in image
analysis, so knowing its generic structure is important.
[PE+ , PS, Ebe96]
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

16. Introduction Definition Structure Question References
Let 0 < d < n.
d-dimensional Relative Critical Set
The x is a critical point iff, at x, ∇f · ei = 0 for i ≤ n − d.
If we specify that λn−d < 0 at x, the x is a point in the function’s
d-dimensional height ridge.
Structure Question
What is the local generic structure of a function’s d-dimensional
ridge in Rn (esp. near partial umbilics)?
The d = 1 dimensional height ridge has applications in image
analysis, so knowing its generic structure is important.
[PE+ , PS, Ebe96]
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

17. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

18. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

19. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

20. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

21. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

22. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

23. Introduction Definition Structure Question References
Theorem ([Dam99, Mil98])
Generically, the closure of the 1-dimensional ridge is
a discrete set of smooth embedded curves, that
has boundary points at partial umbilic points (λn−1 = λn ) or
at singular points (λn−1 = 0) of the Hessian.
Theorem ([Mil98])
Generically, the closure of the 2-dimensional ridge is
a discrete set of smooth embedded surfaces surfaces, that
has boundary curves at partial umbilic points (λn−2 = λn−1 )
or at singular points (λn−2 = 0) of the Hessian, and its
boundary is smooth except at a corner where
λn−2 = λn−1 = 0.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

24. Introduction Definition Structure Question References
This and related genericity result is established by
collecting closed submanifolds and stratified sets of jet space
and then
using a set of mappings,
applying Thom’s Transversality Theorem to get the result.
Theorem (Thom’s Transversality Theorem)
For M and N smooth manifolds with Γ a submanifold of J k (M, N),
let
TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}.
Then TΓ is a residual subset of C ∞ (M, N) in the Whitney
C ∞ -topology. If Γ is closed, then TΓ is open.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

25. Introduction Definition Structure Question References
This and related genericity result is established by
collecting closed submanifolds and stratified sets of jet space
and then
using a set of mappings,
applying Thom’s Transversality Theorem to get the result.
Theorem (Thom’s Transversality Theorem)
For M and N smooth manifolds with Γ a submanifold of J k (M, N),
let
TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}.
Then TΓ is a residual subset of C ∞ (M, N) in the Whitney
C ∞ -topology. If Γ is closed, then TΓ is open.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

26. Introduction Definition Structure Question References
This and related genericity result is established by
collecting closed submanifolds and stratified sets of jet space
and then
using a set of mappings,
applying Thom’s Transversality Theorem to get the result.
Theorem (Thom’s Transversality Theorem)
For M and N smooth manifolds with Γ a submanifold of J k (M, N),
let
TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}.
Then TΓ is a residual subset of C ∞ (M, N) in the Whitney
C ∞ -topology. If Γ is closed, then TΓ is open.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

27. Introduction Definition Structure Question References
This and related genericity result is established by
collecting closed submanifolds and stratified sets of jet space
and then
using a set of mappings,
applying Thom’s Transversality Theorem to get the result.
Theorem (Thom’s Transversality Theorem)
For M and N smooth manifolds with Γ a submanifold of J k (M, N),
let
TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}.
Then TΓ is a residual subset of C ∞ (M, N) in the Whitney
C ∞ -topology. If Γ is closed, then TΓ is open.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

28. Introduction Definition Structure Question References
This and related genericity result is established by
collecting closed submanifolds and stratified sets of jet space
and then
using a set of mappings,
applying Thom’s Transversality Theorem to get the result.
Theorem (Thom’s Transversality Theorem)
For M and N smooth manifolds with Γ a submanifold of J k (M, N),
let
TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}.
Then TΓ is a residual subset of C ∞ (M, N) in the Whitney
C ∞ -topology. If Γ is closed, then TΓ is open.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

29. Introduction Definition Structure Question References
The boundary of the d-dimensional ridge inherits its geometry
from that of the
geometry of the set of partial umbilic matrices (semialgebraic)
geometry of singular (algebraic)
as subsets of in S 2 Rn .
Theorem (The ”ℓ chose two” Test)
There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the
ℓ
property that if d < 2 and another transversality condition holds,
then the closure of a d-dimensional ridge of f misses the partial
umbilics of order ℓ.
Example
The 3-dimensional ridge fails this test for the partial umbilics of
order ℓ = 3.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

30. Introduction Definition Structure Question References
The boundary of the d-dimensional ridge inherits its geometry
from that of the
geometry of the set of partial umbilic matrices (semialgebraic)
geometry of singular (algebraic)
as subsets of in S 2 Rn .
Theorem (The ”ℓ chose two” Test)
There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the
ℓ
property that if d < 2 and another transversality condition holds,
then the closure of a d-dimensional ridge of f misses the partial
umbilics of order ℓ.
Example
The 3-dimensional ridge fails this test for the partial umbilics of
order ℓ = 3.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

31. Introduction Definition Structure Question References
The boundary of the d-dimensional ridge inherits its geometry
from that of the
geometry of the set of partial umbilic matrices (semialgebraic)
geometry of singular (algebraic)
as subsets of in S 2 Rn .
Theorem (The ”ℓ chose two” Test)
There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the
ℓ
property that if d < 2 and another transversality condition holds,
then the closure of a d-dimensional ridge of f misses the partial
umbilics of order ℓ.
Example
The 3-dimensional ridge fails this test for the partial umbilics of
order ℓ = 3.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

32. Introduction Definition Structure Question References
The boundary of the d-dimensional ridge inherits its geometry
from that of the
geometry of the set of partial umbilic matrices (semialgebraic)
geometry of singular (algebraic)
as subsets of in S 2 Rn .
Theorem (The ”ℓ chose two” Test)
There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the
ℓ
property that if d < 2 and another transversality condition holds,
then the closure of a d-dimensional ridge of f misses the partial
umbilics of order ℓ.
Example
The 3-dimensional ridge fails this test for the partial umbilics of
order ℓ = 3.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

33. Introduction Definition Structure Question References
The boundary of the d-dimensional ridge inherits its geometry
from that of the
geometry of the set of partial umbilic matrices (semialgebraic)
geometry of singular (algebraic)
as subsets of in S 2 Rn .
Theorem (The ”ℓ chose two” Test)
There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the
ℓ
property that if d < 2 and another transversality condition holds,
then the closure of a d-dimensional ridge of f misses the partial
umbilics of order ℓ.
Example
The 3-dimensional ridge fails this test for the partial umbilics of
order ℓ = 3.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

34. Introduction Definition Structure Question References
The boundary of the d-dimensional ridge inherits its geometry
from that of the
geometry of the set of partial umbilic matrices (semialgebraic)
geometry of singular (algebraic)
as subsets of in S 2 Rn .
Theorem (The ”ℓ chose two” Test)
There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the
ℓ
property that if d < 2 and another transversality condition holds,
then the closure of a d-dimensional ridge of f misses the partial
umbilics of order ℓ.
Example
The 3-dimensional ridge fails this test for the partial umbilics of
order ℓ = 3.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

35. Introduction Definition Structure Question References
Part of the boundary of the 3-dimensional ridge will coincide with
partial umbilics of order 2 where λn−3 = λn−2
The ”ℓ chose two” Test implies the possibility that this part of the
boundary also contains partial umbilics of order 3.
Question
Knowing the geometry of the set of umbilics of order 2 in a normal
slice to the set of umbilics of order 3 will illuminate the boundary
structure of the 3-dimensional ridge. Analogous information for
higher umbilics could complete the structure theorem for ridges of
all dimension.
In [Arn72], Arnol’d remarks without proof that the structure is of a
cone over projective space.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

36. Introduction Definition Structure Question References
Part of the boundary of the 3-dimensional ridge will coincide with
partial umbilics of order 2 where λn−3 = λn−2
The ”ℓ chose two” Test implies the possibility that this part of the
boundary also contains partial umbilics of order 3.
Question
Knowing the geometry of the set of umbilics of order 2 in a normal
slice to the set of umbilics of order 3 will illuminate the boundary
structure of the 3-dimensional ridge. Analogous information for
higher umbilics could complete the structure theorem for ridges of
all dimension.
In [Arn72], Arnol’d remarks without proof that the structure is of a
cone over projective space.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

37. Introduction Definition Structure Question References
Part of the boundary of the 3-dimensional ridge will coincide with
partial umbilics of order 2 where λn−3 = λn−2
The ”ℓ chose two” Test implies the possibility that this part of the
boundary also contains partial umbilics of order 3.
Question
Knowing the geometry of the set of umbilics of order 2 in a normal
slice to the set of umbilics of order 3 will illuminate the boundary
structure of the 3-dimensional ridge. Analogous information for
higher umbilics could complete the structure theorem for ridges of
all dimension.
In [Arn72], Arnol’d remarks without proof that the structure is of a
cone over projective space.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

38. Introduction Definition Structure Question References
Part of the boundary of the 3-dimensional ridge will coincide with
partial umbilics of order 2 where λn−3 = λn−2
The ”ℓ chose two” Test implies the possibility that this part of the
boundary also contains partial umbilics of order 3.
Question
Knowing the geometry of the set of umbilics of order 2 in a normal
slice to the set of umbilics of order 3 will illuminate the boundary
structure of the 3-dimensional ridge. Analogous information for
higher umbilics could complete the structure theorem for ridges of
all dimension.
In [Arn72], Arnol’d remarks without proof that the structure is of a
cone over projective space.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

39. Introduction Definition Structure Question References
Part of the boundary of the 3-dimensional ridge will coincide with
partial umbilics of order 2 where λn−3 = λn−2
The ”ℓ chose two” Test implies the possibility that this part of the
boundary also contains partial umbilics of order 3.
Question
Knowing the geometry of the set of umbilics of order 2 in a normal
slice to the set of umbilics of order 3 will illuminate the boundary
structure of the 3-dimensional ridge. Analogous information for
higher umbilics could complete the structure theorem for ridges of
all dimension.
In [Arn72], Arnol’d remarks without proof that the structure is of a
cone over projective space.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

40. Introduction Definition Structure Question References
Part of the boundary of the 3-dimensional ridge will coincide with
partial umbilics of order 2 where λn−3 = λn−2
The ”ℓ chose two” Test implies the possibility that this part of the
boundary also contains partial umbilics of order 3.
Question
Knowing the geometry of the set of umbilics of order 2 in a normal
slice to the set of umbilics of order 3 will illuminate the boundary
structure of the 3-dimensional ridge. Analogous information for
higher umbilics could complete the structure theorem for ridges of
all dimension.
In [Arn72], Arnol’d remarks without proof that the structure is of a
cone over projective space.
millerj@truman.edu Truman State University
Relative Critical Sets: Structure and application

41. Introduction Definition Structure Question References
V.I. Arnol’d.
Modes and quasimodes.
Funct. Anal. and Appl., 6(2):94–101, 1972.
James Damon.
Properties of ridges and cores for two-dimensional images.
Journal of Mathematical Imaging and Vision, 10:163–174,
1999.
D. Eberly.
Ridges in Image and Data Analysis, volume 7 of Series Comp.
Imaging and Vision.
Kluwer, 1996.
Jason Miller.
Relative Critical Sets in Rn and Applications to Image
Analysis.
PhD thesis, University of North Carolina, 1998.
millerj@truman.edu Truman State University
S. Pizer, D. Eberly, et al.
Relative Critical Sets: Structure and application