Genericity, Transversality, and Relative Critical Sets

Goals 1-dimensional ridges Genericity Transversality Conclusion
Genericity, Transversality, and Relative Critical
Sets
Dr. Jason Miller
CSU Channel Islands
18 September 2017
Graduate Mathematics Seminar
jason.miller@csuci.edu CSU Channel Islands
Genericity, Transversality, and Relative Critical Sets

About the Talk
1 Goals
2 1-dimensional ridges
3 Genericity
4 Transversality
5 Conclusion

Present the idea of transversality of a mapping between
manifolds.

manifolds.
Present the idea of genericity of a property of functions, and
relate that to some transversailty theorems.

manifolds.
Present the idea of genericity of a property of functions, and
relate that to some transversailty theorems.
Share the idea of a one-dimensional ridge set, and sketch out
how the above approach is used to establish some of its local
generic properties.

1-dimensional ridges
Calc I: use information about the derivative to understand the
geometry of a function.

Calc II & III: extend that to functions of two variables using x f
and Hx (f ), the Hessian of f , its matrix of second partials. These
give geometric information about f near critical points x of f .

Spciﬁcally: vanishing of x f and signs of the eigenvalues of Hx (f ).

Spciﬁcally: vanishing of x f and signs of the eigenvalues of Hx (f ).
Let f : U ⊂ Rm → R be smooth, λi be the eigenvalues of Hx (f )
with unit eigenvectors ei , and order the eigenvalues λi ≤ λi+1.

Deﬁnition (1d Relative Critical Set)
A point x ∈ U is on the 1-dimensional relative critical set of f if
x f · ei = 0 for i ≤ n − 1.

x f · ei = 0 for i ≤ n − 1. If we also specify that λn−1 < 0 at x,
then x is a point in the function’s 1-dimensional (height) ridge.

Theorem (Structure Theorem for 2-Ridges)
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, and that

pass smoothly through critical points of f

Genericity
Deﬁnition (residual)
A residual set is a countable intersection of open dense sets.

Genericity
Residual sets are dense in the space of smooth functions (a Baire
space in the Whitney C∞ topology).

Genericity
Deﬁnition (Generic)
A property of a function is generic if the set of functions with the
property contains a residual set in C∞ and all orbits of the
function under diﬀeomorphism (e.g., change of variables).

Genericity
Deﬁnition (Generic)
A property of a function is generic if the set of functions with the
property contains a residual set in C∞ and all orbits of the
function under diﬀeomorphism (e.g., change of variables).
For us, the set of functions with our properties will be open and
dense. So generic means, if f doesn’t have the property, then an
arbitrarily small perturbation of f will.

Transversality
Deﬁnition
Let X and Y be smooth manifolds, W ⊂ Y a smooth
submanifold, and f : X → Y be smooth. Then f intersect W
transversely at x (denoted f − W at x) if either
1 f (x) /∈ W , or
2 f (x) ∈ W and Tf (x)Y = Tf (x)W + dfx (Tx X).

Transversality
Deﬁnition
1 f (x) /∈ W , or
If f − W , then W behaves under pullback like a regular value of f .

Transversality
Deﬁnition
1 f (x) /∈ W , or
If f − W , then W behaves under pullback like a regular value of f .
Theorem
Let X and Y be smooth manifolds and W ⊂ Y a smooth
submanifold of codimension d. If f : X → Y is smooth and
f − W , then f −1(W ) is a smooth submanifold of X of
codimension d.

Transversality
Deﬁnition
For M and N smooth manifolds, Jk(M, N) is the set of k-jet
extensions, or Taylor polynomials of functions from M to N.

Transversality
Deﬁnition
For M and N smooth manifolds, Jk(M, N) is the set of k-jet
extensions, or Taylor polynomials of functions from M to N.
Theorem (Thom’s Transversality Theorem)
For M and N smooth manifolds with Γ a submanifold of Jk(M, N),
let
TΓ = {f ∈ C∞
(M, N) | jk
(f ) − Γ}.
Then TΓ is a residual subset of C∞(M, N) in the Whitney
C∞-topology. If Γ is closed, then TΓ is open.

This genericity results is established by

using a set of structure mappings,

collecting closed submanifolds and stratiﬁed sets of jet space,
and then

collecting closed submanifolds and stratiﬁed sets of jet space,
and then
applying Thom’s Transversality Theorem to get the result.

Structure Maps

Structure Maps
1 The Hessian map: H(f ) : U → S2Rn by x → H(f )(x)

Structure Maps
2 The preridge map: p(x) = ( x f · e1, . . . , x f · en−1)

Structure Maps
3 The Gauss-type map: G(f , 1) : U → Gr(1, n) × Gr(1, n) by
x → ( f (x) , en )

Structure Maps
3 The Gauss-type map: G(f , 1) : U → Gr(1, n) × Gr(1, n) by
x → ( f (x) , en )
4 The Gauss-Hessian:
(G × H)(f , q) : U → Gr(1, n) × Gr(1, n) × S2Rn

Transversality Conditions
The Hessian map is transverse to an eigenvalue-based stratication
of S2Rn (e.g., singular matrices, repeated eigenvalues)
Name of Manifold Symbol Codimension
simple matrices S2R4 (S ∪ P) 0
simple, corank 1 matrices S(1) 1
double umbilics P(2) 2
corank 2 matrices S(2) 3
multiumbilics of order (2,2) P(2,2) 4
corank 3 matrices S(3) 6
umbilics Um {0} 9
zero {0} 10
Table: List of the submanifolds that stratify S2
R4
.

1 The Gauss-like map is transverse to the partial ﬂag manifold
Fn(1, 1) (codim n − 1).

Fn(1, 1) (codim n − 1).
2 The Gauss-Hessian map is transverse to Fn(1, 1) × S(1)
(codim n).

Fn(1, 1) (codim n − 1).
(codim n).
Others, for example:

Fn(1, 1) (codim n − 1).
(codim n).
1 There is a closed Γi ⊂ J3(U, R) of codimension n + 1 that has
the property if jk(f ) /∈ Γi , then the 1-preridge of f passess
through critical points smoothly. More: the critical points are
Morse critical points, and H(f ) has distinct eigenvalues.

Fn(1, 1) (codim n − 1).
(codim n).
the property if jk(f ) /∈ Γi , then the 1-preridge of f passess
through critical points smoothly. More: the critical points are
Morse critical points, and H(f ) has distinct eigenvalues.
the property if jk(f ) /∈ Γi , then the 1-preridge of f fails to be
regular.

Thus, applying Thom’s Transversailty Theorem to each Γi , we get
an open dense set of f ∈ C∞ with the property described by Γi .

The intersection of these sets in C∞ is open an dense, hence our
properties are generic.

The intersection of these sets in C∞ is open an dense, hence our
properties are generic.
Technicality: the above holds on U0 = U {critical points and
partial umbilics}. Some ﬁnesse is required to ascertain the
structure on the closure.

Thom’s Transversality Theorem is a powerful tool for establishing
genericity of properties of functions.

The generic structure of the d-ridge set of a function is well
understood for small d; how partial umbilics eﬀect ridge structure
is not well understood in higher dimensions.

The generic structure of the d-ridge set of a function is well
understood for small d; how partial umbilics eﬀect ridge structure
is not well understood in higher dimensions.
Ridges are one part of a broader family of relative critical sets
(including, e.g., valley sets) that partition a domain. They have
signiﬁcant potential to describe the geometry of a function.

Genericity, Transversality, and Relative Critical Sets

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Genericity, Transversality, and Relative Critical Sets